[ [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "进位计数制" } ] }, "bbox": [ 144, 148, 240, 164 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "基数 R：允许使用的基本数码个数" } ] }, "bbox": [ 144, 167, 413, 183 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": " 权 Wi：权也称位权，第 i 位上的数码的权重" } ] }, "bbox": [ 146, 186, 502, 200 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "值，即权与数码所处的位置 i 有关" } ] }, "bbox": [ 146, 204, 410, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\mathbf { \\nabla } \\cdot \\mathbf { \\vec { w } } \\mathbf { \\vec { i } } = \\mathbf { \\vec { R } }" } ] }, "bbox": [ 147, 224, 221, 237 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "假设任意数值N用R进制数来表示：" } ], "level": 1 }, "bbox": [ 189, 249, 648, 272 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "N= (Dm-1Dm-2...D。.D.1D-2...D-k） R权重 Di：基本符号， " }, { "type": "equation_inline", "content": "\\mathsf { D } _ { \\mathsf { i } } \\in [ \\mathsf { 0 } ," }, { "type": "text", "content": "R-1],i=-k,-k+1, .….，m-1；小数点在D和D之间" } ] }, "bbox": [ 144, 280, 712, 357 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "数值N的实际值为：加权求和" } ], "level": 1 }, "bbox": [ 189, 363, 591, 386 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "进1 m-1六进制？ 十进制 8N=∑(Di×R')" } ] }, "bbox": [ 144, 388, 685, 468 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "X 何谓十进制、二进制、八进制、十六进制？" } ], "level": 1 }, "bbox": [ 196, 483, 756, 507 ] }, { "type": "table", "content": { "image_source": { "path": "images/da1bad6da64d52421be32b95d9695947bbeed1cb3852e71178dff094c1fe848e.jpg" }, "table_caption": [], "table_footnote": [], "html": "
进制基数R权Wi数码符号
十进制R=1010i0~9
二进制R=22i0、1
八进制R=88i0~7
十六进制R=1616i0~9、A~F
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 184, 510, 694, 670 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "二进制 八进制" } ], "level": 1 }, "bbox": [ 146, 703, 275, 719 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 以小数点为中心分别向两边分组，每三位一组，写出对应的八进制数字（不够位数则在两边加0补足3位 ）" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 二进制 十六进制" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 以小数点为中心分别向两边分组，每四位一组，写出对应的十六进制符号（不够位数则在两边加0补足4位 ）" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 八进制 二进制" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 将每位八进制数展开为3位二进制数，整数的最高位0和小数的最低位0可以略去" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 十六进制 二进制" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 将每位十六进制数展开为4位二进制数，整数的最高位0和小数的最低位0可" } ] } ] }, "bbox": [ 144, 722, 761, 904 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "以略去" } ], "level": 1 }, "bbox": [ 147, 93, 203, 107 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如何在计算机内使用二进制来表示十进制数据？" } ] }, "bbox": [ 144, 130, 512, 145 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "十进制数的编码" } ] }, "bbox": [ 147, 148, 275, 162 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "BCD（Binary Coded Decimal）码：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 使用二进制来编码十进制数字 " }, { "type": "equation_inline", "content": "\\pmb { 0 } \\sim \\pmb { 9 }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "编码方法：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 一般使用 4 位二进制编码来表示 1 位十进制数字" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 在 16 个编码中选用 10 个来表示数字 " }, { "type": "equation_inline", "content": "\\pmb { 0 } \\sim \\pmb { 9 }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 不同的选择构成不同的 BCD 码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "分类：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 有权码：编码的每一位都有固定的权值，加权求和的值即是表示的十进制数字。如 8421 码、2421 码、5211 码、4311 码、84 -2-1 码等" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 无权码：编码的每一位并没有固定的权，主要包括格雷码、余 3 码等十进制数串的表示方法" } ] } ] }, "bbox": [ 146, 168, 727, 367 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数值数据：用于表示数的大小，其值确定，可与数轴上确切的点对应" } ] }, "bbox": [ 146, 370, 680, 386 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 非数值数据：没有值大小的概念，表示字符、汉字、图像、声音等信息" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 计算机中如何表示数值数据？数值数据的格式要素，必须要考虑三点" } ] } ] }, "bbox": [ 147, 390, 717, 423 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 第一，符号如何表示？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 第二，小数点如何表示？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 第三，二进制数字如何排列来表示某个数值？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 机器数：数值数据在计算机中的二进制表示形式。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 机器数的符号位必须被数值化为二进制 0 和 1" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 机器数的小数点用隐含规定的方式来表达" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 表示的数值范围受计算机字长的限制" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 编码方法：不同的排列方式，常见的有原码、反码、补码、移码。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 真值：机器数真正表示的数值数据" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 书写： " }, { "type": "equation_inline", "content": "+ /" }, { "type": "text", "content": "－符号 数值（二进制或十进制）" } ] } ] }, "bbox": [ 147, 426, 663, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算机中参与运算的数值数据有两种：" } ] }, "bbox": [ 146, 649, 440, 664 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": " 无符号数据（Unsigned ）：所有的二进制数据位数均用来表示数值本身，没有正负之分" } ] }, "bbox": [ 146, 668, 734, 701 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 带符号数据（Signed）：则其二进制数据位，包括符号位和数值位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 当机器字长相同时，无符号数和有符号数的表示范围是不同的" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "计算机中：小数点的表示方法？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 计算机中没有专用的部件用来表示小数点" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 机器数中，小数点及其位置是隐含规定的" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 有两种隐含方式：定点和浮点" } ] } ] }, "bbox": [ 147, 705, 734, 812 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "定点机器数（Fixed Point）：小数点的位置是固定不变的，两种位置：" } ] }, "bbox": [ 144, 816, 695, 831 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 定点小数：纯小数，隐含固定在符号位Xs（最高位）之后，整数位是符号位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 定点整数：纯整数，小数点隐含固定在最低位之后，最高位为符号位Xs" } ] } ] }, "bbox": [ 147, 835, 463, 904 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/c7b139b89222151266e017727943407c6624042609160e115030648370c50579.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 156, 124, 473, 290 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "浮点机器数：将实数数值大小表示成科学计数法形式" } ] }, "bbox": [ 144, 332, 557, 350 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 浮点数 N 由三部分构成：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 阶码的底 R：隐含表示，一般约定为 2、8 或 16" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 尾数M：系数，为定点小数" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 阶码 E：R 的指数，为定点整数" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 对尾数所表示的定点小数，进行按比例放大（阶码为正数）或者按比例缩小（阶码为负数），数据真值的小数点位置是浮动的——位置由阶码规定" } ] } ] }, "bbox": [ 144, 351, 769, 461 ] }, { "type": "image", "content": { "image_source": { "path": "images/1daf5d1e2f93f95400319c82fbc9d2ae791ee6d59e6b25c4acc80a7508d40b5d.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 194, 502, 1000, 627 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 负精度-δ：机器数在数轴上最靠近原点0的负数，计算机能表示的最大负数" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 正精度δ：机器数在数轴上最靠近原点0的正数，计算机能表示的最小正数绝对值称为精度δ或分辨率，代表了机器数所能表示的绝对值最小的非零值" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 机器数的表示范围：[最小数，最大负数]、0、[最小正数，最大数]" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 最大数和和最小数：取决于机器数的编码方式、位数和格式" } ] } ] }, "bbox": [ 144, 667, 754, 757 ] }, { "type": "image", "content": { "image_source": { "path": "images/b6134e5eaca7a911410e0badcaa00f52a9436d482092a19623417c4bb70ff044.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 156, 766, 531, 848 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "下溢：位于最大负数和最小正数之间的数据（除 0 外），机器无法表示。" } ] }, "bbox": [ 144, 871, 702, 887 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": " 处理：计算机直接将其视为机器零。" } ] }, "bbox": [ 146, 890, 435, 904 ] } ], [ { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 上溢：当一个数据大于最大（正）数，或者小于最小（负）数时，机器也无法表示，称为上溢，上溢又称溢出。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 处理：计算机置溢出标志位，或者报警。 " } ] } ] }, "bbox": [ 144, 93, 771, 145 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "原码表示法" } ], "level": 1 }, "bbox": [ 147, 149, 238, 162 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "数学定义：真值与编码之间的数学关系" } ], "level": 1 }, "bbox": [ 144, 178, 443, 193 ] }, { "type": "image", "content": { "image_source": { "path": "images/1671e19ddf9f093cf00cf7c15db5550fd45f081c57f5b8e26a12151e29b38513.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 201, 195, 544, 307 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "翻数" } ], "level": 1 }, "bbox": [ 179, 357, 218, 388 ] }, { "type": "image", "content": { "image_source": { "path": "images/c815c6dc8826c731955012e56a7cb6cd015ca599083efa4e0ef91814247f3469.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 230, 349, 416, 393 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "业数" } ], "level": 1 }, "bbox": [ 470, 357, 507, 388 ] }, { "type": "image", "content": { "image_source": { "path": "images/26ff03a141ea9e96a1420fe0eb7111445ecdcbf59b2601a1c00de37b3c4ddc8c.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 515, 350, 697, 395 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "：-（--u）X--" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "//：-（Su-」）XSu " } ] } ] }, "bbox": [ 163, 406, 487, 442 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 507, 407, 633, 438 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "、（5）淌围： " }, { "type": "equation_inline", "content": "\\nleftarrow" } ] }, "bbox": [ 144, 445, 678, 460 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 使用原码进行加减乘除运算：规则类似于真值的加减乘除运算" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 符号位单独处理，不能参加运算，参加运算的是绝对值" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 加减运算：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 根据符号位决定进行的是绝对值的加还是减" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 进行绝对值的加减运算" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 根据绝对值的大小决定结果（和或者差）的符号" } ] } ] }, "bbox": [ 146, 464, 640, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "表示方法：最高位为符号位，其他位为数值位" } ] }, "bbox": [ 144, 630, 504, 645 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 符号位：0－正数，1－负数" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 数值位：与绝对值相同 " } ] } ] }, "bbox": [ 146, 649, 378, 682 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "“”和””只用于助记，在计算机中并无专用部件来表示" } ], "level": 1 }, "bbox": [ 156, 703, 376, 765 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "对于定点整数：" } ], "level": 1 }, "bbox": [ 146, 778, 263, 793 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 若 " }, { "type": "equation_inline", "content": "{ \\bf X } = + { \\bf X } 1 { \\bf X } 2" }, { "type": "text", "content": "……Xn，则[X]原= 0,X1X2 ……Xn" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 若 " }, { "type": "equation_inline", "content": "\\mathbf { \\lambda } \\mathbf { x } ( = \\mathbf { - }" }, { "type": "text", "content": "X1X2 ……Xn，则[X]原 " }, { "type": "equation_inline", "content": "\\varXi" }, { "type": "text", "content": "1,X1X2 ……Xn 。" } ] } ] }, "bbox": [ 146, 797, 539, 831 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": " 对于定点小数：" } ] }, "bbox": [ 147, 834, 285, 848 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 若 X=+0.X1X2 ……Xn，则[X]原 " }, { "type": "equation_inline", "content": "\\backslash =" }, { "type": "text", "content": "0.X1X2 ……Xn" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 若 " }, { "type": "equation_inline", "content": "\\mathbf { \\lambda } \\mathbf { x } ( = \\mathbf { - }" }, { "type": "text", "content": "0.X1X2 ……Xn，则[X]原 " }, { "type": "equation_inline", "content": "\\varXi" }, { "type": "text", "content": "1.X1X2 ……Xn" } ] } ] }, "bbox": [ 146, 853, 542, 887 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "补码" } ], "level": 1 }, "bbox": [ 146, 93, 188, 109 ] }, { "type": "image", "content": { "image_source": { "path": "images/811671a79ae1be5c45f4f715ff2f2fb25f3fcc69b175fff61442343d1a530415.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 178, 134, 670, 233 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2、表示方法：最高位为符号位，其他位为数值位。" } ] }, "bbox": [ 144, 259, 539, 275 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 符号位：0－正数，1－负数。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 数值位：正数时，与绝对值相同；负数时，为绝对值取反后，末位加 1。" } ] } ] }, "bbox": [ 146, 279, 712, 312 ] }, { "type": "image", "content": { "image_source": { "path": "images/ffe44f4fd0268aa31595488035281a89922feca95c9661470ec0bbee0d1405be.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 164, 319, 442, 399 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "真值=加权求和" } ], "level": 1 }, "bbox": [ 189, 412, 386, 435 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于定点整数:" } ] }, "bbox": [ 149, 451, 284, 468 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "■ 若X=+XX..X， 则[X]补= 0.,X.X..Xn" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\not \\equiv \\emptyset \\mathbb { X } = - \\mathbb { X } _ { 1 } \\mathbb { X } _ { 2 } . . . . . . \\mathbb { X } _ { n } , \\mathbb { M } \\subseteq \\mathbb { X } _ { 1 } = \\ 1 , \\overline { { X _ { 1 } } } \\overline { { X _ { 2 } } } . . . . . . . \\ \\overline { { X _ { n } } } \\ + \\ 1" } ] } ] }, "bbox": [ 174, 476, 584, 521 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于定点小数：" } ] }, "bbox": [ 149, 530, 284, 545 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\begin{array} { r } { \\dot { \\Xi } \\mathsf { X } = + 0 . \\mathsf { X } _ { 1 } \\mathsf { X } _ { 2 } . . . . . . . \\mathsf { X } _ { \\mathrm { n } } \\ , \\perp \\sharp [ \\mathsf { X } ] _ { \\sharp \\flat } = 0 . \\mathsf { X } _ { 1 } \\mathsf { X } _ { 2 } . . . . . . . \\mathsf { X } _ { \\mathrm { n } } } \\end{array}" }, { "type": "equation_inline", "content": "{ \\bf \\omega } = + { \\bf 0 }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■ 若X=-0. " }, { "type": "equation_inline", "content": "\\mathbf { X } _ { 1 } \\mathbf { X } _ { 2 } . . . . . . \\mathbf { X } _ { n }" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "[ \\mathsf { X } ] _ { * \\mathsf { h } } = 1 ." }, { "type": "text", "content": ". X X... X +.....1." } ] } ] }, "bbox": [ 174, 555, 692, 600 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "0的表示：0 的补码表示形式是唯一的，即分别按照正数和负数表示均一致，为全零。" } ] }, "bbox": [ 144, 630, 707, 664 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " " }, { "type": "equation_inline", "content": "[ + 0 ]" }, { "type": "text", "content": "补＝ 00…0 [-0]补 " }, { "type": "equation_inline", "content": "= 0 0 . . . 0" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " （2）表示范围：对于 " }, { "type": "equation_inline", "content": "\\mathsf { n } + \\mathsf { 1 }" }, { "type": "text", "content": "位补码机器数 X，它所能表示的数据范围为：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 定点整数： " }, { "type": "equation_inline", "content": "- 2 \\mathsf { n } { \\mathsf { \\leq } } \\mathsf { X } \\leq 2 \\mathsf { n } - 1" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 定点小数： " }, { "type": "equation_inline", "content": "- 1 \\leq x \\leq 1 - 2 - \\mathsf { n }" } ] } ] }, "bbox": [ 146, 668, 717, 738 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "（3）补码运算" } ], "level": 1 }, "bbox": [ 156, 741, 267, 756 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 补码的符号位同数值位一起参加运算 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 可以将减法转化为加法" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 简化了运算器的结构" } ] } ] }, "bbox": [ 146, 760, 445, 812 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": " 计算机中的整型数据（int）均用补码来表示反码" } ] }, "bbox": [ 146, 816, 512, 850 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点整数" } ], "level": 1 }, "bbox": [ 156, 107, 208, 148 ] }, { "type": "equation_interline", "content": { "math_content": "[ \\mathbf {X} ] _ {\\text {反}} = \\left\\{ \\begin{array}{l l} \\mathbf {X} & \\mathbf {X} \\geq 0 \\\\ 2 ^ {n + 1} - 1 + \\mathbf {X} & \\mathbf {X} \\leq 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/09478788883112f3ebac275a3e3567c30351770601fff4e709f9896f71729a42.jpg" } }, "bbox": [ 221, 99, 534, 162 ] }, { "type": "equation_interline", "content": { "math_content": "(\\mod (2 ^ {n + 1} - 1))", "math_type": "latex", "image_source": { "path": "images/0b9d7c1c8d4930124357cbd2163ae386c8d79663140020ee854805e28d1f2dfc.jpg" } }, "bbox": [ 289, 168, 492, 192 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点小数" } ], "level": 1 }, "bbox": [ 603, 112, 655, 152 ] }, { "type": "equation_interline", "content": { "math_content": "[ \\mathrm {X} ] _ {\\text {反}} = \\left\\{ \\begin{array}{l l} \\mathrm {X} & \\mathrm {X} \\geq 0 \\\\ 2 - 2 ^ {- n} + \\mathrm {X} & \\mathrm {X} \\leq 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/2aea1f11a715ec8fea887903f6475e13ad2ed2061063346ed0596916340f8cec.jpg" } }, "bbox": [ 670, 101, 974, 162 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（ mod (2-2-n) )" } ] }, "bbox": [ 737, 167, 919, 190 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "、表示方法：最高位为符号位，其他位为数值位。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 符号位：0－正数，1－负数。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 数值位：正数时，与绝对值相同；负数时，为绝对值取反" } ] } ] }, "bbox": [ 144, 204, 603, 258 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "双值位，正数时，与绝对值怕问，贝数的，为绝对值取汉" } ] }, "bbox": [ 152, 274, 442, 282 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于定点整数：" } ] }, "bbox": [ 161, 287, 245, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "·若 " }, { "type": "equation_inline", "content": "\\mathsf { X } = + \\mathsf { X } _ { 1 } \\mathsf { X } _ { 2 } . . . . . . \\mathsf { X } _ { n } ," }, { "type": "equation_inline", "content": "\\mathbb { M } [ \\mathbf { X } ] _ { E } = 0 , \\mathbf { X } _ { 1 } \\mathbf { X } _ { 2 } . . . . . . . \\mathbf { X } _ { n }" } ] }, "bbox": [ 265, 285, 499, 298 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "·若 " }, { "type": "equation_inline", "content": "\\scriptstyle ( = - \\textsf { X } _ { 1 } , \\textsf { X } _ { 2 } , \\ldots , \\textsf { X } _ { n }" }, { "type": "text", "content": "，则[X]反=1,xx... Xn " } ] }, "bbox": [ 265, 299, 514, 310 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于定点小数：" } ] }, "bbox": [ 161, 322, 245, 332 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·若 " }, { "type": "equation_inline", "content": "\\mathsf { \\pmb { \\Lambda } } = + \\mathbf { 0 } . \\mathsf { \\pmb { X } } _ { 1 } \\mathsf { \\pmb { X } } _ { 2 } . . . . . . . \\mathsf { \\pmb { X } } _ { n }" }, { "type": "equation_inline", "content": "\\mathbb { R } ^ { | | \\mathbf { \\boldsymbol { X } } | } \\mathbb { R } ^ { = } \\mathbf { 0 } . \\mathbf { \\boldsymbol { X } } _ { 1 } \\mathbf { \\boldsymbol { X } } _ { 2 } . . . . . . . \\mathbf { \\boldsymbol { X } _ { n } }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "若X=-0.Xx，则1. " }, { "type": "equation_inline", "content": "\\boldsymbol { \\mathfrak { A } } \\boldsymbol { \\mathfrak { A } } [ \\mathbf { X } ] _ { E } = \\mathbf { 1 } ." } ] } ] }, "bbox": [ 265, 319, 531, 350 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（1）0的表示：0的反码表示有两种形式" } ] }, "bbox": [ 144, 360, 391, 372 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} {[ + 0 ] _ {\\text {反}} = 0 0 \\dots 0} \\end{array}", "math_type": "latex", "image_source": { "path": "images/518fe401000087c650055a23fe44477c4c690ca58923fd0535f5cc1a5997d3c6.jpg" } }, "bbox": [ 163, 376, 265, 387 ] }, { "type": "equation_interline", "content": { "math_content": "[ - 0 ] _ {\\text {反}} = 1 1 \\dots 1", "math_type": "latex", "image_source": { "path": "images/dc59cfd4c6d874afe3061e098bbeae13273490f13db6da829cbb951d46e61849.jpg" } }, "bbox": [ 337, 376, 423, 388 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（2）表示范围：对于n " }, { "type": "equation_inline", "content": "^ +" }, { "type": "text", "content": "1位反码机器数×，它所能表示的数据范围为：" } ] }, "bbox": [ 144, 390, 552, 401 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "定点整数：-(2n-1）≤X≤2n-1" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·定点小数:-（1-2-n）≤X≤1-2-n" } ] } ] }, "bbox": [ 161, 405, 400, 432 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "移码" } ] }, "bbox": [ 144, 445, 186, 458 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点整数" } ], "level": 1 }, "bbox": [ 188, 468, 240, 508 ] }, { "type": "equation_interline", "content": { "math_content": "[ \\mathrm {X} ] _ {\\text {移}} = 2 ^ {\\mathrm {n}} + \\mathrm {X}", "math_type": "latex", "image_source": { "path": "images/cd099e39b47214cdde721263f19a140f282e4937223b97b0846876b33caae6c2.jpg" } }, "bbox": [ 270, 473, 532, 508 ] }, { "type": "equation_interline", "content": { "math_content": "(\\mod 2 ^ {n + 1})", "math_type": "latex", "image_source": { "path": "images/8c3a93fa9528d939a9b2ea2642676792ce251a4879d189806a2c8d5a5bfe087c.jpg" } }, "bbox": [ 334, 523, 482, 546 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2、表示方法：最高位为符号位，其他位为数值位。" } ] }, "bbox": [ 144, 556, 539, 571 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 符号位：1－正数，0－负数。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 数值位：正数时，与绝对值相同；负数时，为绝对值取反后，末位加 1。" } ] } ] }, "bbox": [ 146, 575, 712, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "值位，正效的，与绝对值怕问，贝双的，为绝对值耿反旧，不位加" } ] }, "bbox": [ 144, 627, 976, 640 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "对于定点整数：" } ], "level": 1 }, "bbox": [ 157, 708, 371, 730 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "移码=补码的 符号位取反 " } ] }, "bbox": [ 744, 665, 912, 713 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·若 " }, { "type": "equation_inline", "content": "\\mathfrak { c } { = } { + } \\mathfrak { X } _ { 1 } \\mathfrak { X } _ { 2 } { \\dots } { \\dots } \\mathfrak { X } _ { n }" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "[ \\mathsf { X } ] _ { \\mp \\mathcal { B } } = \\mathsf { 1 } , \\mathsf { X } _ { 1 } \\mathsf { X } _ { 2 } . . . . . . \\mathsf { X } _ { \\mathsf { n } }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·若 " }, { "type": "equation_inline", "content": "\\mathbf { \\ddot { X } } = - \\mathbf { \\nabla } \\mathbf { X } _ { 1 } \\mathbf { X } _ { 2 } \\ldots \\ldots \\mathbf { X } _ { n }" }, { "type": "text", "content": "，则 " }, { "type": "equation_inline", "content": "[ \\mathbf { X } ] _ { \\mathcal { F B } } = \\mathbf { 0 } ," }, { "type": "text", "content": ",xX Xx.. Xn +1. " } ] } ] }, "bbox": [ 196, 747, 900, 820 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（1） 0的表示：0 的移码表示形式是唯一的" } ] }, "bbox": [ 156, 834, 494, 850 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " " }, { "type": "equation_inline", "content": "[ + 0 ]" }, { "type": "text", "content": "移＝ 10…0 [-0]移＝ 10…0" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " （2）表示范围：对于 " }, { "type": "equation_inline", "content": "\\mathsf { n } + \\mathsf { 1 }" }, { "type": "text", "content": "位移码机器数 X，它所能表示的数据范围为：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 定点整数： " }, { "type": "equation_inline", "content": "- 2 \\mathsf { n } { \\mathsf { \\leq } } \\mathsf { X } \\leq 2 \\mathsf { n } - 1" } ] } ] }, "bbox": [ 146, 853, 719, 904 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/17570bda25ff10b972ee9301e89e66c4e3e561a977bba5dfa96f0a63acf2709d.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 174, 97, 231, 131 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "四种定点机器数的对比" } ], "level": 1 }, "bbox": [ 265, 96, 495, 116 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1、表示方法" } ], "level": 1 }, "bbox": [ 552, 99, 682, 117 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点整数" } ], "level": 1 }, "bbox": [ 216, 167, 302, 185 ] }, { "type": "table", "content": { "image_source": { "path": "images/41c63d3523508f4ebc3df1e459527e3c41e3182e257cbced25cee0832cc19663.jpg" }, "table_caption": [], "table_footnote": [], "html": "
真值X=+ X1X2......XnX=- X1X2......Xn
原码[X]原=0 X1X2......Xn[X]原=1 X1X2......Xn
反码[X]反=0 X1X2......Xn[X]反=1 X1X2......Xn
补码[X]补=0 X1X2......Xn[X]补=1 X1X2......Xn+1
移码[X]移=1 X1X2......Xn[X]移=0 X1X2......Xn+1
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真值X=+0.X1X2......XnX=-0.X1X2......Xn
原码[X]原=0.X1X2......Xn[X]原=1.X1X2......Xn
反码[X]反=0.X1X2......Xn[X]反=1.X1X2......Xn
补码[X]补=0.X1X2......Xn[X]补=1. X1 X2...... Xn + 0.00......1
移码[X]移=1.X1X2......Xn[X]移=0. X1 X2...... Xn + 0.00......1
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 255, 237, 831, 357 ] }, { "type": "table", "content": { "image_source": { "path": "images/82e974e6af36d2b511be0bfa3d01f84da4b72e86186e1b9275b73b7ee0287a68.jpg" }, "table_caption": [], "table_footnote": [], "html": "
真值X=+0X=-0
原码[X]原=0 00......0[X]原=1 00......0
反码[X]反=0 00......0[X]反=1 11......1
补码[X]补=0 00......0
移码[X]移=1 00......0
", "table_type": "complex_table", "table_nest_level": 1 }, "bbox": [ 147, 395, 969, 620 ] }, { "type": "table", "content": { "image_source": { "path": "images/12af43b7c38126a6dbfa7217f2e5d569976f2ab0bdb03a07079c2a390a4e7a41.jpg" }, "table_caption": [ { "type": "text", "content": "n+1位机器数" } ], "table_footnote": [], "html": "
定点整数定点小数
原码- (2n-1) ≤X ≤ 2n-1- (1-2-n) ≤X≤1-2-n
反码- (2n-1) ≤X ≤ 2n-1- (1-2-n) ≤X≤1-2-n
补码-2n≤X ≤ 2n-1-1≤X ≤ 1-2-n
移码-2n≤X ≤ 2n-1-1≤X ≤ 1-2-n
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 154, 676, 515, 790 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 154, 103, 268, 120 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\because [ X ] _ {\\text {原}} = X _ {S}, X _ {1} X _ {2} \\dots X _ {n} \\\\ \\cdot X _ {S} = 0: X = + X _ {1} X _ {2} \\dots X _ {n} \\\\ X _ {S} = 1: X = - X _ {1} X _ {2} \\dots X _ {n} \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/1da50c8aec212edaea1e14ff2a3f71cde06fdc87eb3bc5d3dbad1ac89cd8f092.jpg" } }, "bbox": [ 154, 103, 332, 171 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 421, 103, 534, 118 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\because [ X ] _ {\\text {反}} = X _ {S}, X _ {1} X _ {2} \\dots X _ {n} \\\\ = X _ {S} = 0: X = + X _ {1} X _ {2} \\dots X _ {n} \\\\ \\cdot \\mathbf {x} _ {s} = 1: X = - \\overline {{X _ {1}}} \\overline {{X _ {2}}} \\dots \\overline {{X _ {n}}} \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/ae62b8b8d754b264d256ff236caadf0e2d806f49ee7ecc0c815a1c4a7d249109.jpg" } }, "bbox": [ 421, 103, 618, 172 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "[X]补→真值？ " } ] }, "bbox": [ 154, 195, 265, 211 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\because [ X ] _ {\\text {补}} = X _ {S}, X _ {1} X _ {2} \\dots X _ {n} \\\\ \\cdot X _ {S} = 0: X = + X _ {1} X _ {2} \\dots X _ {n} \\\\ \\cdot X _ {S} = 1: X = - \\left(\\overline {{X _ {1}}} \\overline {{X _ {2}}} \\dots \\overline {{X _ {n}}} + 1\\right) \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/d69a591acdb9451f68a6e46ec957680bccbd3a6fd45b217987c4edbf7b23fa0a.jpg" } }, "bbox": [ 154, 214, 386, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "[X]移→真值？ " } ] }, "bbox": [ 421, 196, 534, 212 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\because [ X ] _ {\\text {移}} = X _ {S}, X _ {1} X _ {2} \\dots X _ {n} \\\\ = X _ {S} = 1: X = + X _ {1} X _ {2} \\dots X _ {n} \\\\ \\cdot \\mathbf {X} _ {s} = \\mathbf {0}: X = - (\\overline {{X _ {1}}} \\overline {{X _ {2}}} \\dots \\overline {{X _ {n}}} + 1) \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/fc75d670fc1a7c8e7032dcb815495ed2d13766a2f05a75430bb7cffdc06fce16.jpg" } }, "bbox": [ 421, 216, 653, 265 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "浮点数格式" } ], "level": 1 }, "bbox": [ 147, 278, 240, 293 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "浮点数N的构成：" } ] }, "bbox": [ 149, 302, 265, 316 ] }, { "type": "equation_interline", "content": { "math_content": "N = M \\times R ^ {E}", "math_type": "latex", "image_source": { "path": "images/506b2520b99b0508c9881053fe2005225f52714c4150292ba602b527ab4ec9bb.jpg" } }, "bbox": [ 342, 300, 453, 313 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "阶码E（Exponent）、尾数M（Mantissa）和阶码的底R（Radix)" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "阶码的底R：是一个常数，隐含规定，一般为2、8或16" } ] } ] }, "bbox": [ 149, 319, 574, 348 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "为什么将数符放置在首位？" } ] }, "bbox": [ 157, 357, 203, 385 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "将数符放置 在首位，方 便比较大小 " } ] }, "bbox": [ 156, 401, 211, 428 ] }, { "type": "image", "content": { "image_source": { "path": "images/f318b4e4d2f9ec985b58f83758be6fd1d10255482e3dcd00bc9b8f495c476389.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 278, 357, 579, 400 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "浮点数格式要素：" } ] }, "bbox": [ 265, 405, 383, 417 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "位数：阶码位数 " }, { "type": "equation_inline", "content": "^ +" }, { "type": "text", "content": "尾数位数→总位数" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "编码：阶码和尾数采用的机器数编码" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{3}" }, { "type": "text", "content": "排列顺序：数符是否在最高位" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{4}" }, { "type": "text", "content": "其他特殊的编码规则：譬如隐藏位" } ] } ] }, "bbox": [ 285, 419, 532, 473 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "尾数M：定点小数" } ], "level": 1 }, "bbox": [ 164, 500, 317, 514 ] }, { "type": "equation_interline", "content": { "math_content": "0. 1 0 1 1 1 0 \\times 2 ^ {- 2} = 0. 0 0 1 0 1 1 1", "math_type": "latex", "image_source": { "path": "images/6fed770bf83e60c442d5db60f480c7c5ffa31fbdd84240e62b1e6068aeea5cfd.jpg" } }, "bbox": [ 527, 499, 717, 513 ] }, { "type": "equation_interline", "content": { "math_content": "N = M \\times R ^ {E}", "math_type": "latex", "image_source": { "path": "images/4e9024dc16c33bb4143253ad8c32cc6fc3f89030beced0667b59bef5c494f0d6.jpg" } }, "bbox": [ 563, 519, 695, 538 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "尾数的位数：决定了浮点数有效数值的精度" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·尾数的符号：数符，代表了浮点数的正负 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·尾数的编码：一般采用原码和补码表示" } ] } ] }, "bbox": [ 186, 521, 492, 571 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "阶码E：定点整数" } ], "level": 1 }, "bbox": [ 164, 579, 309, 593 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·阶码的位数：多少决定了浮点数的表示范围" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·阶码的数值：大小决定了数据实际小数点位置与尾数的小数点位置之间的偏移量" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·阶码的符号：阶符，决定了实际小数点位置在尾数的小数点位置的左边还是右边" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·阶码的编码：一般采用移码和补码表示" } ] } ] }, "bbox": [ 186, 599, 732, 670 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "阶码的底R：一般为2、8或16，且隐含规定" } ], "level": 1 }, "bbox": [ 164, 677, 505, 690 ] }, { "type": "image", "content": { "image_source": { "path": "images/6a0fe239362570bed9f3912109c3557d492e9c9e6cb11a12f124138b78f42013.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 453, 703, 478, 714 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "非规格化浮点数：" } ], "level": 1 }, "bbox": [ 154, 110, 329, 128 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "规格化操作：修改阶码和左右移尾数的方法来使其变为规格化浮点数" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "右规：尾数进行右移实现的规格化" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "左规：尾数进行左移实现的规格化" } ] } ] }, "bbox": [ 181, 135, 521, 219 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "什么情况下要做规格化操作？" } ], "level": 1 }, "bbox": [ 526, 110, 806, 128 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "尾数有前导零：即 " }, { "type": "equation_inline", "content": "\\textless 1 / 2" }, { "type": "text", "content": "，需要左规" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "尾数运算溢出：即≥1，需要右规" } ] } ] }, "bbox": [ 554, 135, 870, 175 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "使用规格化浮点数表示数据的优点：" } ], "level": 1 }, "bbox": [ 526, 190, 863, 209 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "提高了浮点数据的精度" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "使程序能够更方便地交换浮点数据" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "可以使浮点数的运算更为简化" } ] } ] }, "bbox": [ 554, 216, 855, 280 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "（自定义格式）规格化浮点数表示方法：" } ], "level": 1 }, "bbox": [ 152, 274, 532, 293 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "（1）写出数据的二进制真值" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "(2）转换为M×2的形式，其中M为没有前导零的定点小数，E为整数" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "（3）按照格式写出M和E的规定机器数编码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "（4）按照格式要求排列E和M" } ] } ] }, "bbox": [ 191, 298, 761, 382 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "浮点数的运算过程中，尾数发生溢出（超出尾数所能表示的范围），进行右规。右规一次即可。" } ] }, "bbox": [ 144, 407, 757, 442 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "阶码发生溢出时，浮点数被判为溢出" } ] }, "bbox": [ 146, 445, 433, 460 ] }, { "type": "image", "content": { "image_source": { "path": "images/a61299ff3ee52a556a1dcdca353af19b9a90d7dae7b2db0fa0bfe27f716b35b7.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 845, 363, 885, 387 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "1. 一些小结论" } ], "level": 1 }, "bbox": [ 147, 504, 317, 526 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于补码表示的最大值和最小值是非对称的" } ] }, "bbox": [ 144, 607, 487, 623 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即最大值的绝对值(0111)的原码比最小值的绝对值的(1000)原码要小" } ] }, "bbox": [ 146, 625, 677, 642 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于 n 位定点整数的机器数则为（ " }, { "type": "equation_inline", "content": "( n + 1" }, { "type": "text", "content": "）位（算上符号位）" } ] }, "bbox": [ 146, 644, 594, 659 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于一个负数X来说" } ] }, "bbox": [ 146, 663, 309, 678 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "[X]原+[X]补 " }, { "type": "equation_inline", "content": "\\cdot = 2 \\wedge _ { \\mathsf { n } }" } ] }, "bbox": [ 146, 682, 280, 697 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Eg.[-2]原 " }, { "type": "equation_inline", "content": "= 1 0 1 0" }, { "type": "text", "content": "；[-2]补 " }, { "type": "equation_inline", "content": "\\mathord { \\left. \\kern - delimiterspace \\right.} 1 1 0 " }, { "type": "text", "content": "，相加为 1000（当成原码并去掉符号位，就是 8）" } ] }, "bbox": [ 146, 699, 761, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "负数的补码就是其绝对值的补数" } ] }, "bbox": [ 164, 718, 416, 734 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "X的绝对值 " }, { "type": "equation_inline", "content": "+ [ x ]" }, { "type": "text", "content": "补但是[x]补当成原码去掉符号位，为10000即 " }, { "type": "equation_inline", "content": "^ { 2 \\wedge }" }, { "type": "text", "content": "（ " }, { "type": "equation_inline", "content": "( n + 1 )" }, { "type": "text", "content": "）" } ] }, "bbox": [ 144, 737, 705, 753 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "还是上个例子。[-2]补 " }, { "type": "equation_inline", "content": "= 1 1 1 0 = 1 4" }, { "type": "text", "content": "， " }, { "type": "equation_inline", "content": "2 + 1 4 = 1 6" }, { "type": "text", "content": "；" } ] }, "bbox": [ 146, 755, 495, 771 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "所以，有符号数和无符号数之间的转换规则是：" } ] }, "bbox": [ 146, 774, 510, 789 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "位模式不变，但是解释这些位的方式改变了" } ] }, "bbox": [ 146, 793, 484, 808 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数值数据的表示" } ] }, "bbox": [ 146, 810, 273, 826 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "进位计数制" } ] }, "bbox": [ 146, 829, 240, 845 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "不同数制之间的转换" } ] }, "bbox": [ 146, 848, 309, 864 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "十进制的编码" } ] }, "bbox": [ 146, 866, 255, 882 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "机器数" } ] }, "bbox": [ 146, 885, 203, 900 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数值数据在计算机中的表示形式称为机器数。机器数的特点是：" } ] }, "bbox": [ 146, 93, 633, 108 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "^ \\ast" }, { "type": "text", "content": "机器数：数值数据在计算机中的二进制表示形式。真值：机器数真正表示的数值数据" } ] }, "bbox": [ 147, 111, 544, 122 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·机器数的符号位必须被数值化为二进制0和1书写：+/-符号数值（二进制或 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·机器数的小数点用隐含规定的方式来表达" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·表示的数值范围受计算机字长的限制" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·编码方法：不同的排列方式，常见的有原码、反码、补码、移码。" } ] } ] }, "bbox": [ 161, 124, 542, 172 ] }, { "type": "image", "content": { "image_source": { "path": "images/c47a7a94827b3662a134a37286e20304c8e0e73f91dd09676afa6229802037f6.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 519, 168, 537, 183 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点机器数的表示方法" } ], "level": 1 }, "bbox": [ 146, 206, 420, 229 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "一个字节由 8 个位组成" } ] }, "bbox": [ 146, 274, 327, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "我们把这种按照一位一位表示数据的方式称为位模式" } ] }, "bbox": [ 146, 292, 556, 307 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于无符号数编码" } ] }, "bbox": [ 146, 311, 292, 326 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在机器中的解释，采用原码表示法（？）" } ] }, "bbox": [ 146, 329, 457, 344 ] }, { "type": "image", "content": { "image_source": { "path": "images/29519d3b05891aec898580f89a8b7d69cdc6b6f2666f40bc361510a60e4b8e61.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 344, 774, 501 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于有符号数的编码，采用补码" } ] }, "bbox": [ 146, 514, 398, 529 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/3bb2c65a3b93dd5f3ba70445f8e1ac6b1845d70eca6912b30ec7b60296858a1a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 89, 808, 335 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于相同的位模式，无符号数和有符号数的数值关系如图所示：" } ] }, "bbox": [ 144, 351, 636, 368 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（这里的 B2U/B2T 是位向量到无符号数和有符号数的函数映射）" } ] }, "bbox": [ 156, 370, 643, 387 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "最高位理解成负权重-2^w-1" } ] }, "bbox": [ 146, 390, 364, 405 ] }, { "type": "equation_interline", "content": { "math_content": "B 2 U _ {w} = x _ {w - 1} \\cdot 2 ^ {w} + B 2 T _ {w}", "math_type": "latex", "image_source": { "path": "images/a4a48fbea7f612dad91bb421ab1b5bca91bc84a9189bcdaefe6f535669af5f99.jpg" } }, "bbox": [ 147, 416, 349, 432 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "用 T2U 来表示有符号数到无符号数的函数映射" } ] }, "bbox": [ 144, 445, 509, 460 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "T 2 U _ { w } ( x ) = \\left\\{ \\begin{array} { c } { { x + 2 ^ { w } , x < 0 } } \\\\ { { x , x > 0 } } \\end{array} \\right." }, { "type": "equation_inline", "content": "T 2 U _ { w } \\left( x \\right) = \\left( x + 2 ^ { w } \\right)" } ] }, "bbox": [ 146, 468, 621, 508 ] }, { "type": "equation_interline", "content": { "math_content": "U 2 T _ {w} (u) = \\left\\{ \\begin{array}{l} x, u \\leq T M \\max _ {u} \\\\ x - 2 ^ {w}, u > T M a x _ {u} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/bebacba3b43ee8a983eac0873165fc32b0d2ed3fbac2c44949f709c7a6c51d81.jpg" } }, "bbox": [ 147, 519, 400, 558 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "W位(包含了符号位！)补码表示的数值" } ] }, "bbox": [ 144, 575, 445, 590 ] }, { "type": "equation_interline", "content": { "math_content": "\\omega = - x _ {\\omega - 1} 2 ^ {\\omega - 1} + \\sum_ {i = 0} ^ {\\omega - 2} x _ {i} 2 ^ {i}", "math_type": "latex", "image_source": { "path": "images/22c58311fdad6d274129624017c5b47b7d10f3e80e3f2d2a4075e54a8bb0c2f9.jpg" } }, "bbox": [ 147, 599, 357, 637 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "也就是说，把第一位作为负数- " }, { "type": "equation_inline", "content": ". 2 \\land \\mathfrak { n }" }, { "type": "text", "content": "来处理；正数为 0,负数为 1；" } ] }, "bbox": [ 144, 649, 636, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "你可能已经发现，二进制数的“互补数”编码规则和补码的定义非常的像，其实不是像，而是它们本来就是等价的。也就是“对应正数的互补数”等于“对应正数按位取反再加1”。下面尝试证明。" } ] }, "bbox": [ 144, 667, 828, 719 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "原命题精确描述为：相对于 w 位的二进制数，整数 " }, { "type": "equation_inline", "content": "\\mathsf { b } \\in ( 0 , ]" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "\\mathsf { b } \\in ( 0 , ]" }, { "type": "text", "content": "，等式 " }, { "type": "equation_inline", "content": "- b = - b + 1" }, { "type": "text", "content": "成立。" } ] }, "bbox": [ 144, 722, 830, 757 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "原码、补码、反码、移码的直接二进制值（机器数的形式值）各自有什么作用？" } ] }, "bbox": [ 144, 760, 761, 775 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "反码、补码的真值为负数的时候机器数形式值（符号位作为 " }, { "type": "equation_inline", "content": "{ 2 } \\Lambda _ { \\mathsf { n } }" }, { "type": "text", "content": "而不是正负号）减去对应的 " }, { "type": "equation_inline", "content": "2 \\sim ( \\mathsf { n } + 1 )" }, { "type": "text", "content": "或者是 " }, { "type": "equation_inline", "content": "_ { 2 ^ { \\wedge } }" }, { "type": "text", "content": "（n+1） " }, { "type": "equation_inline", "content": "+ 1" }, { "type": "text", "content": "就是这个机器数的真值。" } ] }, "bbox": [ 144, 778, 843, 812 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "移码是无论正负直接减去 " }, { "type": "equation_inline", "content": "2 \\sim ( \\mathsf { n } + 1 )" } ] }, "bbox": [ 144, 816, 411, 832 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而原码没有类似的结果" } ] }, "bbox": [ 146, 834, 327, 850 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/78e6ebe0fa6d18716eca1d48f88e453ca73f8c16ae9fe3c381407ee46ef7c2df.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 97, 811, 307 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "有符号数最高位为 0（正数），其余为 1" } ] }, "bbox": [ 147, 315, 460, 332 ] }, { "type": "image", "content": { "image_source": { "path": "images/e3ecdb1331e495f2deeb4e6668c8da5f41e4d66251fb5eee1929cfca02758dec.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 357, 801, 567 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "有符号数的-1和无符号数的最大值一样编码表示" } ] }, "bbox": [ 147, 574, 524, 590 ] }, { "type": "image", "content": { "image_source": { "path": "images/d7ebf2a15fdae3ac42d3466cc5c4d421de2dfae73da3be0f75e1620b2f98e562.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 593, 882, 813 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "2．真值与补码的转换 " } ], "level": 1 }, "bbox": [ 144, 99, 408, 122 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "真值->补码" } ], "level": 1 }, "bbox": [ 144, 166, 238, 181 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "正数的补码：去掉 " }, { "type": "equation_inline", "content": "^ +" }, { "type": "text", "content": "号前面加 0。" } ] }, "bbox": [ 144, 185, 394, 199 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "负数的补码：1、去掉 - 号前面加 1，从右到左找到第一个 1，左边全部取反。" } ] }, "bbox": [ 144, 203, 739, 218 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "2、负数的补码是将原码的符号位保持不变,其余各位取反，然后加1,就得到其补码;" } ] }, "bbox": [ 144, 221, 788, 236 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算一个负数的补码时，通常遵循以下步骤：" } ] }, "bbox": [ 144, 240, 494, 255 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "1. 取绝对值的二进制表示：求出绝对值（即对应正数），求出数值的二进制形式。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "2. 取反：将二进制数的每一位取反，即 0 变为 1，1 变为 " }, { "type": "equation_inline", "content": "0 _ { \\circ }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "3. 加 1：在取反结果的基础上加 1，得到最终的补码表示。" } ] } ] }, "bbox": [ 176, 258, 816, 311 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "也就是说，符号位参与了运算。在第 2 步实际上已经将“符号位为 1“表示成 0-1" } ] }, "bbox": [ 144, 313, 763, 329 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "实际上，这个符号位啊，你无论是先不管符号位取反再上符号位为 1，还是一起从绝对值取反（从0变成1—），不都是符号位为1吗？但参与 " }, { "type": "equation_inline", "content": "+ 1" }, { "type": "text", "content": "运算是必须的。" } ] }, "bbox": [ 144, 332, 847, 366 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "补码- " }, { "type": "equation_inline", "content": "\\cdot >" }, { "type": "text", "content": "真值 " } ], "level": 1 }, "bbox": [ 144, 370, 238, 385 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "符号位 0 的补码的真值：去掉 0 前面加 " }, { "type": "equation_inline", "content": "^ +" }, { "type": "text", "content": "号。" } ] }, "bbox": [ 144, 388, 485, 403 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "符号位1的补码的真值：" } ] }, "bbox": [ 144, 407, 336, 422 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "1、去掉 1 前面加-号，从右到左找到第一个 1，左边全部取反。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "2、负数的原码就是它的补码的补码。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "3、如果补码的符号位为“1”，表示是一个负数，那么补码减1,再将符号位保持不变其余各位取反" } ] } ] }, "bbox": [ 144, 425, 833, 495 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3.对于 " }, { "type": "equation_inline", "content": "( X + 2 \\land ( \\mathbf { n } + 1 ) )" }, { "type": "text", "content": "mod (2^(n+1))的解释 " } ], "level": 1 }, "bbox": [ 176, 558, 678, 581 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对某两个整数a，b，若它们除以正整数 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "所得的余数相等，则称a，b对于模m同余，也就是严格来说，存在整数k使得" } ] }, "bbox": [ 144, 642, 821, 677 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "a - b = k m" }, { "type": "text", "content": "，则称 a,b 对于除数 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "是同余的。一般记做" } ] }, "bbox": [ 144, 680, 561, 695 ] }, { "type": "equation_interline", "content": { "math_content": "a \\equiv b (\\bmod m)", "math_type": "latex", "image_source": { "path": "images/3b36ee0f04f604f2f8b68ab0a13fcabc79e0b7adb7d4065653ecf97005c7d5cb.jpg" } }, "bbox": [ 146, 699, 253, 714 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在模运算中，一个数与它除以“模”后得到的余数是等价的，如 A、B、M 满足" } ] }, "bbox": [ 144, 717, 763, 732 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\mathsf { A } { = } \\mathsf { B } { + } \\mathsf { K } { \\times } \\mathsf { M } ( \\mathsf { K }" }, { "type": "text", "content": "为整数)，则记为 " }, { "type": "equation_inline", "content": "\\mathsf { A } \\equiv \\mathsf { B } ( \\mathsf { m o d } \\mathsf { M } )" }, { "type": "text", "content": ")，即 A、B各除以 M后的余数相同。在补码运算中，[A]补-[B]补=[A]补 " }, { "type": "equation_inline", "content": "+ \\mathsf { M }" }, { "type": "text", "content": "-[B]补，而 M-[B]补 " }, { "type": "equation_inline", "content": "=" }, { "type": "text", "content": "[-B]补，因此补码可以借助加法运算来实现减法运算。从模运算的角度，补码的表示利用了以下规律：" } ] }, "bbox": [ 144, 734, 843, 788 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "模运算的同余性质：对于任意整数 a 和模数 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "，有 " }, { "type": "equation_inline", "content": "\\mathsf { a } \\equiv \\mathsf { a } + \\mathsf { k m } ( \\mathsf { m o d } \\mathsf { m } )" }, { "type": "text", "content": "，其中 k为整数。这意味着在模 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "的运算中，数值是循环的，超过 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "会回到起点。" } ] }, "bbox": [ 144, 791, 843, 825 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "为何常见写法是 " }, { "type": "equation_inline", "content": "^ { 6 6 } x + 2 \\wedge _ { \\mathsf { n } + 1 }" }, { "type": "text", "content": "mod " }, { "type": "equation_inline", "content": "\\mathsf { \\pm } \\mathsf { \\Lambda } \\mathsf { n } \\mathsf { \\pm } ^ { \\prime }" }, { "type": "text", "content": "1”而不是 " }, { "type": "equation_inline", "content": "\\pmb { \\times }" }, { "type": "text", "content": "mod " }, { "type": "equation_inline", "content": "^ { 2 n + 1 }" } ] }, "bbox": [ 144, 828, 631, 843 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数论上， " }, { "type": "equation_inline", "content": "^ { * } \\times" }, { "type": "text", "content": "与 " }, { "type": "equation_inline", "content": "\\mathsf { x } + 2 \\wedge \\{ \\mathsf { n } + 1 \\} \\times ^ { \\prime \\prime }" }, { "type": "text", "content": "在模 " }, { "type": "equation_inline", "content": "2 \\cdot \\{ \\mathsf { n } + 1 \\}" }, { "type": "text", "content": "下等价；" } ] }, "bbox": [ 144, 846, 544, 862 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "但在说明“补码如何把负数映射到二进制的高半区”时，显式地加上 " }, { "type": "equation_inline", "content": "2 \\times \\{ \\mathsf { n } + 1 \\}" }, { "type": "text", "content": "更能直观反映“负数被搬到了高位 " }, { "type": "equation_inline", "content": "\\ O = \\uparrow" }, { "type": "text", "content": "的区间”这个事实。" } ] }, "bbox": [ 144, 865, 842, 898 ] } ], [ { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 非负数直接写成“它自己” (因为落在小半区间 " }, { "type": "equation_inline", "content": "[ 0 , 2 \\land \\mathsf { n } ) ]" }, { "type": "text", "content": ")；" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 负数写成 " }, { "type": "equation_inline", "content": "^ { \\mathfrak { a } } \\mathsf { X } + 2 ^ { \\wedge } \\{ \\mathsf { n } + 1 \\} ^ { \\prime \\prime }" }, { "type": "text", "content": "(因为这样就落在大半区间 " }, { "type": "equation_inline", "content": "[ 2 \\land \\neg , 2 \\land \\{ \\mathsf { n } + 1 \\} - 1 ] ) _ { \\circ }" } ] } ] }, "bbox": [ 144, 93, 707, 128 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在计算机系统中， " }, { "type": "equation_inline", "content": "\\mathsf { n }" }, { "type": "text", "content": "位定点整数（包括符号位）补码的取值范围是 " }, { "type": "equation_inline", "content": "- 2 \\land \\mathsf { n }" }, { "type": "text", "content": "到 " }, { "type": "equation_inline", "content": "2 \\ n _ { \\mathsf { n } } - 1" }, { "type": "text", "content": "。" } ] }, "bbox": [ 144, 130, 794, 145 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这意味着总共有 " }, { "type": "equation_inline", "content": "2 \\cdot \\{ \\mathsf { n } + 1 \\}" }, { "type": "text", "content": "个不同的数值可以表示。因此，模数 " }, { "type": "equation_inline", "content": "2 \\cdot \\{ \\mathsf { n } + 1 \\}" }, { "type": "text", "content": "对应于这些数值的总数量。" } ] }, "bbox": [ 144, 148, 843, 183 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "负数的表示：在模 " }, { "type": "equation_inline", "content": "2 \\cdot \\{ \\mathsf { n } + 1 \\}" }, { "type": "text", "content": "的系统中，负数 X 可以表示为 " }, { "type": "equation_inline", "content": "2 \\wedge \\{ \\mathsf { n } + 1 \\} + \\mathsf { X } ," }, { "type": "text", "content": ",因为 X 为负数，会落在 " }, { "type": "equation_inline", "content": "2 \\land ( \\mathsf { n } )" }, { "type": "text", "content": "到 " }, { "type": "equation_inline", "content": "2 \\cdot \\{ \\mathsf { n } + 1 \\} - 1" }, { "type": "text", "content": "的范围内，符合无符号数的表示范围。" } ] }, "bbox": [ 144, 186, 835, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因为模为 " }, { "type": "equation_inline", "content": "2 \\cdot ( n + 1 ) , k = 1" }, { "type": "equation_inline", "content": "k = 1" }, { "type": "text", "content": "，那么就可以将- " }, { "type": "equation_inline", "content": "2 ^ { \\wedge } \\mathsf { n } - 2 ^ { \\wedge } \\mathsf { n } - 1" }, { "type": "text", "content": "统一成 " }, { "type": "equation_inline", "content": "( X { + } 2 ^ { \\wedge } ( \\mathsf { n } { + } 1 ) )" }, { "type": "text", "content": ") mod " }, { "type": "equation_inline", "content": "( 2 \\land ( \\mathsf { n } + 1 ) )" } ] }, "bbox": [ 144, 223, 811, 239 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "减法（加上一个负数）的运算用加法代替了，我们说的补码（反码）实际上是数对于一个模的同余数。" } ] }, "bbox": [ 144, 241, 836, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（这里的取模运算，请让我们抛弃符号数，eg.-2 mod " }, { "type": "equation_inline", "content": "8 { = } 6" }, { "type": "text", "content": ", " }, { "type": "equation_inline", "content": "[ 6 ] { = } 1 1 0" }, { "type": "text", "content": ",而 [-2]补 " }, { "type": "equation_inline", "content": "\\mathord { = } 1 1 1 0" }, { "type": "text", "content": "，当成原码，去掉符号位）" } ] }, "bbox": [ 144, 278, 831, 312 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "规定：把0当成正数，也即 " }, { "type": "equation_inline", "content": "+ 0" }, { "type": "text", "content": "，这样0的编码就变成：0，0000000。那8位二进制表示的正数范围仍然是： " }, { "type": "equation_inline", "content": "+ 0 -- + 1 2 7 _ { \\mathrm { { ( } } }" }, { "type": "text", "content": "。" } ] }, "bbox": [ 144, 315, 831, 349 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "但是，对于负数就必须要做调整，也即-0必须要让位---1，0000000这个编码不能表示-0。我们可以把负数整体向后“挪动1位”：只要将8位二进制表示的负数范围从：-127—— -0变成：-128 —— -1，就能成功解决问题。" } ] }, "bbox": [ 144, 351, 826, 405 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "-0 的补数是 " }, { "type": "equation_inline", "content": "2 5 6 - 0 = 2 5 6 = 0 \\times 1 0 0" }, { "type": "text", "content": "，由于是“8 位二进制数”，所以-0 的补数是 " }, { "type": "equation_inline", "content": "0 \\times 0 0" }, { "type": "text", "content": "；‘我们知道“- " }, { "type": "equation_inline", "content": "\\cdot 1 = 0 - 1 \\prime" }, { "type": "text", "content": "，所以" } ] }, "bbox": [ 144, 407, 836, 441 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "-1 的补数是 " }, { "type": "equation_inline", "content": "2 5 6 - 1 = 2 5 5 = 0 \\times 1 =" }, { "type": "text", "content": "；这就好比是\"0-1\"不够减，\"借 256\"得到的（这里借的是模数值）。" } ] }, "bbox": [ 144, 445, 836, 479 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "同理，可以-2~-127的补数；" } ] }, "bbox": [ 144, 482, 364, 497 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "注意：这里的补数，就是计算机所讲的补码。" } ] }, "bbox": [ 144, 501, 492, 516 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "-127 的补数是 " }, { "type": "equation_inline", "content": "2 5 6 - 1 2 7 = 1 2 9 = 0 \\times 8 1" } ] }, "bbox": [ 144, 520, 428, 535 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "-1~-127 的补码范围： " }, { "type": "equation_inline", "content": "0 { \\times } \\mathsf { F F } { \\sim } 0 { \\times } 8 1" } ] }, "bbox": [ 144, 539, 416, 552 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "负数-128的补码：" } ] }, "bbox": [ 144, 557, 285, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "我们知道“- " }, { "type": "equation_inline", "content": "\\cdot 1 2 8 = 0 - 1 2 8 ^ { \\prime \\prime }" }, { "type": "text", "content": "，“ " }, { "type": "equation_inline", "content": "- 1 2 8 = - 1 + - 1 2 7 ^ { \\prime \\prime }" }, { "type": "text", "content": "所以" } ] }, "bbox": [ 144, 575, 527, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "-128 的补数是 " }, { "type": "equation_inline", "content": "2 5 6 - 1 2 8 = 1 2 8 = 0 \\times 8 0" }, { "type": "text", "content": "；所以-128 的补码为 0x80，但是-128 没有原码和反码。所以人为地规定 10000000 表示-128" } ] }, "bbox": [ 144, 593, 848, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "总结：" } ] }, "bbox": [ 146, 631, 193, 645 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "补码补码，就是补上去一个模对应的补数" } ] }, "bbox": [ 144, 649, 468, 664 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "小结" } ], "level": 1 }, "bbox": [ 146, 689, 205, 709 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在一个区间长度为 M 内，统一正负数的常用方法是" } ] }, "bbox": [ 144, 756, 546, 771 ] }, { "type": "equation_interline", "content": { "math_content": "X ^ {\\prime} = (X + M) \\mod M", "math_type": "latex", "image_source": { "path": "images/beafd15c2870a35bf6caf29e5deeb098a0a9e02a3d35c6c8cf34e7d138c7fa7d.jpg" } }, "bbox": [ 144, 774, 304, 788 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "其中 M 就是模数。如果你换成一个非 M 的倍数 N，则只有在 " }, { "type": "equation_inline", "content": "\\Nu \\equiv 0" }, { "type": "text", "content": "(mod M)的情况下才能达到同样效果。因此，负数加上的部分必须是 M 的倍数，通常取最小正倍数 M 来实现统一。" } ] }, "bbox": [ 144, 791, 843, 844 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "可以这么理解：如果一个区间（或集合）恰好包含 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "个整数，并且这 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "个整数构成了一个模 " }, { "type": "equation_inline", "content": "\\mathsf { m }" }, { "type": "text", "content": "的完全剩余系，那么对任意整数 X，用" } ] }, "bbox": [ 144, 848, 835, 882 ] }, { "type": "equation_interline", "content": { "math_content": "X \\bmod m", "math_type": "latex", "image_source": { "path": "images/b7fa56ca12cf469ff5d4d066afb576e47268073ab4fc06cc30d6109b3342fda2.jpg" } }, "bbox": [ 146, 885, 223, 898 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "总能得到落在这个集合中的唯一代表。换句话说，只要这个集合包含的元素个数为 m（而且互不同余），那么无论 X是正数还是负数，都有" } ] }, "bbox": [ 144, 93, 842, 127 ] }, { "type": "equation_interline", "content": { "math_content": "X \\equiv X \\bmod m", "math_type": "latex", "image_source": { "path": "images/92ff28ad2ac2d39604ac401c45d8fdc10fe10ad7aa4d8cebca64fb6437194e65.jpg" } }, "bbox": [ 146, 131, 248, 143 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "而且 Xmodm 就是该模 m 意义下的标准表示。" } ] }, "bbox": [ 144, 148, 505, 164 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "为什么定点小数的模 2？" } ], "level": 1 }, "bbox": [ 146, 174, 428, 196 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "至于模数为 2 的说法，这是因为补码表示中，数值的表示是循环的。例如，-1 和 " }, { "type": "equation_inline", "content": "1 - 2 \\land ( - \\mathsf { n } )" }, { "type": "text", "content": "之间的差值正好是 " }, { "type": "equation_inline", "content": "2 - 2 \\land ( - \\mathsf { n } )" }, { "type": "text", "content": "，但由于补码的循环特性，这个差值在模 2 的意义下是等价的。" }, { "type": "equation_inline", "content": "- 2 \\land ( - \\mathsf { n } )" }, { "type": "text", "content": "并没有被省略，而是补码表示的范围已经包含了从 -1 到 " }, { "type": "equation_inline", "content": "1 - 2 \\land ( - \\mathsf { n } )" }, { "type": "text", "content": "的所有可能值。补码的模数为 2 是因为补码表示中，数值的表示是循环的，且循环周期为 2。" } ] }, "bbox": [ 144, 204, 853, 275 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "关于数的扩展" } ], "level": 1 }, "bbox": [ 146, 319, 314, 341 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "基本上是这样的。对两类数来说：" } ] }, "bbox": [ 144, 385, 406, 400 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "高位扩展（符号扩展）：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "低位扩展（增加精度）：" } ] } ] }, "bbox": [ 176, 403, 396, 437 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点整数的符号扩展：" } ], "level": 1 }, "bbox": [ 146, 458, 317, 474 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于无符号数，称为零扩展" } ] }, "bbox": [ 144, 478, 366, 493 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于有符号数，称为符号位扩展" } ] }, "bbox": [ 144, 495, 400, 511 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在原符号位和数值位中间添加新位，正数都添0，负数原码添0，负数反、补码、移码添1。" } ] }, "bbox": [ 144, 514, 830, 548 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} B 2 T _ {w} \\left(\\left[ x _ {w - 1}, x _ {w - 2}, \\dots , x _ {0} \\right]\\right) (1) \\\\ B 2 T _ {w + k} \\left(\\left[ x _ {w - 1}, \\dots , x _ {w - 1}, x _ {w - 1}, \\dots , x _ {0} \\right]\\right) (2) \\\\ B 2 T _ {w + 1} - B 2 T _ {w} = 0 \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/aed113e2721cf447af94ac7c3d992bb566d046b88cda848b052ccf30d002f3c4.jpg" } }, "bbox": [ 147, 555, 463, 615 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "定点小数的符号扩展：" } ], "level": 1 }, "bbox": [ 146, 626, 317, 640 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在原符号位和数值位后面添加新位，正数都添0，负数原、补码、移码添0，负数反码添1。" } ] }, "bbox": [ 144, 644, 830, 677 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "低位截断 " } ], "level": 1 }, "bbox": [ 146, 681, 221, 696 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "将 int 类型强制转换为 short 类型，高 16 位被丢弃" } ] }, "bbox": [ 144, 700, 547, 715 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "将一个 " }, { "type": "equation_inline", "content": "\\pmb { \\mathsf { W } }" }, { "type": "text", "content": "位的无符号数，截断成k位的方法是，丢弃最高的w-k位。截断操作可以对应于取模运算。" } ] }, "bbox": [ 144, 718, 845, 752 ] }, { "type": "equation_interline", "content": { "math_content": "B 2 U _ {w} \\left(\\left[ x _ {w - 1}, x _ {w - 2}, \\dots , x _ {0} \\right]\\right) \\mod 2 ^ {k} = B 2 U _ {k} \\left(\\left[ x _ {k - 1}, x _ {k - 2}, \\dots , x _ {0} \\right]\\right)", "math_type": "latex", "image_source": { "path": "images/8319129a0d5fe876e86a131988b6e7ba39efdfa539360f865dc676c3b6d6bff7.jpg" } }, "bbox": [ 147, 763, 623, 780 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "截断有符号数" } ], "level": 1 }, "bbox": [ 146, 791, 258, 807 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "先转换成无符号数，再截断，再转换成有符号数" } ] }, "bbox": [ 144, 810, 522, 826 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即 " } ] }, "bbox": [ 146, 829, 169, 844 ] }, { "type": "equation_interline", "content": { "math_content": "T ^ {2} U _ {w} (x) = \\left\\{ \\begin{array}{c} x + 2 ^ {w}, x < 0 \\\\ x, x > 0 \\end{array} \\right., \\text {即} T 2 U _ {w} (x) = (x + 2 ^ {w}) \\mod 2 ^ {w}", "math_type": "latex", "image_source": { "path": "images/f1c88bc310fe1e2e37ba83dd693b9624742a44553e7d967fccfd571cc58dd954.jpg" } }, "bbox": [ 300, 852, 600, 878 ] }, { "type": "equation_interline", "content": { "math_content": "B 2 T _ {k} \\left(\\left[ x _ {k - 1}, \\dots , x _ {0} \\right]\\right) = U 2 T _ {w} \\left(\\left[ B 2 T _ {w} \\left(\\left[ x _ {w - 1}, x _ {w - 2}, \\dots , x _ {0} \\right]\\right) + 2 ^ {w} \\right] \\mod 2 ^ {w} \\right] \\mod 2 ^ {k}) = U 2 T _ {w} \\left(B 2 U _ {w} \\left(\\left[ x _ {w - 1}, x _ {w - 2}, \\dots , x _ {0} \\right]\\right) \\mod 2 ^ {k}\\right)", "math_type": "latex", "image_source": { "path": "images/01e4f66698ba2ac43fa05920478fcf77b0dd4053bffd5595316e9249499cf7e7.jpg" } }, "bbox": [ 147, 882, 754, 896 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "4、其他运算符" } ], "level": 1 }, "bbox": [ 144, 170, 329, 195 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "布尔运算（位级运算）" } ], "level": 1 }, "bbox": [ 144, 243, 406, 267 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "~/NOT 非（取反）（单目运算）" } ] }, "bbox": [ 144, 309, 396, 325 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "&/AND 与" } ] }, "bbox": [ 147, 328, 230, 343 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "|/OR 或" } ] }, "bbox": [ 149, 346, 216, 362 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "^/EXCLUSIVE-OR 异或 " } ] }, "bbox": [ 147, 365, 327, 380 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "取反的时候：" } ] }, "bbox": [ 147, 382, 247, 399 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此，可以总结出~按位取反的计算结论是： " }, { "type": "equation_inline", "content": "\\sim \\mathsf { n } = - ( \\mathsf { n } + 1 )" } ] }, "bbox": [ 147, 401, 584, 418 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "例如本例中， " }, { "type": "equation_inline", "content": "\\mathord { \\sim } 5 = - ( 5 + 1 )" }, { "type": "text", "content": "，即 " }, { "type": "equation_inline", "content": "\\mathord { \\sim } 5 = - 6" } ] }, "bbox": [ 147, 420, 435, 436 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "逻辑运算 " } ], "level": 1 }, "bbox": [ 146, 461, 258, 483 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "所有非零参数为 true，参数 0 表示 false" } ] }, "bbox": [ 144, 527, 462, 543 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "结果只有两个，true与false 惰性运算" } ] }, "bbox": [ 147, 545, 448, 561 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "|| OR " } ] }, "bbox": [ 149, 564, 201, 579 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "&& AND " } ] }, "bbox": [ 147, 583, 218, 596 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "! NOT " } ] }, "bbox": [ 147, 602, 196, 615 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "移位运算 " } ], "level": 1 }, "bbox": [ 144, 640, 258, 664 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "逻辑移位。将移位的数据视为无符号数据，移位的结果知识数据位置发生了变化。移出的数据位一般置入CF标志位(进位、借位 标志)" } ] }, "bbox": [ 144, 707, 850, 741 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "算术移位，对象是有符号数，算术移位的结果在绝对值上进行放大缩小；符号位保持不变。" } ] }, "bbox": [ 144, 744, 847, 760 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算机的算术移位指令大多采用补码的 移位规则" } ] }, "bbox": [ 147, 763, 527, 778 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "循环移位，左移时最高位移入最低位，右移时最高位移入最低为，若和CF标志位一起循环，则称为大循环，否则小循环" } ] }, "bbox": [ 144, 781, 855, 816 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "左移相当于乘以2，右移相当于除以二" } ] }, "bbox": [ 144, 819, 448, 834 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "CF: 进位/借位标志。最近的操作使最高位产生了进位。可用来检査无符号操作的溢出。 加法运算时 " }, { "type": "equation_inline", "content": "\\textstyle \\sum = 1" }, { "type": "text", "content": ",则CF置1表示进位；减法运算时 " }, { "type": "equation_inline", "content": "C { = } 0 , C F" }, { "type": "text", "content": "置1表示借位。CF只对无符号数运算有意义。" } ] }, "bbox": [ 144, 837, 850, 891 ] } ], [ { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "ZF: 零标志。最近的操作得出的结果为 0（全零）则 ZF 置 1。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "SF: 符号标志。记录运算结果的符号，它与运算结果的最高位相同。采用补码表示，运算结果为正数置0，负数置1。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "OF: 溢出标志。最近的操作导致一个补码溢出——正溢出或负溢出。有符号数有意义。有溢出则置1，否则置0；" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "PF:奇偶标志位。操作数 1 的个数为偶数置 1，否则置 0；" } ] } ] }, "bbox": [ 144, 93, 845, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "左移" } ], "level": 1 }, "bbox": [ 146, 229, 200, 247 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "逻辑左移，高位移出，低位补0 " } ] }, "bbox": [ 144, 288, 394, 303 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "左移不区分算术还是逻辑" } ] }, "bbox": [ 144, 307, 344, 321 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于原码的算术左移，符号位不变，高位移出，低位补 0；" } ] }, "bbox": [ 144, 326, 594, 340 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当左移移出的数据位为“1”时，发生溢出。" } ] }, "bbox": [ 146, 344, 473, 357 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于补码的算数左移。符号位不变，高位移出，低位补0；当左移移出的数据位正数为1负数为0时（符号位与被移出的位不同），发生溢出。一次。为保证不发生溢出，移位数据的最高有效位必须与符号位相同" } ] }, "bbox": [ 144, 363, 835, 414 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因此，在硬件补码实现算数左移时，直接将数据的最高有效位移入符号位，当不发生溢出时，不会改变机器数的符号" } ] }, "bbox": [ 144, 418, 836, 451 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "右移" } ], "level": 1 }, "bbox": [ 146, 498, 200, 517 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "逻辑右移，低位移出，高位补0 " } ] }, "bbox": [ 144, 558, 394, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于原码的算数右移，符号位不变，低位移出，高位补0" } ] }, "bbox": [ 144, 577, 586, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "算数右移 " } ] }, "bbox": [ 146, 595, 221, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于补码，符号位不变，低位移出，高位补符号位" } ] }, "bbox": [ 144, 614, 537, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于原码，符号位不变，低位移出，高位补0" } ] }, "bbox": [ 144, 633, 499, 646 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "C语言中一个特性就是支持按位进行布尔运算（位运算）" } ] }, "bbox": [ 144, 688, 576, 702 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "位运算的用法就是实现掩码运算" } ] }, "bbox": [ 146, 707, 396, 721 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "特定的位序列 " } ] }, "bbox": [ 146, 725, 255, 739 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "比如说和0xFF，进行与运算，就是其本身" } ] }, "bbox": [ 146, 744, 472, 758 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5、机器数编码与真值 " } ], "level": 1 }, "bbox": [ 144, 803, 408, 824 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "为什么原码、补码、反码、移码 作为机器数的时候，符号位算作 " }, { "type": "equation_inline", "content": "2 \\land \\mathsf { n ? }" } ] }, "bbox": [ 144, 869, 692, 883 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "关于移码" } ] }, "bbox": [ 146, 887, 221, 902 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "以 8 位数为例，总共能表示 256 个数，把这些数加上一个固定的值，从而形成新的码值（还是 8 位），这就是移码的定义。例如，可以把有符号数映射到 0 - 255 的段上。" } ] }, "bbox": [ 144, 93, 847, 127 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "移码的优势：原来的大小关系不变，且一一对应（没有两种表示的 0），很容易判断大小（如判断是最小值或者最大值）" } ] }, "bbox": [ 144, 130, 847, 165 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "64 为机器上，不同的数据类型，所占的字节数不同，数值范围取值不同" } ] }, "bbox": [ 144, 167, 702, 183 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Long 在 64 位机器上占 8 个字节 32 位机器上，long 占 4 个字节" } ] }, "bbox": [ 144, 186, 643, 202 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Unsigned关键字 数字只能为非负数，无符号数" } ] }, "bbox": [ 144, 204, 522, 219 ] }, { "type": "table", "content": { "image_source": { "path": "images/d4b7fb7f45843a9c1fd0ec69762e3eee3776f4c1067cb3e72e15e7fcb0f4a688.jpg" }, "table_caption": [], "table_footnote": [], "html": "
C data typeMinHexadecimalMaxHexadecimalBytes
[signed] char-2^70x802^7-10x7F1
Unsigned char02^8-10xFF1
Short-2^150x80002^15-10x7FFF2
Unsigned short02^16-10xFFFFFF2
Int-2^310x8(7个0)2^31-10x7(7个F)4
Unsigned int02^32-10x(8个F)4
Long-2^630x8(15个0)2^63-10x7(15个F)8
Unsigned long02^64-10x(16个F)8
Long long
Int32_t
Uint 32_t
Int 64_t
Uint64_t
Float
Double
Long double
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 139, 221, 1000, 583 ] }, { "type": "image", "content": { "image_source": { "path": "images/1a41381584803073f6746fb38769bedaa0499f690915a2f9652a86f1e498a1d5.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 601, 776, 827 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "假设一个整数的数据类型有w位，用向量x表示" } ] }, "bbox": [ 144, 845, 524, 860 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "For vector " }, { "type": "equation_inline", "content": "\\vec { x } = [ x _ { w - 1 } , x _ { w - 2 } , \\cdots , x _ { 0 } ]" } ] }, "bbox": [ 146, 870, 410, 890 ] } ], [ { "type": "equation_interline", "content": { "math_content": "B 2 U _ {w} (\\vec {x}) \\doteq x _ {w - 1} \\cdot 2 ^ {w - 1} + x _ {w - 2} \\cdot 2 ^ {w - 2} \\dots", "math_type": "latex", "image_source": { "path": "images/de3c46b45a0ebed828429b94f86f6fe59c6e34673c3717d5ce1a39510700d598.jpg" } }, "bbox": [ 147, 91, 448, 108 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "B2U 的意思是 binary to unsigned" } ] }, "bbox": [ 144, 130, 420, 146 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于有符号数的编码采取补码的方式" } ] }, "bbox": [ 146, 149, 433, 164 ] }, { "type": "equation_interline", "content": { "math_content": "B 2 T _ {w} (\\vec {x}) \\dot {=} x _ {w - 1} \\cdot - 2 ^ {w - 1} + x _ {w - 2} \\cdot 2 ^ {w - 2} \\dots", "math_type": "latex", "image_source": { "path": "images/48f6fa4f5c81ae014437f8ab7922d9f65073a33216929d86a90f64b6ad571642.jpg" } }, "bbox": [ 147, 166, 465, 181 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "这里最高位为1时表示负数，最高位为0时表示非负数" } ] }, "bbox": [ 144, 204, 574, 219 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "最高位也叫做符号位" } ] }, "bbox": [ 146, 223, 310, 237 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "关于符号位，需要理解负权重的概念，而不能简单的当成一个负号" } ] }, "bbox": [ 144, 241, 662, 256 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "记住对于无符号数，最高位的1表示的不是权重" } ] }, "bbox": [ 146, 260, 521, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "也就是说， " }, { "type": "equation_inline", "content": "- 2 \\land 3 = - 8" } ] }, "bbox": [ 146, 279, 305, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "-1在有符号数全为1，但在无符号数为最大值" } ] }, "bbox": [ 146, 297, 502, 312 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如果在计算的时候一个是有符号数，一个是无符号数，" } ] }, "bbox": [ 144, 316, 563, 332 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "C语言会隐式地将有符号数转换为无符号数" } ] }, "bbox": [ 144, 334, 482, 349 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "6、溢出与取模运算的数学关系" } ], "level": 1 }, "bbox": [ 144, 411, 515, 434 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "设我们有两个 w 位无符号整数 a 和 b，其和为 S，即" } ] }, "bbox": [ 144, 476, 556, 493 ] }, { "type": "equation_interline", "content": { "math_content": "S = a + b", "math_type": "latex", "image_source": { "path": "images/9f132641c8488ca004b4e1ba029ef363bff54e9952cd7d821c7e8188ecf11372.jpg" } }, "bbox": [ 146, 497, 221, 510 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于二进制表示限定为 w 位，系统只能表示 0 到 2w−1 内的数。因此，任何 S 超出这一范围的部分都会被丢弃。用数学语言描述就是：实际存储的结果 R 为" } ] }, "bbox": [ 144, 514, 847, 548 ] }, { "type": "equation_interline", "content": { "math_content": "R = S \\bmod 2 ^ {\\wedge} w", "math_type": "latex", "image_source": { "path": "images/c08c9bb2aee4c18808b8b0e6863e5d4dbb6b96d59beb5873e6fdc6f526fd4ba1.jpg" } }, "bbox": [ 146, 552, 265, 565 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "也就是说，在加法运算中，丢弃溢出位的操作等同于计算 S 对 " }, { "type": "equation_inline", "content": "_ { 2 } \\wedge _ { \\mathsf { W } }" }, { "type": "text", "content": "取模。这正是计算机硬件中处理无符号整数溢出的方式" } ] }, "bbox": [ 144, 570, 847, 602 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "很多人可能会疑惑，既然阶码字段有 8 位，那么为什么不选择 " }, { "type": "equation_inline", "content": "2 ^ { \\wedge } 7 = 1 2 8" }, { "type": "text", "content": "作为偏移量？原因在于：" } ] }, "bbox": [ 144, 607, 847, 640 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "对称分布的需求：以 127 为偏移量，可以使得存储阶码为 127 时实际指数为 0。这种设计使得正指数和负指数在规格化数中分布较为对称，便于硬件实现和算法设计。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "历史约定与标准化：IEEE754 标准规定的偏移量就是 127，它的选择经过广泛验证并且与现有的硬件架构和软件设计高度契合。采用 127 而非 128 使得编码与译码的过程更加简便，同时避免了 0 不对称的问题。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "具体来说，设若采用 128 作为偏移量，则当存储阶码 E 为 128 时，实际指数为0；但这会使得整个指数范围整体右移，从而使得可表示的最小负指数与最大正指数的数量不均衡，不符合 IEEE754 对于规格化数的要求。选择 127 则能使得 0 成为一个中心值，既便于表达接近1 的数值，又能合理覆盖较大范围的指数。" } ] } ] }, "bbox": [ 174, 644, 847, 824 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3. 关于阶码真值为 -127 和 128" } ], "level": 1 }, "bbox": [ 144, 829, 406, 844 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "一般我们正常探讨的阶码区间是 " }, { "type": "equation_inline", "content": "- 1 2 6 \\mathsf { D } \\sim 1 2 7 \\mathsf { D }" }, { "type": "text", "content": "，而真值 -127D 的阶码为 0000 0000B，真值 128D 的阶码为 1111 1111B。" } ] }, "bbox": [ 144, 848, 836, 882 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.1. 阶码真值为 -127" } ], "level": 1 }, "bbox": [ 144, 93, 329, 108 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当阶码全为 0 ，尾数不全为 0，表示 非规格化小数 ，用来表示比最小绝对值还要小的数，即" } ] }, "bbox": [ 144, 112, 836, 143 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "尾数码 隐含的最高位不是 1，而是 0；" } ] }, "bbox": [ 144, 148, 436, 164 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "阶码真值 固定为 -126，而非 -127；" } ] }, "bbox": [ 144, 168, 420, 183 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "当阶码全为 0 ，尾数全为 0，表示 真值 " }, { "type": "equation_inline", "content": "+ / - 0" } ] }, "bbox": [ 144, 186, 505, 200 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "5.3.2. 阶码真值为 128" } ], "level": 1 }, "bbox": [ 144, 204, 324, 219 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "1. 当阶码全为 1 ，尾数全为 0，表示 正负无穷大 +/- ∞" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "2. 当阶码全为 1 ，尾数不全为 0，表示非数值 NaN（Not a Number）" } ] } ] }, "bbox": [ 174, 223, 727, 256 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如 0/0， " }, { "type": "equation_inline", "content": "\\infty - \\infty" }, { "type": "text", "content": "等非法运算的结果即为 NaN" } ] }, "bbox": [ 144, 260, 478, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "汉字输入法程序：将汉字输入计算机转换成汉字内码" } ] }, "bbox": [ 144, 278, 557, 294 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "1、输入—汉字输入码，也称作外码， 由西文字符编码而成（拼音编码、字形编码等）" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "2、存储、交换和检索—内码 ，两个字节表示。汉字内码在计算机中是唯一的。（区位码、国标码、汉字内码2个字节最高位均为“1”）" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "3、交换—交换码，是指不同的具有汉字处理功能的计算机系统之间在交换汉字信息所用的代码标准。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "4、输出——字形码 点阵表示法 字模码：用于汉字的显示和打印" } ] } ] }, "bbox": [ 144, 297, 853, 405 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "汉字字库——— 所有汉字的字模点阵代码按顺序集中存放根据内码查表" } ] }, "bbox": [ 174, 407, 727, 423 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "字形检索程序：将汉字的机内代码转化为汉字的字形码" } ] }, "bbox": [ 174, 426, 603, 442 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "整数的运算 " } ], "level": 1 }, "bbox": [ 146, 463, 238, 478 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算值： " }, { "type": "equation_inline", "content": "x { + } y" } ] }, "bbox": [ 146, 482, 250, 498 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "实际值：计算机中实际上保存的结果" } ] }, "bbox": [ 144, 501, 433, 516 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "{ } _ { w y } ^ { w }" }, { "type": "equation_inline", "content": "w" } ] }, "bbox": [ 144, 538, 818, 577 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "溢出时，计算值 " }, { "type": "equation_inline", "content": "\\mathop { }" }, { "type": "text", "content": "实际值 " }, { "type": "equation_inline", "content": "\\pm 2 \\land \\_" }, { "type": "text", "content": "（这里相当于是原码！）" } ] }, "bbox": [ 144, 593, 564, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如何判定运算结果是否发生了溢出" } ] }, "bbox": [ 146, 612, 415, 627 ] }, { "type": "algorithm", "content": { "algorithm_caption": [], "algorithm_content": [ { "type": "text", "content": "Int uadd.ok(unsigned x,unsigned y) \n{ unsigned sum " }, { "type": "equation_inline", "content": "\\equiv" }, { "type": "text", "content": " x+y; If (sum " }, { "type": "equation_inline", "content": "\\rightharpoondown" }, { "type": "text", "content": " x) \nReturn 1; else return 0; \n} " } ] }, "bbox": [ 144, 633, 442, 778 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "有符号整数" } ], "level": 1 }, "bbox": [ 146, 785, 238, 800 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "为了避免数据大小的扩张，最终结果将截断为w位；即实际上数字只占w-1位两个异号的数字肯定不会溢出" } ] }, "bbox": [ 144, 804, 754, 838 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "两个同号的才有可能溢出" } ] }, "bbox": [ 146, 841, 344, 856 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/3390337029ead83d93ec3a52189a1c769867939c392af0161ac824c78e907384.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 89, 682, 312 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "正溢出/上溢" } ] }, "bbox": [ 147, 315, 248, 331 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "负溢出/下溢 " } ] }, "bbox": [ 147, 332, 247, 349 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "减法转换成加法来做 " } ] }, "bbox": [ 147, 351, 310, 368 ] }, { "type": "algorithm", "content": { "algorithm_caption": [], "algorithm_content": [ { "type": "equation_inline", "content": "x - y = x + (-y) = x + (-y + 1)" } ] }, "bbox": [ 147, 388, 334, 406 ] }, { "type": "image", "content": { "image_source": { "path": "images/28aaa54c06c24037d9b1537962c7c3302913f8ecd66b2e073b85e38b433cb4c3.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 406, 677, 602 ] }, { "type": "equation_interline", "content": { "math_content": "F o r x a n d y, x + \\mathbf {\\Lambda} _ {w} ^ {t} y = \\left\\{ \\begin{array}{l l} x + y - 2 ^ {w - 1} - 2 ^ {w - 1} = x + y - 2 ^ {w}, 2 ^ {w - 1} \\leq x + y, \\text {正 溢 出} \\\\ x + y, - 2 ^ {w - 1} \\leq x + y < 2 ^ {w - 1} \\\\ x + y + 2 ^ {w}, x + y < - 2 ^ {w - 1}, \\text {负 溢 出} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/f17b1400a5437fc9428c98f60b2bf7260bea9e983f048508ff4ff0d48ababef6.jpg" } }, "bbox": [ 147, 686, 764, 744 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于正溢出，正溢出实际上是让符号位从 1变成0，计算值 " }, { "type": "equation_inline", "content": "\\equiv" }, { "type": "text", "content": "实际值 " }, { "type": "equation_inline", "content": "+ 2 \\land _ { \\mathsf { W } , }" }, { "type": "text", "content": ",它没被截断，最高位由0变成了1，这两个数本来就没有- " }, { "type": "equation_inline", "content": "\\mathbf { \\mathcal { Z } ^ { \\wedge } }" }, { "type": "text", "content": "（w-1)的权重结果有了，那么实际值就要加上2^（w-1），又因为补码的定义，又得减去 " }, { "type": "equation_inline", "content": "2 \\setminus ( \\mathsf { w } - 1 )" } ] }, "bbox": [ 144, 760, 848, 813 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于负溢出，计算值 " }, { "type": "equation_inline", "content": "=" }, { "type": "text", "content": "实际值- " }, { "type": "equation_inline", "content": "{ \\mathcal { 2 } } ^ { \\wedge } \\mathsf { w }" }, { "type": "text", "content": "，这里因为原来两个最高位符号位从 1变成0，两个数本来有- " }, { "type": "equation_inline", "content": "{ \\mathcal { 2 } } ^ { \\wedge } \\mathsf { w }" }, { "type": "text", "content": "的权重结果没了。那么实际值就要减去个- " }, { "type": "equation_inline", "content": ". 2 \\land _ { \\mathsf { W } }" } ] }, "bbox": [ 144, 816, 848, 850 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如何判断溢出？" } ] }, "bbox": [ 146, 852, 265, 868 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "两个正数相加，得到结果为负，正溢出" } ] }, "bbox": [ 144, 871, 450, 887 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "两个负数相加，得到结果 " }, { "type": "equation_inline", "content": "\\geqslant 0" }, { "type": "text", "content": "，负溢出" } ] }, "bbox": [ 146, 890, 443, 904 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "加法逆元" } ] }, "bbox": [ 144, 93, 221, 108 ] }, { "type": "equation_interline", "content": { "math_content": "F o r x, 0 \\leq x < 2 ^ {w}, x + x ^ {\\prime} = x ^ {\\prime} + x = 0", "math_type": "latex", "image_source": { "path": "images/2029c22c026c4fa30e7e761fb82f81e61eaa09fee23c12b123a8799e763f16e8.jpg" } }, "bbox": [ 147, 110, 436, 126 ] }, { "type": "equation_interline", "content": { "math_content": "y - x - > y + x ^ {\\prime}.", "math_type": "latex", "image_source": { "path": "images/598b78d35c62d0ae336bf7c134154459802dc823a2167e0142f6b811df97facd.jpg" } }, "bbox": [ 144, 149, 226, 165 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于无符号整型来说， " }, { "type": "equation_inline", "content": "\\mathsf { x } + \\mathsf { x } ^ { \\prime } { = } 2 \\mathsf { \\Lambda } \\mathsf { w } { = } 0" } ] }, "bbox": [ 144, 167, 423, 183 ] }, { "type": "equation_interline", "content": { "math_content": "x ^ {\\prime} = \\left\\{ \\begin{array}{l} x, x = 0 \\\\ 2 ^ {w} - x, x > 0 \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/6404e5f3b47c1aa33ac05ac957a5d74af45e51c64c456cbee4abb38e221f6f20.jpg" } }, "bbox": [ 147, 186, 305, 225 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于补码表示的有符号数的逆元比较简单" } ] }, "bbox": [ 144, 241, 468, 256 ] }, { "type": "equation_interline", "content": { "math_content": "x ^ {\\prime} = \\left\\{ \\begin{array}{l} - x, x > T M i n _ {w} \\\\ T M i n _ {w}, x = T M i n _ {w} \\end{array} \\right.", "math_type": "latex", "image_source": { "path": "images/25d619fbd5211e844c40d4c1cf83b0b73bf6e4be6171c18abfbfcbbb6121a54b.jpg" } }, "bbox": [ 147, 260, 349, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "由于补码最大值比最小值的绝对值要小" } ] }, "bbox": [ 144, 315, 452, 331 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即 " }, { "type": "equation_inline", "content": "0 1 1 1 < 1 0 0 0" } ] }, "bbox": [ 146, 334, 263, 348 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "关于 最小值的逆元要通过负溢出 的方法来实现" } ] }, "bbox": [ 144, 351, 515, 368 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即 " } ] }, "bbox": [ 147, 370, 169, 385 ] }, { "type": "equation_interline", "content": { "math_content": "T \\min _ {w} + T \\min _ {w} = - 2 ^ {w} = 0", "math_type": "latex", "image_source": { "path": "images/3e4a3262de20fec5fe5b778f9103bd42c979de6c88ad08934a7fa4929cc139ab.jpg" } }, "bbox": [ 147, 388, 371, 404 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "X、y 都为 w 位，乘积结果可能为 2w 位，c 语言定义无符号数乘法所产生的结果是 w 位因此，z=xy mod " }, { "type": "equation_inline", "content": "_ { 2 } \\wedge _ { \\mathsf { W } }" } ] }, "bbox": [ 144, 426, 823, 461 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于有符号数，我们有 z=U2T(xymod2^w)" } ] }, "bbox": [ 146, 464, 484, 480 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "X’y’ mod 2^w=(xy) mod " }, { "type": "equation_inline", "content": "{ 2 } ^ { \\wedge } \\mathsf { w } ." }, { "type": "text", "content": ".即有符号整数和无符号整数高位截断后结果的位级表示相同整数的乘法往往利用移位运算（拆解成与多个 " }, { "type": "equation_inline", "content": "2 ^ { \\land } \\mathsf { k }" }, { "type": "text", "content": "相乘再相加减）" } ] }, "bbox": [ 144, 482, 842, 516 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "整数的除法遇到除不尽的情况，总是向0舍入" } ] }, "bbox": [ 146, 519, 505, 535 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "但是，移位的时候会导致向下舍入。为此，我们要在移位之前加入偏置。" } ] }, "bbox": [ 146, 538, 705, 552 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即如果要右移 k 位，在低 k 位加上 " }, { "type": "equation_inline", "content": "2 \\land k - 1" }, { "type": "text", "content": ", 向 " }, { "type": "equation_inline", "content": "k { + 1 }" }, { "type": "text", "content": "产生一个进位，实现了向 0 取整。" } ] }, "bbox": [ 144, 556, 779, 571 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "即用整除实现了向上取整 " } ] }, "bbox": [ 146, 575, 346, 590 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "举个例子：" } ] }, "bbox": [ 146, 593, 228, 608 ] }, { "type": "equation_interline", "content": { "math_content": "K = 0", "math_type": "latex", "image_source": { "path": "images/a29db2a794701a100c0b44039aed46a131e1a905155a3365fed051fc4e450f80.jpg" } }, "bbox": [ 147, 612, 183, 626 ] }, { "type": "equation_interline", "content": { "math_content": "1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 0 - 1 2 3 4 0", "math_type": "latex", "image_source": { "path": "images/ae2b079a304117ba6b2332725c7f682d79ef921b7df0b701b11aa92650856927.jpg" } }, "bbox": [ 147, 630, 374, 644 ] }, { "type": "equation_interline", "content": { "math_content": "K = 4 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 - 7 7 2 \\text {而} 1 2 3 4 0 / 2 ^ {\\wedge} k = - 7 7 1. 2 5;", "math_type": "latex", "image_source": { "path": "images/91c104b874db5721d63a7201467a2bcf161076331b50f00aebbf8e09535e4b20.jpg" } }, "bbox": [ 146, 649, 576, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "加上偏置，1100111111011011，右移四位有 1111110011111101,即-771。" } ] }, "bbox": [ 144, 667, 737, 683 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "division by a power of 2 " } ] }, "bbox": [ 184, 684, 381, 700 ] }, { "type": "equation_interline", "content": { "math_content": "(x < 0? x + (1 < < k) - 1: x) > > K", "math_type": "latex", "image_source": { "path": "images/ec86c5a0179520e72388c0b2997a88bf941d1d67f2002173a8e06d37b94f8f02.jpg" } }, "bbox": [ 147, 705, 416, 722 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "但是，除以2不能推广到除以任意整数浮点数" } ] }, "bbox": [ 144, 741, 450, 776 ] } ], [ { "type": "table", "content": { "image_source": { "path": "images/daa5fa94b901bcc90c2370bb8aa5a419ea390821b2b4d9cefc8f3c2732dcf0f0.jpg" }, "table_caption": [], "table_footnote": [], "html": "
类型总 位 数尾数阶码真值计算
短 实 数32248\\( N = {\\left( -1\\right) }^{M} \\times \\left( {{1.5}{M}_{1}M{2\\ldots }{M}_{n}}\\right) \\times {2}^{E - {127}} \\)
长 实 数645311\\( N = {\\left( -1\\right) }^{M} \\times \\left( {{1.5}{M}_{1}M{2\\ldots }{M}_{n}}\\right) \\times {2}^{E - {1023}} \\)
临 时 实数806515
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 137, 154, 858, 305 ] }, { "type": "image", "content": { "image_source": { "path": "images/3caf6d389607f56b60584b40fcc4fecb687f2270ed1d9b484c2da9a93b7e2872.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 149, 367, 890, 633 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "符号位 阶码 尾数" } ] }, "bbox": [ 147, 649, 284, 665 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "7、IEEE754 " } ], "level": 1 }, "bbox": [ 147, 728, 295, 747 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Float32bit " } ] }, "bbox": [ 147, 791, 236, 807 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Double64bit " } ] }, "bbox": [ 147, 810, 253, 826 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 410, 174, 492, 192 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "我梦里的财富：+302657264526￥" } ] }, "bbox": [ 176, 210, 557, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 719, 174, 821, 192 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "阶码" } ] }, "bbox": [ 877, 181, 921, 199 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "y " } ] }, "bbox": [ 589, 237, 640, 259 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "符号" } ] }, "bbox": [ 662, 237, 704, 255 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "尾数" } ] }, "bbox": [ 759, 237, 803, 254 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 850, 237, 895, 254 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "符号：决定数值的正负性" } ] }, "bbox": [ 168, 294, 391, 311 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "尾数：影响数值的精度。尾数的位数越多，精度越高" } ] }, "bbox": [ 169, 312, 626, 328 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "阶码：反映小数点的实际位置" } ] }, "bbox": [ 169, 329, 428, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "基数：K进制通常默认基数为K" } ] }, "bbox": [ 169, 346, 433, 362 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "规格化：确保尾数的最高位非0数位刚好在小数点之前" } ] }, "bbox": [ 169, 363, 636, 379 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "阶码确定范围 " } ] }, "bbox": [ 147, 388, 257, 405 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "尾数影响精度" } ] }, "bbox": [ 147, 407, 257, 423 ] }, { "type": "image", "content": { "image_source": { "path": "images/79f181b47490d71aa33641df7b5c4485f39a62657deff27543ee4ad27dd2b53e.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 439, 704, 543 ] }, { "type": "equation_interline", "content": { "math_content": "V = (- 1) ^ {S} \\times M \\times 2 ^ {E}", "math_type": "latex", "image_source": { "path": "images/5d840cc43b3cb748816ef8c0a608d069f3cfb943c1eba1242c7308d12b963ba2.jpg" } }, "bbox": [ 147, 609, 307, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "尾数用原码定点小数表示" } ] }, "bbox": [ 144, 649, 347, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "阶码用移码，规定偏置值为127 " } ] }, "bbox": [ 144, 667, 396, 683 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\mathsf { S } { = } 0 _ { - }" }, { "type": "text", "content": ",正数； " }, { "type": "equation_inline", "content": "s { = } 1" }, { "type": "text", "content": "负数；M 为小数部分，尾数部分，frac。E 为指数部分，阶码，exp。" } ] }, "bbox": [ 144, 686, 794, 702 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Emin=1.emax " }, { "type": "equation_inline", "content": "= 2 5 4" }, { "type": "text", "content": "，对半分，所以要减去 " }, { "type": "equation_inline", "content": "2 5 4 / 2 { = } 1 2 7" }, { "type": "text", "content": "；" } ] }, "bbox": [ 146, 705, 579, 720 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "第一位总是1，因此没必要显示出来，所以尾数部分要加1" } ] }, "bbox": [ 147, 722, 600, 738 ] }, { "type": "image", "content": { "image_source": { "path": "images/e50dc1d79cdffb23905de3ed3b7c0658a8254a152177a16d4d0e0392f652513b.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 149, 740, 489, 857 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/bd23b66fa0ae5d5aa0cccc7dda113d13d37aca406c550155e1c82dd36201b0e2.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 143, 680, 332 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "如何将十进制真值转换为偏置值为M的移码？" } ] }, "bbox": [ 184, 338, 571, 356 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "②按“无符号整数”规则转换为指定位数" } ] } ] }, "bbox": [ 186, 372, 546, 405 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "小数点前的 1 省略了（隐藏了 1）" } ], "level": 1 }, "bbox": [ 146, 445, 400, 460 ] }, { "type": "image", "content": { "image_source": { "path": "images/d598243f6708b4a1a023068860747b875a9ef2e74e3149a5cb201ad75a8e9e35.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 461, 858, 705 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "1、当阶码字段二进制位不全为 0，也不全为 1 时，（0 和 255），规格化" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "2、当阶码字段全为 0，非规格化.1、尾数部分也全为 0，则当 " }, { "type": "equation_inline", "content": "\\mathsf { S } { = } 0" }, { "type": "text", "content": "，表示正 0， " }, { "type": "equation_inline", "content": "s { = } 1" }, { "type": "text", "content": "，负0；" } ] } ] }, "bbox": [ 144, 741, 845, 794 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "表示非常接近 0 的数；如果尾数不为 0，则 " }, { "type": "equation_inline", "content": "\\mathsf { E } { = } \\mathsf { 1 }" }, { "type": "text", "content": "-bias， " }, { "type": "equation_inline", "content": "\\mathsf { M } = \\mathsf { f }" }, { "type": "text", "content": "不加 1" } ] }, "bbox": [ 146, 797, 655, 813 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "单精度记得减去 127e-bias" } ] }, "bbox": [ 147, 93, 359, 108 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/650d5305479445f75aede3bdffbb25a65884bf601e7cfc1f140fcc4261742161.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 89, 781, 305 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "3、当阶码字段全为 1，特殊值；当尾数为 0 时， " }, { "type": "equation_inline", "content": "\\mathsf { S } { = } 0" }, { "type": "text", "content": "，正无穷； " }, { "type": "equation_inline", "content": "s { = } 1" }, { "type": "text", "content": "，负无穷；当尾数不为0时，表示NaN表示不是一个数" } ] }, "bbox": [ 146, 315, 848, 348 ] }, { "type": "image", "content": { "image_source": { "path": "images/c7544ae05a23243dd7e02061d4e04648f67639f1185b2a400c3cdfb32356d57d.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 349, 776, 549 ] }, { "type": "table", "content": { "image_source": { "path": "images/14c33bdab70e592aaf0ccde21c71194aa3140aa51f19aa17576c208ea689fdc7.jpg" }, "table_caption": [ { "type": "text", "content": "单精度为例" } ], "table_footnote": [ { "type": "text", "content": "NaN:非数 (not a number)" } ], "html": "
MsEM意义
0/100±0
0/10非0非规格化数
0/11~254任意规格化数
0/12550±无穷大
0/1255非0NaN
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0J J J J J JJ J………J JH9N
0J J J J J J……H9N
0J J J J J J00………0JH9N
0J J J J J J00………00+∞
0J J J J J JJ J………J J粤羊锁怀正貌
0…………锁怀正貌
00000000J00………00粤羊锁怀正貌
000000000J J………J J粤羊非锁怀正貌
0…………非锁怀正貌
00000000000………0J粤羊非锁怀正貌
00000000000………00+0
W2EW解锁
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JJ J J J J J JJ J………J JU9U
JJ J J J J J J……U9U
JJ J J J J J J00………0JU9U
JJ J J J J J J00………00--∞
JJ J J J J J JJ J………J J晋1非顿报快改新
J………………顿报快改新
J0000000J00………00晋1非顿报快改新
J00000000J J………J J晋1非顿报快改新
J………………非顿报快改新
J0000000000………0J晋1非顿报快改新
J0000000000………00-0
W2EW熟瑶
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 431, 741, 678, 892 ] } ], [ { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·阶码：是2n-1的移码， " }, { "type": "equation_inline", "content": "\\mathtt { E } =" }, { "type": "text", "content": "[E']移=2n-1+E' " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·隐藏位：单、双精度浮点数，尾数规格化的“1”，放置于整数位，并将其隐藏；临时实数无隐藏位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■符号位：尾数的符号位Ms在最高位" } ] } ] }, "bbox": [ 149, 91, 655, 155 ] }, { "type": "table", "content": { "image_source": { "path": "images/a8de82aae0dab15c57a2a4b539d59e65ebc8094ec0467130528b436ab139841c.jpg" }, "table_caption": [], "table_footnote": [], "html": "
类型真值计算
单精度规格化: \\( \\mathrm{N} = \\left( {-1}\\right) {}^{\\mathrm{{Ms}} \\times }\\left( {{1.}{\\mathrm{M}}_{1}{\\mathrm{M}}_{2}\\cdots {\\mathrm{M}}_{\\mathrm{n}}}\\right) \\times {2}^{\\mathrm{E} - {127}} \\) 非规格化: \\( \\mathrm{N} = \\left( {-1}\\right) {}^{\\mathrm{{Ms}} \\times }\\left( {{0.}{\\mathrm{M}}_{1}{\\mathrm{M}}_{2}\\cdots {\\mathrm{M}}_{\\mathrm{n}}}\\right) \\times {2}^{-{126}} \\)
双精度规格化: \\( \\mathrm{N} = \\left( {-1}\\right) {}^{\\mathrm{{Ms}} \\times }\\left( {{1.}{\\mathrm{M}}_{1}{\\mathrm{M}}_{2}\\cdots {\\mathrm{M}}_{\\mathrm{n}}}\\right) \\times {2}^{\\mathrm{E} - {1023}} \\) 非规格化: \\( \\mathrm{N} = \\left( {-1}\\right) {}^{\\mathrm{{Ms}} \\times }\\left( {{0.}{\\mathrm{M}}_{1}{\\mathrm{M}}_{2}\\cdots {\\mathrm{M}}_{\\mathrm{n}}}\\right) \\times {2}^{-{1022}} \\)
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 203, 164, 611, 254 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "例3.10若X和Y均是IEEE754标准的单精度浮点数。" } ] }, "bbox": [ 164, 280, 549, 294 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "（1）若× " }, { "type": "equation_inline", "content": "=" }, { "type": "text", "content": "-135.625，求X的规格化浮点数表示。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "(2）若浮点数Y的存储形式为41360000H，求Y的真值。" } ] } ] }, "bbox": [ 184, 298, 603, 330 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（1）真值→IEEE754规格化单精度浮点数" } ] }, "bbox": [ 157, 335, 472, 350 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "写出数据的二进制真值 " }, { "type": "equation_inline", "content": "\\mathbf { x } =" }, { "type": "text", "content": "(-10000111.101） 2" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "转换为M " }, { "type": "equation_inline", "content": "\\times 2 ^ { E }" }, { "type": "text", "content": "的形式：M' " }, { "type": "equation_inline", "content": "{ \\bf \\delta } = \\pm 1 . \\times { \\bf x } \\ldots \\times" }, { "type": "text", "content": "，E'为整数" } ] } ] }, "bbox": [ 157, 351, 571, 384 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "X=-1.0000111101×27 M'=-1.0000111101 E'=+7 " } ] }, "bbox": [ 176, 386, 673, 400 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\textcircled{3}" }, { "type": "text", "content": "写出尾数M'的原码，尾数数值隐藏（舍去）整数位的“1\"" } ] }, "bbox": [ 157, 403, 589, 419 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\boldsymbol { \\ M } _ { S } = 1" }, { "type": "text", "content": "尾数数值M " }, { "type": "equation_inline", "content": "\\mathtt { \\mathtt { = 0 0 0 } }" }, { "type": "text", "content": "0111 1010 0000 0000 0000" } ] }, "bbox": [ 176, 420, 633, 435 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\textcircled{4}" }, { "type": "text", "content": "写出阶码E'的移码E： " }, { "type": "equation_inline", "content": "\\mathsf { E } = [ \\mathsf { E } ^ { \\prime } ] _ { \\scriptscriptstyle 1 \\mathscr { B } } = 1 2 7 + \\mathsf { E } ^ { \\prime } = 1 2 7 + 7 = 1 3 4 = 1 0 0 (" }, { "type": "text", "content": "00110B" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\textcircled{5}" }, { "type": "text", "content": "按照格式排列E和M：即MsEM" } ] } ] }, "bbox": [ 157, 437, 668, 469 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "[ \\mathsf { X } ] _ { \\overrightarrow { \\jmath } } = \\uparrow" }, { "type": "text", "content": "10000110 000 0111 1010 0000 0000 0000B=C307A000H " } ] }, "bbox": [ 176, 470, 623, 483 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "例3.10（2）浮点数 真值" } ], "level": 1 }, "bbox": [ 168, 530, 405, 546 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "阶码8位，2n-1的移码" } ], "level": 1 }, "bbox": [ 431, 519, 527, 546 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "浮点数的二进制编码：按照格式分，写出M、阶码E和尾数数值M " } ] }, "bbox": [ 189, 550, 687, 565 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\cdot [ \\mathrm {Y} ] _ {\\text {浮}} = 4 1 3 6 0 0 0 0 \\mathrm {H} = 0 1 0 0 / 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B \\\\ \\cdot \\mathrm {M} _ {\\mathrm {s}} = 0 \\quad \\mathrm {E} = 1 0 0 0 0 0 1 0 \\mathrm {B} = 1 3 0 \\quad \\mathrm {M} = 0 1 1 0 1 1 0 0 0 0 0 0 0 \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/4bb7b4d97d7247a48a292b6172ecc23648d7a5b7d4857813729d6eba329859dc.jpg" } }, "bbox": [ 213, 569, 729, 602 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "添加隐 ·阶码编码E不是全0也不是全1，可判定Y是规格化浮点数" } ] }, "bbox": [ 147, 606, 608, 620 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "藏位1写出尾数的真值M’： " }, { "type": "equation_inline", "content": "\\pmb { M } _ { S } \\mathbf { = } 0" }, { "type": "text", "content": "，表明是正数添加隐藏位1 " } ] }, "bbox": [ 147, 623, 626, 639 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数符+ " } ] }, "bbox": [ 157, 651, 544, 664 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "求出阶码的真值E'：Y是单精度浮点数，故E'=E-127=130-127=3 " } ] }, "bbox": [ 210, 661, 675, 675 ] }, { "type": "paragraph", "content": { "paragraph_content": [] }, "bbox": [ 189, 678, 445, 694 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} Y = M ^ {\\prime} \\times 2 ^ {E ^ {\\prime}} = + 1. 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \\times 2 ^ {3} \\\\ = + 1 0 1 1. 0 1 1 B = + 1 1. 3 7 5 \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/5c9230f5b3b2ef382259d17038e03be5ba5cfeecb2986baeab96434081a7c95d.jpg" } }, "bbox": [ 189, 678, 596, 728 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "但是，浮点数只能近似表示实数运算" } ] }, "bbox": [ 146, 741, 433, 757 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "1、向上舍入" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "2、向下舍入" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "3、向零舍入" } ] } ] }, "bbox": [ 147, 760, 252, 812 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "4、四舍六入五取偶 " } ] }, "bbox": [ 146, 816, 305, 831 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Int ->float 同样 32 位不会发生溢出但是会失去精度" } ] }, "bbox": [ 146, 834, 544, 848 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Int/float->double 不发生溢出，可以保留精度" } ] }, "bbox": [ 146, 853, 509, 868 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Double->float float 数值更小会溢出；float 精度更小会舍入" } ] }, "bbox": [ 146, 872, 615, 887 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Float 、double->int 1、向 0 舍入 2、发生溢出" } ] }, "bbox": [ 146, 890, 514, 904 ] }, { "type": "image", "content": { "image_source": { "path": "images/bfa78e06b0d3913c28c391fec9983e9560c657cdba21f6186a404da3be4f5b19.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 709, 719, 741, 740 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/c18707fdc6690cbbeb12755c25664031e3864720f676bdd884409cae44b965d7.jpg" }, "image_caption": [ { "type": "text", "content": "大端法与小端法" } ], "image_footnote": [] }, "bbox": [ 147, 108, 932, 394 ] }, { "type": "image", "content": { "image_source": { "path": "images/fc5f82e5e9be2a5c3b02fbe81cb70ad9ef2ad7623210873829e13a3b41d79e02.jpg" }, "image_caption": [ { "type": "text", "content": "全加器" } ], "image_footnote": [] }, "bbox": [ 147, 423, 934, 740 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "串行进位加法器（行波进位加法器）" } ] }, "bbox": [ 147, 778, 423, 795 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/97427437348e9fdabf7e1f1dd98d4ee87c5cacd20c88a229ec6fe0060a545d10.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 166, 112, 710, 225 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "先行进位加法器" } ] }, "bbox": [ 146, 259, 275, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "乘法运算 " } ] }, "bbox": [ 147, 278, 221, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "除法运算" } ] }, "bbox": [ 147, 296, 221, 313 ] }, { "type": "image", "content": { "image_source": { "path": "images/20dbaf06f5585bbdff9ba7e4458ceb025cb910d7536b1db144cebd7cca1b0b28.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 156, 331, 865, 554 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字输入：在西文标准键盘上，通过不同的字符组合，将汉字输入计算机，转换为汉字内码的过程" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字输入码：也称外码，由不同的西文字符组合成的编码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字输入码种类：" } ] } ] }, "bbox": [ 146, 593, 668, 665 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "数字编码：区位码、国标码、电报码等 " } ] }, "bbox": [ 146, 684, 453, 702 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "拼音编码：搜狗拼音、微软拼音等 " } ] }, "bbox": [ 146, 722, 416, 739 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "字形编码：五笔字型输入法、郑码" } ] }, "bbox": [ 146, 760, 416, 776 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "音形编码：自然码、钱码" } ] }, "bbox": [ 146, 796, 347, 813 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "汉字内码：用于汉字信息的存储、交换、检索等操作的机内代码" } ] }, "bbox": [ 144, 815, 645, 832 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 一般采用两个字节表示，汉字内码在计算机中是唯一的" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字内码涉及以下几种编码：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " （1）区位码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字编码的国家标准：GB2312-1980《信息交换用汉字编码字符集·基本集》" } ] } ] }, "bbox": [ 146, 834, 747, 906 ] } ], [ { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": " 共收录了 6763 个汉字、682 个图形符号" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 一级常用汉字：3755 个，按拼音字母顺序排列" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 二级次常用汉字：3008 个，按偏旁部首排列" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 两个字节编码汉字，每个字节各取 7 位，可编码 " }, { "type": "equation_inline", "content": "1 2 8 \\times 1 2 8 = 1 6 3 8 4" }, { "type": "text", "content": "个字符" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 将所有的汉字及符号组成一个 " }, { "type": "equation_inline", "content": "9 4 \\times 9 4" }, { "type": "text", "content": "的方阵，行 “区”，列 “位”" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 每个汉字与符号都位于某个区的某个位上，区号和位号 “区位码”" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 例如：“中”字位于 54 区 48 位 “中”字的区位码即为“5448  （2）国标码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 国标码 " }, { "type": "equation_inline", "content": "\\mid =" }, { "type": "text", "content": "区位码 " }, { "type": "equation_inline", "content": "+ 2 0 2 0 \\mathsf { H }" }, { "type": "text", "content": "，占用两个字节" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 例如“中”字：区位码 " }, { "type": "equation_inline", "content": "\\mathbf { = } 5 4 4 8" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 区码和位码转化为 16 进制， " }, { "type": "equation_inline", "content": "{ \\bf = 3 6 3 0 H }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 国标码 " }, { "type": "equation_inline", "content": "\\lvert = 3 6 3 0 \\rvert + 2 0 2 0 \\ H = 5 6 5 0 \\ H" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " （3）汉字内码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 西文字符：七位的 ASCII 码，当用一个字节表示时，最高位为“0”" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字内码：汉字内码 " }, { "type": "equation_inline", "content": "=" }, { "type": "text", "content": "汉字国标码 " }, { "type": "equation_inline", "content": "\\pm 8 0 8 0 \\mathsf { H }" }, { "type": "text", "content": "，2 个字节的最高位均为“1”" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 例如“中”字的机内码 " }, { "type": "equation_inline", "content": "\\lvert = 5 6 5 0 \\rvert + 8 0 8 0 \\ H = \\mathrm { D } 6 \\mathrm { D } 0 \\mathsf { H }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 文本文件中储存的是汉字内码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "汉字处理系统中字形信息： " }, { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "用活字或文字版的母体字形形式； " }, { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "点阵表示法、" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "矢量表示法等形式，最基本的，应用也最广泛的是字模码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 字模码：用点阵形式表示与存储汉字字形代码，它是汉字的输出形式，用于汉字的显示和打印" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 根据汉字输出的要求不同，点阵的规模也不同：" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 简易型汉字： " }, { "type": "equation_inline", "content": "1 6 \\times 1 6" }, { "type": "text", "content": "， 32 字节/汉字" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 普及型汉字： " }, { "type": "equation_inline", "content": "2 4 \\times 2 4" }, { "type": "text", "content": "， 72 字节/汉字" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 提高型汉字： " }, { "type": "equation_inline", "content": "3 2 \\times 3 2" }, { "type": "text", "content": "，128 字节/汉字" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "• 精密型汉字： " }, { "type": "equation_inline", "content": "4 8 \\times 4 8" }, { "type": "text", "content": "，288 字节/汉字" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 汉字字库：将所有汉字的字模点阵代码按顺序集中存放，构成了汉字库" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": " 字形检索程序：将汉字的机内代码转化为汉字" } ] } ] }, "bbox": [ 146, 93, 794, 590 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "补码加减 " } ], "level": 1 }, "bbox": [ 146, 612, 221, 627 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "补码的加法运算公式：" } ] }, "bbox": [ 176, 634, 359, 650 ] }, { "type": "equation_interline", "content": { "math_content": "= [ X + Y ] _ {\\text {补}} = [ X ] _ {\\text {补}} + [ Y ] _ {\\text {补}}", "math_type": "latex", "image_source": { "path": "images/16152e56ad039c3a60e7e161d0186a30d2a31b140d8d3a52512fcda2f04b72f0.jpg" } }, "bbox": [ 196, 653, 359, 670 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "补码的减法运算公式：" } ] }, "bbox": [ 436, 633, 618, 648 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} - [ X - Y ] _ {\\text {补}} = [ X + (- Y) ] _ {\\text {补}} \\\\ = [ \\mathrm {X} ] _ {\\text {补}} + [ - \\mathrm {Y} ] _ {\\text {补}} \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/d601aad040f3da0f676d517a595008665e004685db26eaaf694f8007a33844ca.jpg" } }, "bbox": [ 457, 652, 635, 684 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "证明：" } ] }, "bbox": [ 188, 676, 228, 689 ] }, { "type": "equation_interline", "content": { "math_content": "[ X ] _ {\\text {补}} = 2 ^ {n + 1} + X \\quad (\\mathrm {m o d} 2 ^ {n + 1})", "math_type": "latex", "image_source": { "path": "images/ff07a017411944f1ae1e23f721fcfef0a5d4879e94c43f42bb91ae4076b0e5de.jpg" } }, "bbox": [ 194, 690, 421, 706 ] }, { "type": "equation_interline", "content": { "math_content": "[ Y ] _ {\\text {补}} = 2 ^ {n + 1} + Y \\quad (\\mathrm {m o d} 2 ^ {n + 1})", "math_type": "latex", "image_source": { "path": "images/cbd22d9a28cb9f43bd443474b627349e6b3ccbfd0e59579155d7e55217c64159.jpg" } }, "bbox": [ 194, 711, 421, 726 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} [ X ] _ {\\text {补}} + [ Y ] _ {\\text {补}} = 2 ^ {n + 1} + X + 2 ^ {n + 1} + Y \\quad (\\mathrm {m o d} 2 ^ {n + 1}) \\\\ = 2 ^ {n + 1} + (X + Y) \\quad (\\text {m o d} 2 ^ {n + 1}) \\\\ = [ X + Y ] _ {\\text {补}} \\quad (\\mathrm {m o d} 2 ^ {n + 1}) \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/6aad23c4096e7f809a235f33ef7d754f099269ccb0889185b7dffeae6ca74b1b.jpg" } }, "bbox": [ 194, 730, 564, 787 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "任意两数之和的补码 " }, { "type": "equation_inline", "content": "=" }, { "type": "text", "content": "两数的补码之和" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "任意两数之差的补码 " }, { "type": "equation_inline", "content": "=" }, { "type": "text", "content": "被减数的补码与减数相反数的补码之和" } ] } ] }, "bbox": [ 166, 799, 608, 829 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "求补运算：[Y]补→[-Y]补 " } ], "level": 1 }, "bbox": [ 206, 114, 453, 137 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "求补规则：将 " }, { "type": "equation_inline", "content": "[ Y ] _ { \\ast \\mathrm { h } }" }, { "type": "text", "content": "包括符号位在内每一位取反，末位加1。" } ] }, "bbox": [ 157, 148, 665, 167 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若 " }, { "type": "equation_inline", "content": "[ \\boldsymbol { \\Upsilon } _ { \\sharp \\flat } = \\boldsymbol { \\Upsilon } _ { \\mathsf { s } }" }, { "type": "equation_inline", "content": "\\mathsf { Y } _ { 1 } . . . . . \\mathsf { Y } _ { \\mathsf { n } }" }, { "type": "text", "content": "，则：" } ] }, "bbox": [ 157, 172, 438, 191 ] }, { "type": "equation_interline", "content": { "math_content": "- [ - Y ] _ {\\text {补}} = \\overline {{Y _ {S}}} \\overline {{Y _ {1}}} \\dots \\dots \\overline {{Y _ {n}}} + 1", "math_type": "latex", "image_source": { "path": "images/2682be6a595e12b8bb9a0fa345e503fc16b8eea8df3305199ff85fa6c2d16fd2.jpg" } }, "bbox": [ 181, 196, 428, 216 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "若[Y]补 " }, { "type": "equation_inline", "content": "\\mathbf { \\Psi } = \\mathsf { Y } _ { \\mathsf { S } } . \\mathsf { Y } _ { 1 } . . . . . \\mathsf { Y } _ { \\mathsf { n } }" }, { "type": "text", "content": "，则：" } ] }, "bbox": [ 157, 221, 430, 239 ] }, { "type": "equation_interline", "content": { "math_content": "- [ - Y ] _ {\\text {补}} = \\overline {{Y _ {S}}} \\overline {{Y _ {1}}} \\dots \\dots \\overline {{Y _ {n}}} + 0. 0 \\dots \\dots 0 1", "math_type": "latex", "image_source": { "path": "images/a00512c6c13063d6f0d24745175a8bb187e230002084848bd824d798a7574aa3.jpg" } }, "bbox": [ 181, 244, 507, 265 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "：溢出：运算结果超出机器数的表示范围" } ] }, "bbox": [ 144, 274, 433, 288 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "计算机硬件必须具备：" } ] }, "bbox": [ 146, 294, 312, 307 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·能够检测运算结果是否发生溢出；" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·能够指示运算结果是否发生溢出；" } ] } ] }, "bbox": [ 164, 312, 411, 344 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "补码加减运算判溢方法：" } ] }, "bbox": [ 519, 278, 705, 292 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·单符号位判溢 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·进位判溢 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■双符号位判溢 " } ] } ] }, "bbox": [ 544, 297, 657, 349 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "对于加减运算，可能发生溢出的情况：" } ] }, "bbox": [ 144, 350, 421, 365 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "■同号（两数）相加" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·异号（两数）相减 " } ] } ] }, "bbox": [ 164, 369, 310, 401 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "Why? " } ] }, "bbox": [ 183, 416, 231, 430 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "一定会发生溢出的情况：" } ] }, "bbox": [ 337, 382, 521, 397 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "正数相加，且运算结果符号位为1 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·负数相加，且运算结果符号位为0 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·正数－负数，且运算结果符号位为1 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■负数－正数，且运算结果符号位为0 " } ] } ] }, "bbox": [ 359, 401, 625, 472 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "18 " } ], "level": 1 }, "bbox": [ 694, 472, 715, 489 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "（1）单符号位判溢方法" } ], "level": 1 }, "bbox": [ 211, 505, 400, 521 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\overline { { A D D } } / S U B = 0" }, { "type": "text", "content": "：做加法" } ] }, "bbox": [ 223, 529, 418, 546 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\overline { { A D D } } / S U B = 1" }, { "type": "text", "content": "：做减法" } ] }, "bbox": [ 225, 552, 418, 568 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "【例4.3】的4种情况：" } ] }, "bbox": [ 490, 505, 689, 521 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} [ \\mathrm {X} ] _ {\\text {补}} = \\mathrm {X} _ {S} \\mathrm {X} _ {1} \\dots \\dots \\mathrm {X} _ {n} \\\\ [ \\mathrm {Y} ] _ {\\text {补}} = \\mathrm {Y} _ {S} \\mathrm {Y} _ {1} \\dots \\dots \\mathrm {Y} _ {n} \\\\ [ X \\pm Y ] _ {\\text {补}} = F _ {s} F _ {1} \\dots \\dots F _ {n} \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/af33e894e0bc5bc9f1c8958716fb93b6806ae55dfec4cbe0afd5ccb203a48277.jpg" } }, "bbox": [ 468, 526, 667, 575 ] }, { "type": "table", "content": { "image_source": { "path": "images/ea812e338c51b36ff98fc816caadb34d0e22ff55762a1c584f92915146ff3477.jpg" }, "table_caption": [], "table_footnote": [], "html": "
运算操作数XXS操作数YYS运算结果FFS溢出情况
X+Y正数0正数0负数1正溢出
X+Y负数1负数1正数0负溢出
X-Y正数0负数1负数1正溢出
X-Y负数1正数0正数0负溢出
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 179, 583, 872, 730 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\boldsymbol { \\mathsf { V } } = \\boldsymbol { \\mathsf { 1 } }" }, { "type": "text", "content": "：溢出" } ] }, "bbox": [ 179, 740, 277, 757 ] }, { "type": "page_number", "content": { "page_number_content": [] }, "bbox": [ 828, 741, 838, 750 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "（3）双符号位判溢方法" } ], "level": 1 }, "bbox": [ 166, 90, 352, 108 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "·X和Y采用双符号位补码参加运算，正数的双符号位为00，负数的双符号位为11；当运算结果的两位符号Fs1 " }, { "type": "equation_inline", "content": "\\bar { \\mathsf { F } } _ { 5 2 }" }, { "type": "text", "content": "不同时（01或10），发生溢出。" } ] }, "bbox": [ 181, 112, 786, 148 ] }, { "type": "equation_interline", "content": { "math_content": "\\frac {\\mathrm {X} _ {\\mathrm {S}} \\mathrm {X} _ {\\mathrm {S}} \\mathrm {X} _ {1} \\mathrm {X} _ {2} \\dots \\dots \\mathrm {X} _ {\\mathrm {n}}}{\\mathrm {Y} _ {\\mathrm {S}} \\mathrm {Y} _ {\\mathrm {S}} \\mathrm {Y} _ {1} \\mathrm {Y} _ {2} \\dots \\dots \\mathrm {Y} _ {\\mathrm {n}}}", "math_type": "latex", "image_source": { "path": "images/8e4471389a14d592cef0763a38ba5b9149e96fb3d023ce5d82fe1d5ecec37bb1.jpg" } }, "bbox": [ 231, 158, 415, 212 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{r l} & \\mathrm {X} _ {\\mathrm {S}} \\mathrm {X} _ {1} \\mathrm {X} _ {2} \\dots \\dots \\mathrm {X} _ {\\mathrm {n}} \\\\ + & \\mathrm {Y} _ {\\mathrm {S}} \\mathrm {Y} _ {1} \\mathrm {Y} _ {2} \\dots \\dots \\mathrm {Y} _ {\\mathrm {n}} \\\\ \\hline & \\mathrm {F} _ {\\mathrm {S}} \\mathrm {F} _ {1} \\mathrm {F} _ {2} \\dots \\dots \\mathrm {F} _ {\\mathrm {n}} \\end{array} \\quad \\begin{array}{r l} & \\mathrm {F} _ {\\mathrm {S} 1} = \\mathrm {X} _ {\\mathrm {S}} \\oplus \\mathrm {Y} _ {\\mathrm {S}} \\oplus \\mathrm {C} _ {\\mathrm {S}} \\\\ & \\mathrm {F} _ {\\mathrm {S} 2} = \\mathrm {F} _ {\\mathrm {S}} \\end{array}", "math_type": "latex", "image_source": { "path": "images/33ca0463674fbdfdea12da1f4fc85e2b564531bea634d6ae07c5e5b2fd83b560.jpg" } }, "bbox": [ 467, 159, 803, 212 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{l} \\mathbf {V} = \\mathbf {F} _ {\\mathbf {S} 1} \\oplus \\mathbf {F} _ {\\mathbf {S} 2} \\\\ \\cdot \\mathbf {V} = \\mathbf {F} _ {\\mathbf {S} 1} \\oplus \\mathbf {F} _ {\\mathbf {S} 2} = \\mathbf {X} _ {\\mathbf {S}} \\oplus \\mathbf {Y} _ {\\mathbf {S}} \\oplus \\mathbf {C} _ {\\mathbf {S}} \\oplus \\mathbf {F} _ {\\mathbf {S}} \\\\ \\end{array}", "math_type": "latex", "image_source": { "path": "images/24d011e1161e6bf4c821bf0139a75899371b23eae04990612853b12e8d8ef95d.jpg" } }, "bbox": [ 253, 231, 709, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "=Fs1 " }, { "type": "equation_inline", "content": "\\bar { \\mathsf { F } } _ { S 2 } = 0 1" }, { "type": "text", "content": "，则正溢出； " }, { "type": "equation_inline", "content": "\\bar { \\mathsf { F } } _ { 5 1 } \\mathsf { F } _ { 5 2 } = 1 0" }, { "type": "text", "content": "，则负溢出。 " } ] }, "bbox": [ 255, 256, 650, 275 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "核心部件：一个普通的二进制并行加法器。" } ] }, "bbox": [ 144, 329, 389, 342 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "A:累加器，存放[X]补 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "B:寄存器，存放 " }, { "type": "equation_inline", "content": "[ \\boldsymbol { \\mathsf { Y } } ] _ { \\sharp \\flat }" } ] } ] }, "bbox": [ 146, 344, 280, 373 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "取反电路：" } ] }, "bbox": [ 147, 376, 208, 388 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·控制信号P " }, { "type": "equation_inline", "content": "\\cdot = 1" }, { "type": "text", "content": "，取反电路工作；" }, { "type": "equation_inline", "content": "\\mathsf { P } \\mathsf { = } \\mathsf { 0 }" }, { "type": "text", "content": "：不工作 (不取反)" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "· P= ADD/SUB " } ] } ] }, "bbox": [ 149, 391, 352, 434 ] }, { "type": "image", "content": { "image_source": { "path": "images/ae54ce06282a8e6bd6869a6dae889b392853294a3eb61a65b62d54b612cdaac3.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 357, 349, 652, 456 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "工作原理：" } ], "level": 1 }, "bbox": [ 146, 453, 206, 464 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "- ADD/SUB " }, { "type": "equation_inline", "content": "{ \\tt = } 0" }, { "type": "text", "content": "：补码加法器，B寄存器→并行加法器 " }, { "type": "equation_inline", "content": "[ \\mathsf { P } \\mathsf { \\tau } ] = \\mathsf { 0 } ]" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·ADD/SUB=1:补码减法器，B取反→并行加法器（ " }, { "type": "equation_inline", "content": "\\scriptstyle \\mathbf { P } = 1" }, { "type": "text", "content": "），且并行加法器" } ] } ] }, "bbox": [ 149, 468, 603, 495 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "的最低位的进位-1 即R取反加1：相当干rY1..加r-V1.. →减法运算" } ] }, "bbox": [ 164, 498, 568, 506 ] }, { "type": "image", "content": { "image_source": { "path": "images/d9d5837f8365fb673cb70ee9ba3a74bd16f123df640119ed41a51adfb2ae366a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 589, 494, 640, 507 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "乘法实现 " } ], "level": 1 }, "bbox": [ 147, 519, 221, 533 ] }, { "type": "image", "content": { "image_source": { "path": "images/4e246a115c30e7de0c4f12d19cb99bf90bc9cd7e8afda36ca1d83dca0bfc5c41.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 201, 557, 643, 768 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "昇法少骤 " } ], "level": 1 }, "bbox": [ 154, 89, 228, 98 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "X [X]原=Xs X, Xx....Xn " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "[Y]原=Ys Y, Y....Yn " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\mathbf { P } { = } \\mathbf { X } { \\cdot } \\mathbf { Y }" }, { "type": "text", "content": "，Ps是积的符号" } ] } ] }, "bbox": [ 154, 104, 356, 158 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "符号位单独处理 Ps=Xs Ys" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "绝对值进行数值运算|P|=|X|*|Y| " } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{3}" }, { "type": "text", "content": "初始部分积为0，从乘数最低位开始累加、右移： " }, { "type": "equation_inline", "content": "\\yen 123" }, { "type": "text", "content": "，部分积加|X|， " }, { "type": "equation_inline", "content": "\\scriptstyle \\pmb { \\ Y } _ { \\mathbf { l } = \\mathbf { 0 } }" }, { "type": "text", "content": "部分积加0，累加结果右移一位，得新部分积。" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{4}" }, { "type": "text", "content": "累加右移n次，即 " }, { "type": "equation_inline", "content": "\\mathbf { \\beta } = \\mathbf { n }" } ] } ] }, "bbox": [ 154, 162, 739, 253 ] }, { "type": "image", "content": { "image_source": { "path": "images/cae87f09a7a831e5a8b40bbdeee629097938a3a1a74a22e443e586a8ee6afeb5.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 149, 291, 181, 303 ] }, { "type": "image", "content": { "image_source": { "path": "images/e9f01da7655ec11fc4a057bacb897b4b59ba6b38198840dcaf3782c713afc75c.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 174, 303, 534, 501 ] }, { "type": "image", "content": { "image_source": { "path": "images/c9d69a7ab9cac99299f1038b5f374410d4c9dd5a62edfb63afd19f1e01341604.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 544, 357, 779, 429 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "控制逻辑电路" } ], "level": 1 }, "bbox": [ 566, 444, 684, 458 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\scriptstyle \\mathbf { Q } _ { n } = \\mathbf { 0 }" }, { "type": "text", "content": ": +0 (0...0&B) " } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\scriptstyle \\mathbf { Q } _ { n } = 1" }, { "type": "text", "content": ": +|X| (11...1&B) " } ] } ] }, "bbox": [ 584, 464, 757, 494 ] }, { "type": "image", "content": { "image_source": { "path": "images/55167d2f731515103cb555f2a8a983360cf03536b770a98fecac758e276be506.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 749, 502, 779, 521 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "3、原码乘法的硬件实现 " } ], "level": 1 }, "bbox": [ 205, 558, 453, 579 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "A:累加寄存器" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "B:被乘数寄存器" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "Q:乘数寄存器" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "桶形移位器：F、Q联合逻辑右移" } ] } ] }, "bbox": [ 159, 593, 468, 683 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "A-{C，F.F1)Q-{Fn1) " } ] }, "bbox": [ 184, 687, 490, 703 ] }, { "type": "image", "content": { "image_source": { "path": "images/50c123f3e8175837087fdfcd305333c9c9036c5008f2054a02668e8566c45fd6.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 515, 558, 800, 713 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "初始：" } ], "level": 1 }, "bbox": [ 201, 720, 252, 737 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "A=0 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "B=|X| " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "Q=IYI " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "计数器=n " } ] } ] }, "bbox": [ 176, 741, 284, 818 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "累加、右移n次" } ], "level": 1 }, "bbox": [ 312, 747, 452, 766 ] }, { "type": "image", "content": { "image_source": { "path": "images/cee718a471c69c493caac3a6ae32770ed50a1dfbd57d645b620c0ef6a559157a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 310, 768, 453, 796 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "结果：" } ], "level": 1 }, "bbox": [ 539, 720, 589, 737 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "A=乘积绝对值高位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "B =｜X) " } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\pmb { \\mathrm { Q } } =" }, { "type": "text", "content": "乘积绝对值低位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "计数器 " }, { "type": "equation_inline", "content": "\\mathtt { \\mathtt { = 0 } }" } ] } ] }, "bbox": [ 473, 741, 663, 818 ] }, { "type": "image", "content": { "image_source": { "path": "images/35618de2ffe45452a153e0af96cb4963364157976dd9ea747ea1ac8ccbe444b9.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 752, 804, 784, 827 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "原码一位乘法流程：" } ], "level": 1 }, "bbox": [ 169, 92, 384, 112 ] }, { "type": "image", "content": { "image_source": { "path": "images/0c453e8e70a0be8534b121ac4c245278583c49001e1d8fb673fd9262cba1b395.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 152, 128, 584, 368 ] }, { "type": "image", "content": { "image_source": { "path": "images/a36d031b37fdd38c2990cb53473d9ab5f781d7bc3d220d4963c8e5341bc4d63e.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 645, 87, 779, 379 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "除法" } ], "level": 1 }, "bbox": [ 147, 407, 186, 423 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "X=+0.1011，Y=-0.1101，求X÷Y " } ] }, "bbox": [ 159, 434, 440, 449 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "手工除法：" } ], "level": 1 }, "bbox": [ 159, 454, 252, 468 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·判断被除数是否大于除数，决定商0/1。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■若商1，则减除数（得部分余数），再添加一位数或者添加0" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·若商0，则直接添加一位数或者添加0 " } ] } ] }, "bbox": [ 183, 470, 557, 518 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "改进手工算法 适合机器运算：" } ], "level": 1 }, "bbox": [ 159, 530, 400, 544 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·计算机通过做减法测试来实现判断：结果大于等于0，表明够商1；结果小于0，表明不够减，商0。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■计算机将部分余数左移一位，再直接与不右移的除数相减。" } ] } ] }, "bbox": [ 181, 548, 583, 592 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "即：减（+除数相反数的补码）" } ], "level": 1 }, "bbox": [ 159, 596, 394, 609 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "左移，构成循环" } ] }, "bbox": [ 206, 615, 324, 630 ] }, { "type": "image", "content": { "image_source": { "path": "images/080881940dddf1c0a012c30525b5c825fe4016a68c3cf727c989870893f7a3b6.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 445, 634, 473, 646 ] }, { "type": "image", "content": { "image_source": { "path": "images/bc67a26ef8c12a855344bd8b9a0214ec1762b77d8e934cbed0ba98eaad5f8e08.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 559, 426, 764, 576 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/f60008b096af71284ca25b36cf55312ff0d5181018ae1d26bb5b13719529cfef.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 144, 89, 794, 357 ] }, { "type": "image", "content": { "image_source": { "path": "images/913d33a05eded314c13c28fbf1daa186986245ed66e8eb12b6d67956866f8a7e.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 221, 419, 719, 561 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "不恢复余数算法" } ], "level": 1 }, "bbox": [ 147, 630, 275, 645 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "又称为加减交替法：当不够减时，不恢复余数，用加上除数（+IYI）的办法来继续求下一位商，其他操作不变。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "加减交替法的规则如下：" } ] } ] }, "bbox": [ 171, 659, 922, 724 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "余数为正：商上1，求下一位商的办法，是余数左移一位，再减去除数；" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "余数为负：商上0，求下一位商的办法，是余数左移一位，再加上除数。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "若最后一次上商为0，而又需得到正确余数，则最后一次仍需恢复余数。" } ] } ] }, "bbox": [ 200, 728, 880, 800 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "\"Why? " } ] }, "bbox": [ 176, 808, 270, 829 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/52d547c67d42a9203c76f16d5b457286133f23e717cfb269e410a277f0f94a48.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 191, 93, 675, 304 ] }, { "type": "image", "content": { "image_source": { "path": "images/1635b02d62a20c8866fd2f847b7be51a7eb63e2baf3207aed8f554ea8f18d46a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 149, 360, 537, 596 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "如何控制加减交替？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "如何上商？" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{3}" }, { "type": "text", "content": "加减与移位次数如何控制，结果在哪里？" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "加减交替：控制逻辑" } ] } ] }, "bbox": [ 568, 351, 858, 456 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\scriptstyle \\mathbf { Q } _ { \\mathrm { n } } = 1" }, { "type": "text", "content": "表明够减一下一次做减法→B中的|Y|取反（11...11B)" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\scriptstyle \\mathbf { Q } _ { \\mathrm { n } } = \\mathbf { 0 }" }, { "type": "text", "content": "表明不够减一下一次做加法→B中的|Y|不取反 (0...00B)" } ] } ] }, "bbox": [ 589, 461, 852, 529 ] }, { "type": "title", "content": { "title_content": [ { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "上商：Fs取反" } ], "level": 1 }, "bbox": [ 549, 535, 705, 551 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·Fs=0表明够减 " }, { "type": "equation_inline", "content": "\\scriptstyle \\mathbf { Q } _ { \\mathsf { n } } =" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\bar { \\mathsf { F } } _ { \\mathsf { S } } \\mathsf { = }" }, { "type": "text", "content": "1表明不够减 →Qn=0 " } ] } ] }, "bbox": [ 569, 555, 744, 607 ] }, { "type": "image", "content": { "image_source": { "path": "images/3f2316993b77cc87c3b76fb52c1b9a866d3d27a79298726c073a1420897aecdd.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 739, 543, 873, 615 ] }, { "type": "page_footer", "content": { "page_footer_content": [ { "type": "text", "content": "定点运算器" } ] }, "bbox": [ 147, 667, 240, 683 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "构成：" } ], "level": 1 }, "bbox": [ 171, 99, 235, 117 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "内部总线IB" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "· ALU " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·2个暂存器：LA和LB " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·通用寄存器堆GPR：n+1个 " } ] } ] }, "bbox": [ 181, 123, 426, 206 ] }, { "type": "image", "content": { "image_source": { "path": "images/e985aecc3e9eb9e76340b5366b6fa387f7ab170ac283a9cc5778b77f921838fd.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 416, 102, 875, 227 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "· " }, { "type": "equation_inline", "content": "\\mathrm { L A } _ { \\mathrm { i n } } = 1" }, { "type": "text", "content": "：IB上数据写入LA " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\mathsf { L B } _ { \\mathrm { i n } } = 1" }, { "type": "text", "content": ":IB上数据写入LB" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "ALUOP：功能码/运算码，选择ALU执行的运算" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathtt { A L U } _ { \\mathrm { o u t } } = \\mathbb { 1 }" }, { "type": "text", "content": "：ALU的运算结果（即输出端F）→IB" } ] } ] }, "bbox": [ 168, 229, 447, 353 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "RA：执行读操作的寄存器地址 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "WA：执行写操作的寄存器地址 " } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathtt { G R } _ { \\mathtt { o u t } } = 1" }, { "type": "text", "content": "：读出RA指定的寄存器数据送到IB上(Reg[RA]→IB)" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathsf { G R } _ { \\mathrm { i n } } = 1" }, { "type": "text", "content": "：将IB上的数据写入由写地址WA选中的寄存器 (IB→Reg[WA])" } ] } ] }, "bbox": [ 468, 230, 860, 355 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\gtrdot" }, { "type": "text", "content": "数据通路：指数据在硬件系上流动的过程与控制步骤" } ] }, "bbox": [ 146, 420, 420, 454 ] }, { "type": "image", "content": { "image_source": { "path": "images/125f1f976a45bde61707512608d443be1397a84c51bebbeef84d0565bd4ae8b7.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 363, 423, 779, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "在总线上传送数据的数据通路：需要一个CPU周期" } ] }, "bbox": [ 144, 544, 542, 558 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "源部件将数据送上总线；目的部件从总线上接收数据" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "任何要把数据送上总线的源部件，其数据输出端必须以三态输出的形式与总线连接，并由使能信号控制" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·对于某个总线系统来说，各个总线源部件的数据输出使能信号不能同时有效：任何时刻只能允许一个部件将数据送上总线以保证分时高用总线" } ] } ] }, "bbox": [ 161, 563, 729, 636 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/8c546a5f5a7aab5fcbb5bf8368f8ffbbd6e20baab9c2b514f1bcfa342a332678.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 178, 129, 198, 151 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "【例4.10】假设在单总线运算器中，完成一条指令ADDR，R作 " }, { "type": "equation_inline", "content": "\\mathbb { R } _ { 2 } + \\mathbb { R } _ { 7 } \\mathbb { R } _ { 3 }" }, { "type": "text", "content": "，则分析该操作的数据通路，需要几个CPU周期？" } ] }, "bbox": [ 206, 131, 1000, 186 ] }, { "type": "image", "content": { "image_source": { "path": "images/8195232877e38e022ce58354bf3e0f8e7017782e37ce393611d5c1f08bbd1c6a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 181, 193, 805, 361 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "需要" } ] }, "bbox": [ 907, 286, 996, 309 ] }, { "type": "table", "content": { "image_source": { "path": "images/676656b11afbd4bb4ed24c75e2c3bbdba810d2f3ee73f6b7d4aa3f20b3344706.jpg" }, "table_caption": [], "table_footnote": [], "html": "
步骤操作发送的信号
1R2→LARA=2, GRout=1, LAin=1
2R7→LBRA=7, GRout=1, LBin=1
3ALU执行F=A+B运算，结果F→R3ALU_OP=加法功能码，ALU_out=1,
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 149, 363, 1000, 502 ] }, { "type": "image", "content": { "image_source": { "path": "images/d814906f52ca58133470d1ec963e40201e7792ebe345adbefb39a1651947d1be.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 514, 186, 545 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "X 【例4.10】假设在单总线运算器中，完成一条指令ADD " }, { "type": "equation_inline", "content": "\\pmb { \\mathsf { R } } _ { 3 }" }, { "type": "equation_inline", "content": "{ \\sf R } _ { 2 }" }, { "type": "text", "content": "，R的运算操" }, { "type": "equation_inline", "content": "\\mathsf { E R } _ { 2 } + \\mathsf { R } _ { 7 } \\to \\mathsf { R } _ { 3 }" }, { "type": "text", "content": "，则分析该操作的数据通路，需要几个CPU周期？" } ] }, "bbox": [ 169, 542, 880, 582 ] }, { "type": "image", "content": { "image_source": { "path": "images/a7fb5924317ec14cc0f9c5e3f8fb93545692880d72c48a901eeeba9aab0db9c9.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 171, 587, 613, 705 ] }, { "type": "table", "content": { "image_source": { "path": "images/9b13f97b15ecbf42a4b3fbc38d6a0264d9e3f9b7d55a93c3776bd8f1ba8fc74f.jpg" }, "table_caption": [], "table_footnote": [], "html": "
步骤操作发送的信号
1R2→LARA=2, GRout=1, LAin=1
2R7→LBRA=7, GRout=1, LBin=1
3ALU执行F=A+B运算,结果F→R3ALU_OP=加法功能码, ALU_out=1, GRin=1, WA=3
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 149, 707, 902, 807 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "需要3个CPU周期" } ] }, "bbox": [ 685, 651, 868, 670 ] }, { "type": "image", "content": { "image_source": { "path": "images/1ea251a75cc631e8da1edd057cec79441fb72193971d3eef097821a86412e2de.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 465, 801, 495, 815 ] }, { "type": "page_header", "content": { "page_header_content": [ { "type": "text", "content": "（）车芯线给构" } ] }, "bbox": [ 280, 91, 552, 109 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "构成：" } ], "level": 1 }, "bbox": [ 168, 99, 226, 115 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·2条内部总线：IB1、IB2 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "· ALU " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·暂存器：LC " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·通用寄存器堆GPR：n+1个 " } ] } ] }, "bbox": [ 176, 118, 400, 196 ] }, { "type": "image", "content": { "image_source": { "path": "images/a87051a1213d96b6ac50fd128c53e03bcfdec6caf2a3592e7cd20df2d14f35b1.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 458, 92, 791, 212 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "： " }, { "type": "equation_inline", "content": "\\iota C _ { \\mathrm { i n } } = 1" }, { "type": "text", "content": "：ALU运算结果写入LC，即F→LC" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\dot { } \\cdot \\mathsf { L C } _ { \\mathsf { o u t } } = 1" }, { "type": "text", "content": "：LC中数据送上总线IB1，即LC→IB1" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\ast" }, { "type": "text", "content": "ALU_OP:ALU功能码/运算码" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\ast" }, { "type": "text", "content": "RA1：执行读出数据到IB1的寄存器地址" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\ast" }, { "type": "text", "content": "RA2：执行读出数据到IB2的寄存器地址" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "^ \\ast" }, { "type": "text", "content": "WA：执行写操作的寄存器地址 " } ] } ] }, "bbox": [ 159, 221, 477, 319 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathsf { G R } _ { \\mathsf { o u t 1 } } = 1" }, { "type": "text", "content": "：读出RA1指定的寄存器数据送到IB1上，即Reg[RA1]→IB1" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathsf { G R } _ { \\mathsf { o u t } 2 } = 1" }, { "type": "text", "content": "：读出RA2指定的寄存器数据送到IB2上，即Reg[RA2]→IB2" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathsf { G R } _ { \\mathrm { i n } } = 1" }, { "type": "text", "content": "：将IB1上的数据写入由写地址WA中的寄存器，即IB1→Reg[WA]" } ] } ] }, "bbox": [ 489, 222, 789, 313 ] }, { "type": "image", "content": { "image_source": { "path": "images/75625929eabda9a925a66bd9baf77befd3d36f41294bd86703c1f96680c42d1e.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 336, 200, 382 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "（2）双总线结构" } ], "level": 1 }, "bbox": [ 255, 338, 455, 363 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "【例4.11】假设在双总线运算器中，同样完成指令ADD " }, { "type": "equation_inline", "content": "{ \\sf R } _ { 3 }" }, { "type": "equation_inline", "content": "\\mathsf { R } _ { 2 }" }, { "type": "equation_inline", "content": "\\mathsf { R } _ { 7 }" }, { "type": "text", "content": "的运算操作 " }, { "type": "equation_inline", "content": "R _ { 2 } + R _ { 7 } R _ { 3 }" }, { "type": "text", "content": "，分析数据通路，需要几个CPU周期？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "需要2个CPU周期" } ] } ] }, "bbox": [ 179, 379, 519, 505 ] }, { "type": "image", "content": { "image_source": { "path": "images/2f5b4666b8610e3c937ac6d9db58dffad8bf5997aa2c849c2a86b02e3a84305b.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 531, 370, 954, 514 ] }, { "type": "table", "content": { "image_source": { "path": "images/27e6b4c904af645b814918bd9000a1238d85a03c91d2ebcc857a47d91d031796.jpg" }, "table_caption": [], "table_footnote": [], "html": "
步骤操作发送的信号
1R2→IB1, R7→IB2, F=A+B, F→LCRA1=2, RA2=7, ALU_OP=加法功能码,GRout1=1, GRout2=1, LCin=1
2LC→R3LCout=1, GRin=1, WA=3
", "table_type": "simple_table", "table_nest_level": 1 }, "bbox": [ 166, 519, 939, 608 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "思考：如果LC暂存器输出端连接到IB2上，能否完成上述操作？" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "总结：在分析总线结构的运算器数据通路时，基本的原则：在一个CPU周期内，某条总线上的数据必须是唯一的，且不能保留至下一个CPU周期。" } ] } ] }, "bbox": [ 173, 609, 910, 663 ] } ], [ { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "特征：按照运算操作的过程与控制需求，部件之间由专用的独立线路进行连接" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "特点：线路比较复杂，但是传输效率高，适用于流水线CPU" } ] } ] }, "bbox": [ 161, 99, 952, 147 ] }, { "type": "image", "content": { "image_source": { "path": "images/a7e3272769accbdabd6a76fd92b15f114f238570452f7eb9ae49f9f25ff03f10.jpg" }, "image_caption": [ { "type": "text", "content": "单周期CPU的运算器" }, { "type": "text", "content": "每条指令只需一个CPU周期完成" } ], "image_footnote": [] }, "bbox": [ 169, 193, 463, 344 ] }, { "type": "image", "content": { "image_source": { "path": "images/dd24d6faebf55dcbf9ad5485c350bf3bc364b5b8f7b499d47a4200f6e1f95dd1.jpg" }, "image_caption": [ { "type": "text", "content": "多周期CPU的运算器" }, { "type": "text", "content": "每条指令需要多个CPU周期完成" } ], "image_footnote": [] }, "bbox": [ 566, 197, 969, 347 ] }, { "type": "image", "content": { "image_source": { "path": "images/314d6cadb0bfe6466e978c155fecfc534eaf9f467fd04c79c80fc0bce91e1af4.jpg" }, "image_caption": [ { "type": "text", "content": "单周期CPU的运算器" } ], "image_footnote": [] }, "bbox": [ 149, 425, 366, 536 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "·只需1个clk周期：" } ] }, "bbox": [ 147, 546, 284, 562 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "RAA=2,RBA=7,WA=3 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·ALU_OP=加法功能码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·Reg_W=1，在clk的边沿完成R2+R→R3操作" } ] } ] }, "bbox": [ 152, 565, 359, 626 ] }, { "type": "image", "content": { "image_source": { "path": "images/2a77f41d31aa31d653f723077162ad45fbdb4b02309f3ee781bfbda011629076.jpg" }, "image_caption": [ { "type": "text", "content": "执行ADD R3, " } ], "image_footnote": [] }, "bbox": [ 403, 426, 764, 537 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "需3个节拍（clk周期）：" } ] }, "bbox": [ 408, 539, 601, 555 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·第1个clk（读寄存器）： " }, { "type": "equation_inline", "content": "\\mathsf { R } _ { 2 } { \\longrightarrow } \\mathsf { A }" }, { "type": "equation_inline", "content": "\\scriptstyle \\mathsf { R } _ { 7 } \\to \\mathsf { B }" }, { "type": "text", "content": ":RAA " }, { "type": "equation_inline", "content": "^ { 1 = 2 }" }, { "type": "text", "content": "RB A=7" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·第2个clk（执行）：A+B→F:ALU_OP=加法功能码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·第3个clk（写回）： " }, { "type": "equation_inline", "content": "\\mathsf { F } { } \\mathsf { R } _ { 3 }" }, { "type": "text", "content": ":WA " }, { "type": "equation_inline", "content": "^ { = 3 }" }, { "type": "text", "content": "，Reg_W=1" } ] } ] }, "bbox": [ 421, 558, 768, 619 ] }, { "type": "image", "content": { "image_source": { "path": "images/59c466468b213ccd281befcd1eebfaec5971995dd519d811b30d31d64f190ad7.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 727, 620, 757, 633 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "特征：能够对一组寄存器同时读出或者写入多个数据 " } ], "level": 1 }, "bbox": [ 166, 99, 759, 122 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "三端口寄存器堆可以同时进行" } ], "level": 1 }, "bbox": [ 168, 134, 346, 175 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·读A端口" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "读B端口" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·写入数据 " } ] } ] }, "bbox": [ 200, 181, 309, 249 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "寄存器堆：4×8位" } ], "level": 1 }, "bbox": [ 146, 259, 342, 278 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\mathbb { R } _ { 0 } \\sim \\mathbb { R } _ { 3 } ," }, { "type": "text", "content": "，地址2位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·RA_A:A口地址 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·RA_D:A口读出数据 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "RB_A:B口地址" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "RB_D:B口读出数据" } ] } ] }, "bbox": [ 176, 282, 366, 382 ] }, { "type": "image", "content": { "image_source": { "path": "images/7f04dad3f2cd8523941ddb3b2e5073b147d88189f7b4c6de8b40c926dd617f13.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 383, 124, 957, 388 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "标志寄存器FR：又称为状态寄存器" } ], "level": 1 }, "bbox": [ 151, 493, 485, 511 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "PSW:程序状态字" } ], "level": 1 }, "bbox": [ 586, 492, 759, 510 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "■用来保存ALU操作结果的某些状态，可作为外界对运算结果进行分析的依据，也可以用于判断程序是否要转移的条件" } ] }, "bbox": [ 179, 514, 847, 552 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "最基本的5种运算结果标志：" } ], "level": 1 }, "bbox": [ 151, 557, 418, 574 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "ZF：结果为零标志" } ] }, "bbox": [ 179, 577, 371, 596 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "·运算结果为全0，ZF置1 " } ] }, "bbox": [ 210, 601, 438, 619 ] }, { "type": "equation_interline", "content": { "math_content": "Z F = \\overline {{F _ {n} + F _ {n - 1} + . . . . . . + F _ {0}}}", "math_type": "latex", "image_source": { "path": "images/a5c4729e85b29a0b726dcecde4c1d03d27fe1aefc0dd56e96ae2290f3555b2b9.jpg" } }, "bbox": [ 502, 599, 788, 621 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "·运算结果不全为0，ZF置0 " } ] }, "bbox": [ 210, 623, 465, 640 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "CF：进位/借位标志位，CF标志只对无符号数运算有意义 " } ] }, "bbox": [ 179, 645, 705, 662 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·加法运算时：C=1则CF置1（表示有进位），否则置0" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·减法运算时： " }, { "type": "equation_inline", "content": "\\subset = \\pmb { 0 }" }, { "type": "text", "content": "则CF置1（表示不够减，有借位），否则置0" } ] } ] }, "bbox": [ 210, 667, 781, 707 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "最高位的进位" } ] }, "bbox": [ 240, 715, 295, 744 ] }, { "type": "equation_interline", "content": { "math_content": "C F = \\overline {{\\overline {{A D D}} / S U B}} \\cdot C + \\overline {{A D D}} / S U B \\cdot \\overline {{C}}", "math_type": "latex", "image_source": { "path": "images/198799afad5d19dbe98bc49f9730bc20c3033499a7e0974fc0dab2d21a8a3897.jpg" } }, "bbox": [ 403, 717, 789, 746 ] }, { "type": "image", "content": { "image_source": { "path": "images/131355cd6254e9395745464151013638000f1f37cc5df5cc5192d7004b5949af.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 848, 744, 887, 759 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "③UF：溢出标志，反映有符亏数加减运算所得结果是否溢出；UF标志只对带" } ] }, "bbox": [ 213, 87, 902, 104 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "符号数运算有意义。 " } ] }, "bbox": [ 233, 105, 410, 123 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·运算溢出：OF=1 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·运算没有溢出：OF " }, { "type": "equation_inline", "content": "\\mathtt { \\mathtt { \\mathtt { = 0 } } }" } ] } ] }, "bbox": [ 245, 127, 463, 166 ] }, { "type": "equation_interline", "content": { "math_content": "\\begin{array}{c c} O F = \\overline {{A D D / S U B}} \\left(\\overline {{X _ {S}}} \\overline {{Y _ {S}}} F _ {S} + X _ {S} Y _ {S} \\overline {{F _ {S}}}\\right) \\\\ + \\overline {{A D D}} / S U B \\left(\\overline {{X _ {S}}} Y _ {S} F _ {S} + X _ {S} \\overline {{Y _ {S}}} \\overline {{F _ {S}}}\\right) \\\\ \\hline O F = C _ {1} \\oplus C _ {S} & O F = F _ {S 1} \\oplus F _ {S 2} \\end{array}", "math_type": "latex", "image_source": { "path": "images/a28dcbba0653634f55ff16ba3e06fa763adc8d040d9001cf50a1d904a7b4b743.jpg" } }, "bbox": [ 527, 112, 882, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\textcircled{4}" }, { "type": "text", "content": "SF：符号标志，记录运算结果的符号。" } ] }, "bbox": [ 213, 196, 576, 215 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·为运算结果的最高位 SF =Fn " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·运算结果为正数时： " }, { "type": "equation_inline", "content": "\\pmb { S F = 0 }" }, { "type": "text", "content": "；为负数时： " }, { "type": "equation_inline", "content": "s F { = } 1" }, { "type": "text", "content": "。 (无溢出时才正确)" } ] } ] }, "bbox": [ 245, 219, 860, 263 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "equation_inline", "content": "\\textcircled{5}" }, { "type": "text", "content": "PF：奇偶标志，反映运算结果中“1”的个数的奇偶性" } ] }, "bbox": [ 213, 268, 724, 287 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·结果中“1”的个数为偶数：PF=1 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·结果中“1”的个数为奇数：PF=0 " } ] } ] }, "bbox": [ 245, 291, 573, 331 ] }, { "type": "equation_interline", "content": { "math_content": "P F = \\overline {{F _ {n} \\oplus F _ {n - 1} \\oplus \\dots . \\oplus F _ {0}}}", "math_type": "latex", "image_source": { "path": "images/7c93d59d5353deea7dc93ce4178f1d5db92fe203d58ff6893f32dd9086d3f0f0.jpg" } }, "bbox": [ 593, 297, 922, 326 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "假设两个浮点数X和Y " } ], "level": 1 }, "bbox": [ 171, 435, 472, 458 ] }, { "type": "equation_interline", "content": { "math_content": "X = M _ {X} \\times 2 ^ {E _ {X}}", "math_type": "latex", "image_source": { "path": "images/fc2ec54c2995e1381192330e4733e8db4f3b32307ef1e25ce2f16a081d1839cd.jpg" } }, "bbox": [ 376, 474, 588, 506 ] }, { "type": "equation_interline", "content": { "math_content": "Y = M _ {Y} \\times 2 ^ {E _ {Y}}", "math_type": "latex", "image_source": { "path": "images/86cb4df029cc660adee008112e1ea0073e028a6315d8dd849ab6c401adf302a0.jpg" } }, "bbox": [ 695, 473, 902, 506 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "对齐小数点：使得阶码小的那个数变得与阶码大的那个数的阶码" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "然后：对尾数（有效数位）做加减运算" } ] } ] }, "bbox": [ 171, 539, 996, 606 ] }, { "type": "equation_interline", "content": { "math_content": "Z = X \\pm Y = \\left(M _ {X} \\cdot 2 ^ {\\left(E _ {X} - E _ {Y}\\right)} \\pm M _ {Y}\\right) \\times 2 ^ {E _ {Y}} \\quad E _ {X} \\leq", "math_type": "latex", "image_source": { "path": "images/31a12e7058a4d903852aab62b787f3f8e53dd7bd69f6616fc35cae13cf3d45c5.jpg" } }, "bbox": [ 262, 627, 998, 661 ] }, { "type": "page_number", "content": { "page_number_content": [ { "type": "text", "content": "74 " } ] }, "bbox": [ 887, 338, 927, 357 ] } ], [ { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(2）对阶：" } ] }, "bbox": [ 154, 93, 238, 110 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "目标是：对齐小数点" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "原则是：小阶对大阶" } ] } ] }, "bbox": [ 147, 117, 363, 164 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "求阶差△E=Ex-Eγ，若△E≠0，即Ex≠E时需要对阶。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "若△E>0，则Ex>Eγ，M每右移一位， " }, { "type": "equation_inline", "content": "\\mathbb { E } _ { \\mathsf { Y } ^ { + } }" }, { "type": "text", "content": "1，直至 " }, { "type": "equation_inline", "content": "\\mathsf { E } _ { \\mathsf { v } } = \\mathsf { E } _ { \\mathsf { X } }" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "若△E<0，则 " }, { "type": "equation_inline", "content": "\\mathsf { E } _ { \\mathsf { X } } { < } \\mathsf { E } _ { \\mathsf { Y } }" }, { "type": "equation_inline", "content": "\\pmb { M } _ { \\mathbf { X } }" }, { "type": "text", "content": "每右移一位， " }, { "type": "equation_inline", "content": "\\mathtt { E _ { X } } + 1" }, { "type": "text", "content": "，直至 " }, { "type": "equation_inline", "content": "\\therefore x = E _ { y }" } ] } ] }, "bbox": [ 169, 167, 653, 224 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "(3）尾数相加减 " } ] }, "bbox": [ 154, 227, 287, 243 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·若做减法，先将操作数Y的尾数取相反数，再与Y的尾数相加 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "■对阶时，右移多出的数位不丢弃（因为一般设置位数更宽的ALU进行计算，同时设置更宽的寄存器来存放中间运算结果)" } ] } ] }, "bbox": [ 169, 247, 779, 300 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "82 " } ] }, "bbox": [ 774, 303, 808, 317 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "浮点加减运算步骤 " } ], "level": 1 }, "bbox": [ 196, 347, 445, 366 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "【4）结果规格化：尾数运算的结果可能出现两种非规格化情况：" } ] }, "bbox": [ 144, 382, 773, 401 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·尾数溢出：需要右规（1次），即尾数右移1位，阶码+1" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "尾数| " }, { "type": "equation_inline", "content": "< 2 ^ { - 1 }" }, { "type": "text", "content": "：需要左规，即尾数左移1位，阶码-1，左规可能多次，直到尾数变为规格化形式。" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·先判断是否溢出，如果没有，再判断是否有前导零" } ] } ] }, "bbox": [ 147, 407, 927, 524 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "[5）舍入：将尾数多余的位数舍去，变成规定的浮点数格式" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "舍入策略：舍去多余位数的方法，简单的舍入策略：" } ] } ] }, "bbox": [ 144, 555, 742, 600 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·截断法：多余的全舍去 " } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·0舍1入法：多余位的最高位为1时，在尾数最低位上+1，否则舍去多余位" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·末位恒置1：始终在舍入结果的最末位置1" } ] } ] }, "bbox": [ 196, 604, 833, 662 ] } ], [ { "type": "title", "content": { "title_content": [ { "type": "text", "content": "IEEE754标准：规定了4种可选的舍入模式" } ], "level": 1 }, "bbox": [ 169, 104, 544, 120 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{1}" }, { "type": "text", "content": "向上舍入（总是朝+）：为正数时，只要多余位不全为0，就向最低有效位进1；为负数时，则采用简单的截断法" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{2}" }, { "type": "text", "content": "向下舍入（总是朝－）：为正数时，只要多余位不全为0，就简单地截尾为负数时，则向最低有效位进1" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{3}" }, { "type": "text", "content": "向0舍入：即朝数轴的原点方向舍入，就是无论正数还是负数，都采用简单的截尾，从而使得绝对值总是变小。这种方法容易累积误差" } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "\\textcircled{4}" }, { "type": "text", "content": "就近舍入：即舍入到最接近的数，“四舍五入”。多余位=R,...Rn" } ] } ] }, "bbox": [ 186, 124, 737, 239 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "text", "content": "·当R " }, { "type": "equation_inline", "content": "\\scriptstyle 1 = 0" }, { "type": "text", "content": "：截尾（即舍去）；" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "·当R " }, { "type": "equation_inline", "content": "_ 1 = 1" }, { "type": "text", "content": ",且R...R不全为0：向最低有效位进1；" } ] }, { "item_type": "text", "item_content": [ { "type": "text", "content": "· " }, { "type": "equation_inline", "content": "\\yen 123,456,7" }, { "type": "text", "content": "：即等于一半（中点），则若最低有效位为0就截尾，若最低有效位为1就进1，以使得最低有效位总是为0" } ] } ] }, "bbox": [ 213, 241, 734, 294 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "中间结果设置了：保护位(guard)、舍入位(round)、粘滞位(sticky)" } ] }, "bbox": [ 171, 298, 680, 313 ] }, { "type": "image", "content": { "image_source": { "path": "images/55a92cad7bcb03d99177cc82bcc618a4c22ad6fac867143503578508456fb4c6.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 173, 361, 647, 600 ] }, { "type": "image", "content": { "image_source": { "path": "images/7749aefff1f44fe5f12bbb0be1645640461baee14490e7fc1b26359b4e945f52.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 700, 577, 734, 598 ] } ], [ { "type": "image", "content": { "image_source": { "path": "images/5f43ed1e5bce18a593cd23ba0865b39f93ba309eb7af27bdb85047d14e10b90e.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 147, 89, 191, 126 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "浮点加减运算例题 " } ], "level": 1 }, "bbox": [ 228, 87, 443, 107 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "(2）尾数相加：" } ], "level": 1 }, "bbox": [ 196, 120, 347, 140 ] }, { "type": "equation_interline", "content": { "math_content": "\\left[ \\mathrm {M} _ {2} \\right] _ {\\text {补}} = 1. 1 0 0 0 0 0 1 (1 0 1)", "math_type": "latex", "image_source": { "path": "images/c7be7abbcc0bc100173cbcaaba5d5d2d45f6f8b9123aff410916dbdbd6c648e9.jpg" } }, "bbox": [ 243, 146, 497, 166 ] }, { "type": "equation_interline", "content": { "math_content": "[ \\mathrm {M} _ {\\mathrm {X}} ] _ {\\text {补}} 1 1. 0 1 1 0 1 0 1", "math_type": "latex", "image_source": { "path": "images/bc981b49a259ec216935ded9e350ec42d27a63b5c885a8c25604293f278692ac.jpg" } }, "bbox": [ 643, 123, 823, 141 ] }, { "type": "equation_interline", "content": { "math_content": "+ \\left[ \\mathbf {M} _ {\\mathbf {Y}} \\right] _ {\\text {补}} 0 0. 0 0 0 1 1 0 0 (1 0 1)", "math_type": "latex", "image_source": { "path": "images/7e3e81e5162c15ee0aced41ad675a22a62b16a6ca66ac6ac296cc2210324465e.jpg" } }, "bbox": [ 615, 145, 868, 164 ] }, { "type": "equation_interline", "content": { "math_content": "\\left[ \\mathbf {M} _ {\\mathrm {X}} + \\mathbf {M} _ {\\mathrm {Y}} \\right] _ {\\text {补}} 1 1. 1 0 0 0 0 0 1 (1 0 1)", "math_type": "latex", "image_source": { "path": "images/09e233bfccbb5497efe17d44d89d40cb5f63330ab1e6aa4ada242c3f2020ee76.jpg" } }, "bbox": [ 605, 168, 868, 187 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "（3）结果规格化：首先判断：是否溢出？" } ] }, "bbox": [ 200, 196, 606, 215 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因为双符号位为11，没有溢出" } ] }, "bbox": [ 421, 216, 660, 231 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "再检查：有无前导零？" } ] }, "bbox": [ 393, 233, 603, 250 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "因为 " }, { "type": "equation_inline", "content": "\\boldsymbol { \\mathsf { M } } _ { \\mathrm { Z } } \\boldsymbol { \\mathrm { l } } _ { * \\mathrm { f } }" }, { "type": "text", "content": "符号位和最高有效位相同 " }, { "type": "equation_inline", "content": "= 1 . 1" }, { "type": "text", "content": "故有一个前导零，须左规一位" } ] }, "bbox": [ 421, 252, 746, 284 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "左规一位：尾数左移1位，阶码-1" } ] }, "bbox": [ 393, 286, 702, 303 ] }, { "type": "list", "content": { "list_type": "text_list", "list_items": [ { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "[ \\mathbf { M } _ { 2 } ] _ { * \\mathrm { h } } = 1 . 0 0 0 0 0 1 1" }, { "type": "text", "content": " (01) " } ] }, { "item_type": "text", "item_content": [ { "type": "equation_inline", "content": "[ \\ E _ { \\mathrm { Z } } ] _ { * \\mathrm { h } } = 0 0" }, { "type": "text", "content": "，001 + 11, " }, { "type": "equation_inline", "content": "1 1 1 = 0 0" }, { "type": "text", "content": "，000" } ] } ] }, "bbox": [ 421, 306, 744, 338 ] }, { "type": "title", "content": { "title_content": [ { "type": "text", "content": "(4）舍入：按照0舍1入法，尾数多余位舍去 " } ], "level": 1 }, "bbox": [ 200, 342, 645, 363 ] }, { "type": "paragraph", "content": { "paragraph_content": [ { "type": "text", "content": "结果为： " }, { "type": "equation_inline", "content": "[ \\mathsf { X } + \\mathsf { Y } ] _ { \\dag \\Xi } = 0" }, { "type": "text", "content": "，000 1.0000011 " } ] }, "bbox": [ 220, 367, 584, 384 ] }, { "type": "image", "content": { "image_source": { "path": "images/cd1e8e31644d4311c85ff19418a4b8474f57ffd8021141b8b4cc57db704dfb8a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 779, 366, 811, 384 ] }, { "type": "image", "content": { "image_source": { "path": "images/3696e23690406c549b23470efcee5693b5f60e0db09eef2133e90f7a62811e0a.jpg" }, "image_caption": [], "image_footnote": [] }, "bbox": [ 852, 366, 870, 384 ] } ] ]