IEEE 浮点数格式

明亮 posted @ 2010年7月06日 23:42 in 【生活/随想】 , 1773 阅读
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IEEE 浮点数格式

转自：http://liyanrui.is-programmer.com/posts/3806.html
 

IEEE 754 标准规定了三种浮点数格式：单精度、双精度、扩展精度。《编程卓越之道》第一部的 4.2 节对这些浮点数格式已进行了详细的讲解，为了让读书笔记更像读书笔记，本文只道出个人的一些理解以及一些疑惑之处。

 

IEEE 浮点数标准的由来

话说 Intel 计划给最早的 8086 增加浮点运算单元 (FPU) 时，他们请来了最好的数值分析专家来为 8087 FPU 设计浮点数格式，这位专家接着又请来了该领域的另外两位专家，这三个人 (Kahn, coonan 与 Stone) 设计了 Intel 的浮点格式，即 KCS 浮点数标准。这个标准实在太出色了，因此 IEEE 组织将 KCS 选作为 IEEE 浮点数格式的基础，即 IEEE 标准 754。

单精度浮点数

IEEE 754 标准所定义的单精度浮点数的长度为 32 位，按位域可划分为：符号位、阶码位与尾数位，如下：

		31----------------------22---------------------------------------------------------0
  |                       |                                                          |
  X X X X    X X X X    X X X X    X X X X    X X X X    X X X X    X X X X    X X X X
  | |-------------------| |----------------------------------------------------------|
符号        阶码                                     尾数

符号位取 0 表示正数，取 1 表示负数。

阶码位是 8 位，这里有一点小门道需要注意，那就是$2^n$的指数 n 并不能直接当作阶码来处理，需要将其与 127 (0x7f) 相加才可得到 $2^n$ 的阶码表示。

尾数的位域长度在图示中是 23 位，但实际上却是 24 位，这个位是“不可见”的，其值固定为 1，这也就是说 IEEE 754 标准所定义的浮点数，其有效数字是介于 1 与 2 之间的小数。

可以尝试写一下 1.0 这个数的二进制单精度浮点格式，这有助于更好地理解单精度浮点数格式的位域分布。

		1.0 的二进制单精度浮点格式：0 0111 1111 000 0000 0000 0000 0000 0000

值得注意的一个问题是：书上说之所以要将指数加上 127 来得到阶码，是为了简化浮点数的比较运算，这一点我没有体会出来。但是通过 127 这个偏移量 (移码)，可以区分出指数的正负。阶码为 127 时表示指数为 0；阶码小于 127 时表示负指数；阶码大于 127 时表示正指数。

第二个值得思考的问题是：使用 24 位尾数，大概可以得到 $$6\frac{1}{2}$$ 个十进制数字的精度，其中的“半个”数字由 FPU 的好意而产生的一个随机数字，这个数字通常接近 5 (四舍五入？)。

第三个问题是我经常要碰到的：IEEE 754 标准所定义的单精度浮点数所表示的数的范围是多少？书上给出的答案是大约为 $2^{\pm128}$ 或者大约 $10^{\pm38}$。这个比较好理解，因为尾数的最大值是接近 2，而指数的范围是 [-127, 127]，那么这个范围就可以表示为$2\times2^{\pm127}$。

双精度浮点数

相对于单精度浮点数格式，双精度的阶码变为 11 位，移码变为为 1,023，尾数变为 53 位 (包含那个固定为 1 的隐含位)。这样，再加上符号位，双精度浮点数的长度为 64 位，提供了大约 $10^{\pm308}$ 的动态范围以及 $$14\frac{1}{2}$$ 个数字的精度。

 

		63-62------------51-------------------------------------------------------------0
  |  |             |                                                              |
  X  XXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX  
  |  |-------------|--------------------------------------------------------------| 
符号       阶码                                    尾数

 

扩展精度浮点数

为了追求更高的浮点运算精度，Intel 又搞出来扩展精度格式。扩展精度的浮点数长度为 80 个位，相对于双精度浮点数所多出来的 16 个位，有 12 位加入到尾数位中，有 4 位加入到阶码位中。

 

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온라인 카지노 커뮤니티 온카허브 입니다. 온카허브는 카지노 먹튀 사이트들과 안전한 카지노 사이트 정보를 공유하고 있습니다. 카지노 먹튀검증 전문팀을 자체적으로 운영함으로써 철저한 검증을 진행하고 있습니다.

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카지노사이트 说:
2024年1月24日 10:12

카지노사이트 바카라사이트 우리카지노 카지노는 바카라, 블랙잭, 룰렛 및 슬롯 등 다양한 게임을 즐기실 수 있는 공간입니다. 게임에서 승리하면 큰 환호와 함께 많은 당첨금을 받을 수 있고, 패배하면 아쉬움과 실망을 느끼게 됩니다.s

69
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하노이 가라오케 说:
2024年1月24日 10:31

하노이 꼭 가봐야 할 베스트 업소 추천 안내 및 예약, 하노이 밤문화 에 대해서 정리해 드립니다. 하노이 가라오케, 하노이 마사지, 하노이 풍선바, 하노이 밤문화를 제대로 즐기시기 바랍니다. 하노이 밤문화 베스트 업소 요약 베스트 업소 추천 및 정리. https://hanoi-nightlife.com/

70
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베트남 유흥 说:
2024年1月25日 12:07

베트남 남성전용 커뮤니티❣️ 베트남 하이에나 에서 베트남 밤문화를 추천하여 드립니다. 베트남 가라오케, 베트남 VIP마사지, 베트남 이발관, 베트남 황제투어 남자라면 꼭 한번은 경험 해 봐야할 화끈한 밤문화로 모시겠습니다. https://vn-hyena.com/

71
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안전놀이터 说:
2024年1月25日 12:20

No.1 먹튀검증 사이트, 먹튀사이트, 검증사이트, 토토사이트, 안전사이트, 메이저사이트, 안전놀이터 정보를 제공하고 있습니다. 먹튀해방으로 여러분들의 자산을 지켜 드리겠습니다. 먹튀검증 전문 커뮤니티 먹튀클린만 믿으세요!!


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