浮点标准 IEEE754 精要
目录 |
1 概述
IEEE754 标准是 IEEE 对浮点数表示的规范,目的在于统一浮点数的编码,提高浮点运算程序的可移植性。
IEEE754有3种浮点数格式:单精度、双精度、扩展双精度。
每种格式皆由3部分组成: 符号位(s)、指数(e)和尾数(m)。
single-precision : | 31 | 30:23 | 22:0 | (Ns=1, Ne=8, Nm=23) double-precision: | 63 | 62:52 | 51:0 | (Ns=1, Ne=11, Nm=52) double-extended: | 79 | 78:64 | 63:0 | (以x86之80位为例)
所表示值按指数域分为归一化值和未归一化值。
IEEE754-2008 标准引入了 半精度浮点(Half-Precision Float) float16 类型:'
gcc 中在 arm/AArch64 (64-bit execution state of the ARMv8 ISA) 中支持这个类型 __fp16,ARM 编译时带参数 -mfp16-format=ieee 即可,AArch64 无需此参数。
arm 需包头文件 <arm_fp16.h>,编译是带参数 -mfpu=neon-fp16 -mfloat-abi=softfp
- -mfp16-format=ieee, selects the IEEE 754-2008 format. Normalized values in the range of 2^{-14} to 65504. There are 11 bits of significand precision, approximately 3 decimal digits
- -mfp16-format=alternative, selects the ARM alternative format. Normalized values in the range of 2^{-14} to 131008. Similar to the IEEE format, but does not support infinities or NaNs
2 归一化值
当 e != 0 && e != ~0 (全0与全1)所表示值为归一化值
V = (-1)^s * 2^E * (M+1)
其中 E = e - Bias, Bias = 2^(Ne-1)-1
如单精度浮点数 Bias = 127, V = (-1)^s * 2^(e-127) * (M+1)
3 未归一化值
当 e == 0 || e == ~0 时,所表示值为未归一化值
- 1. e == 0
m == 0, s == 0 ---> +0.0 m == 0, s == 1 ---> -0.0 m != 0 则V = (-1)^s * 2^E * M,其中E = 1 - Bias, Bias = 2^(Ne-1)-1
如单精度浮点数的话,e==0, m!=0, 则 E = 1-127 = -126
- 2. e == ~0
m == 0, s == 0 ---> +INFINITY m == 0, s == 1 ---> -INFINITY 如果 m != 0 ----> NaN, Not a Number
例1 二进制单精度浮点数转十进制数
0x80480000 1000 0000 0100 1000 0000 0000 0000 0000
1 00000000 10010000000000000000000
s = 1
e = 0, E = 1 - 127 = -126
因e == 0,则:尾数部分M为(无须加1):
0.10010000000000000000000=0.5625
该浮点数的十进制为:
(-1)^1 * 2^(-126) * 0.5625 = -6.612156e-39
可以使用如下 C 程序验证之:
#include <stdio.h> union FI { float f; struct { unsigned char b0; unsigned char b1; unsigned char b2; unsigned char b3; }; }u; int main() { u.b3 = 0x80; u.b2 = 0x48; u.b1 = 0x00; u.b0 = 0x00; printf ("x = %e ", u.f); return 0; }
更简洁的:
#include <stdio.h> int main() { int x = 0x80480000; float y = *(float *)&x; printf ("x = %e ", y); return 0; }
4 Reference