我正在阅读一本关于C语言的书，其中讨论了浮点数的范围，作者给出了如下表格:
Type Smallest Positive Value Largest value Precision
==== ======================= ============= =========
float 1.17549 x 10^-38 3.40282 x 10^38 6 digits
double 2.22507 x 10^-308 1.79769 x 10^308 15 digits
我不知道“最小正数”和“最大值”这两列中的数字从哪里来。
32位浮点数有23 + 1位的尾数和8位的指数(-126到127被使用)，因此您可以表示的最大数字为:
(1 + 1 / 2 + ... 1 / (2 ^ 23)) * (2 ^ 127) =
(2 ^ 23 + 2 ^ 23 + .... 1) * (2 ^ (127 - 23)) =
(2 ^ 24 - 1) * (2 ^ 104) ~= 3.4e38
p + BIAS，其中BIAS是127，尾数有23位和第24位隐藏位，假定为1。这个隐藏位是尾数的最高有效位 (MSB)，指数必须被选择为使其为1。01000000000000000000000000000000，即 1x2^-126 = 1.17549435E-38
最大值是 011111111111111111111111111111111，尾数为2 * (1 - 1/65536)，指数为127，得到(1 - 1 / 65536) * 2 ^ 128 = 3.40277175E38。无穷大、NaN和次正规数
这些是其他答案中尚未提到的重要注意事项。
首先阅读IEEE 754和次正规数的介绍：什么是次正规浮点数？
然后，对于单精度浮点数（32位）：
IEEE 754 says that if the exponent is all ones (0xFF == 255), then it represents either NaN or Infinity.
This is why the largest non-infinite number has exponent 0xFE == 254 and not 0xFF.
Then with the bias, it becomes:
254 - 127 == 127
FLT_MIN is the smallest normal number. But there are smaller subnormal ones! Those take up the -127 exponent slot.
#include <assert.h>
#include <float.h>
#include <inttypes.h>
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
float float_from_bytes(
uint32_t sign,
uint32_t exponent,
uint32_t fraction
) {
uint32_t bytes;
bytes = 0;
bytes |= sign;
bytes <<= 8;
bytes |= exponent;
bytes <<= 23;
bytes |= fraction;
return *(float*)&bytes;
}
int main(void) {
/* All 1 exponent and non-0 fraction means NaN.
* There are of course many possible representations,
* and some have special semantics such as signalling vs not.
*/
assert(isnan(float_from_bytes(0, 0xFF, 1)));
assert(isnan(NAN));
printf("nan = %e\n", NAN);
/* All 1 exponent and 0 fraction means infinity. */
assert(INFINITY == float_from_bytes(0, 0xFF, 0));
assert(isinf(INFINITY));
printf("infinity = %e\n", INFINITY);
/* ANSI C defines FLT_MAX as the largest non-infinite number. */
assert(FLT_MAX == 0x1.FFFFFEp127f);
/* Not 0xFF because that is infinite. */
assert(FLT_MAX == float_from_bytes(0, 0xFE, 0x7FFFFF));
assert(!isinf(FLT_MAX));
assert(FLT_MAX < INFINITY);
printf("largest non infinite = %e\n", FLT_MAX);
/* ANSI C defines FLT_MIN as the smallest non-subnormal number. */
assert(FLT_MIN == 0x1.0p-126f);
assert(FLT_MIN == float_from_bytes(0, 1, 0));
assert(isnormal(FLT_MIN));
printf("smallest normal = %e\n", FLT_MIN);
/* The smallest non-zero subnormal number. */
float smallest_subnormal = float_from_bytes(0, 0, 1);
assert(smallest_subnormal == 0x0.000002p-126f);
assert(0.0f < smallest_subnormal);
assert(!isnormal(smallest_subnormal));
printf("smallest subnormal = %e\n", smallest_subnormal);
return EXIT_SUCCESS;
}
编译并运行：
gcc -ggdb3 -O0 -std=c11 -Wall -Wextra -Wpedantic -Werror -o subnormal.out subnormal.c
./subnormal.out
输出：
nan = nan
infinity = inf
largest non infinite = 3.402823e+38
smallest normal = 1.175494e-38
smallest subnormal = 1.401298e-45
这是类型指数部分大小的结果，例如在IEEE 754中。您可以使用float.h中的FLT_MAX、FLT_MIN、DBL_MAX和DBL_MIN来检查大小。