C++关于double精度失真的原因？

2014-11-25 10:27:15 856 阅读 AI 写同款 · GEO 优化AI 选题数据监测一键分发 ›

C++浮点精度问题：从根源到实战的完整解决方案浮点数是计算机中表示实数的主流方式，遵循IEEE 754标准，其核心原理是利用有限的二进制位来近似无限的实数。由于二进制无法精确表示某些十进制小数，这种近似性导致了固有的精度丢失问题。在工程实践中，浮点误差的累积和放大可能引发灾难性后果，例如在图形渲染中造成深度冲突，在金融计算中导致金额比较错误，或在科学计算中使迭代算法无法收敛。理解误差的传播机制是设计健壮系统的关键。本文聚焦于C++中的浮点精度挑战，深入剖析了二进制表示的固有缺陷，并系统性地介绍了容差比较、提升精度与改变数据表示、以及设计数值稳定算法等三阅读详情

	int b = 256;

		double a = 0.090196078431372548;

		a = a * (b - 1);

		int index = (int)a;

		printf("a=%lf, b=%d\n", a, b);

我这样定义代码，执行到第三步的时候a=23.00000000000000，为什么小数部分全变0了呢？望大牛解答。编译器是VS2013

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赵4老师 2014-11-25

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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
float f;
double d;
void showf() {
    char bs[65];
    char b[65];
    unsigned char *p;
    char e[12];
    char *t;
    int ex;

    printf("\nf=%.6f\n",f);
    p=(unsigned char *)&f;
    printf("hex=%02X %02X %02X %02X\n",p[3],p[2],p[1],p[0]);
    ltoa(*(long *)&f,b,2);
    sprintf(bs,"%032s",b);
    printf("bin=%s\n",bs);
    printf("bin=%.1s %.8s   %s\n",bs,bs+1,bs+9);
    strncpy(e,bs+1,8);e[8]=0;
    ex=strtol(e,&t,2);
    printf("    %c %-4d-127 1.%s\n",(bs[0]=='0')?'+':'-',ex,bs+9);
    ex-=127;
    printf("    %c %-8d 1.%s\n",(bs[0]=='0')?'+':'-',ex,bs+9);
}
void showd() {
    char bs[65];
    char b[65];
    unsigned char *p;
    char e[12];
    char *t;
    int ex;

    printf("\nd=%.15lf\n",d);
    p=(unsigned char *)&d;
    printf("hex=%02X %02X %02X %02X %02X %02X %02X %02X\n",p[7],p[6],p[5],p[4],p[3],p[2],p[1],p[0]);
    _i64toa(*(__int64 *)&d,b,2);
    sprintf(bs,"%064s",b);
    printf("bin=%s\n",bs);
    printf("bin=%.1s %.11s   %s\n",bs,bs+1,bs+12);
    strncpy(e,bs+1,11);e[11]=0;
    ex=strtol(e,&t,2);
    printf("    %c %-6d-1023 1.%s\n",(bs[0]=='0')?'+':'-',ex,bs+12);
    ex-=1023;
    printf("    %c %-11d 1.%s\n",(bs[0]=='0')?'+':'-',ex,bs+12);
}
int main() {

    for (f=0.0f;f<=2.2f;f+=0.2f) showf();
    printf("------------\n");
    f=2.0f;showf();
    printf("-----------------------------------\n");
    for (d=0.0 ;d<=2.2 ;d+=0.2 ) showd();
    printf("------------\n");
    d=2.0 ;showd();

    return 0;
}
//
//f=0.000000
//hex=00 00 00 00
//bin=00000000000000000000000000000000
//bin=0 00000000   00000000000000000000000
//  + 0   -127 1.00000000000000000000000
//  + -127     1.00000000000000000000000
//
//f=0.200000
//hex=3E 4C CC CD
//bin=00111110010011001100110011001101
//bin=0 01111100   10011001100110011001101
//  + 124 -127 1.10011001100110011001101
//  + -3       1.10011001100110011001101
//
//f=0.400000
//hex=3E CC CC CD
//bin=00111110110011001100110011001101
//bin=0 01111101   10011001100110011001101
//  + 125 -127 1.10011001100110011001101
//  + -2       1.10011001100110011001101
//
//f=0.600000
//hex=3F 19 99 9A
//bin=00111111000110011001100110011010
//bin=0 01111110   00110011001100110011010
//  + 126 -127 1.00110011001100110011010
//  + -1       1.00110011001100110011010
//
//f=0.800000
//hex=3F 4C CC CD
//bin=00111111010011001100110011001101
//bin=0 01111110   10011001100110011001101
//  + 126 -127 1.10011001100110011001101
//  + -1       1.10011001100110011001101
//
//f=1.000000
//hex=3F 80 00 00
//bin=00111111100000000000000000000000
//bin=0 01111111   00000000000000000000000
//  + 127 -127 1.00000000000000000000000
//  + 0        1.00000000000000000000000
//
//f=1.200000
//hex=3F 99 99 9A
//bin=00111111100110011001100110011010
//bin=0 01111111   00110011001100110011010
//  + 127 -127 1.00110011001100110011010
//  + 0        1.00110011001100110011010
//
//f=1.400000
//hex=3F B3 33 34
//bin=00111111101100110011001100110100
//bin=0 01111111   01100110011001100110100
//  + 127 -127 1.01100110011001100110100
//  + 0        1.01100110011001100110100
//
//f=1.600000
//hex=3F CC CC CE
//bin=00111111110011001100110011001110
//bin=0 01111111   10011001100110011001110
//  + 127 -127 1.10011001100110011001110
//  + 0        1.10011001100110011001110
//
//f=1.800000
//hex=3F E6 66 68
//bin=00111111111001100110011001101000
//bin=0 01111111   11001100110011001101000
//  + 127 -127 1.11001100110011001101000
//  + 0        1.11001100110011001101000
//
//f=2.000000
//hex=40 00 00 01
//bin=01000000000000000000000000000001
//bin=0 10000000   00000000000000000000001
//  + 128 -127 1.00000000000000000000001
//  + 1        1.00000000000000000000001
//------------
//
//f=2.000000
//hex=40 00 00 00
//bin=01000000000000000000000000000000
//bin=0 10000000   00000000000000000000000
//  + 128 -127 1.00000000000000000000000
//  + 1        1.00000000000000000000000
//-----------------------------------
//
//d=0.000000000000000
//hex=00 00 00 00 00 00 00 00
//bin=0000000000000000000000000000000000000000000000000000000000000000
//bin=0 00000000000   0000000000000000000000000000000000000000000000000000
//  + 0     -1023 1.0000000000000000000000000000000000000000000000000000
//  + -1023       1.0000000000000000000000000000000000000000000000000000
//
//d=0.200000000000000
//hex=3F C9 99 99 99 99 99 9A
//bin=0011111111001001100110011001100110011001100110011001100110011010
//bin=0 01111111100   1001100110011001100110011001100110011001100110011010
//  + 1020  -1023 1.1001100110011001100110011001100110011001100110011010
//  + -3          1.1001100110011001100110011001100110011001100110011010
//
//d=0.400000000000000
//hex=3F D9 99 99 99 99 99 9A
//bin=0011111111011001100110011001100110011001100110011001100110011010
//bin=0 01111111101   1001100110011001100110011001100110011001100110011010
//  + 1021  -1023 1.1001100110011001100110011001100110011001100110011010
//  + -2          1.1001100110011001100110011001100110011001100110011010
//
//d=0.600000000000000
//hex=3F E3 33 33 33 33 33 34
//bin=0011111111100011001100110011001100110011001100110011001100110100
//bin=0 01111111110   0011001100110011001100110011001100110011001100110100
//  + 1022  -1023 1.0011001100110011001100110011001100110011001100110100
//  + -1          1.0011001100110011001100110011001100110011001100110100
//
//d=0.800000000000000
//hex=3F E9 99 99 99 99 99 9A
//bin=0011111111101001100110011001100110011001100110011001100110011010
//bin=0 01111111110   1001100110011001100110011001100110011001100110011010
//  + 1022  -1023 1.1001100110011001100110011001100110011001100110011010
//  + -1          1.1001100110011001100110011001100110011001100110011010
//
//d=1.000000000000000
//hex=3F F0 00 00 00 00 00 00
//bin=0011111111110000000000000000000000000000000000000000000000000000
//bin=0 01111111111   0000000000000000000000000000000000000000000000000000
//  + 1023  -1023 1.0000000000000000000000000000000000000000000000000000
//  + 0           1.0000000000000000000000000000000000000000000000000000
//
//d=1.200000000000000
//hex=3F F3 33 33 33 33 33 33
//bin=0011111111110011001100110011001100110011001100110011001100110011
//bin=0 01111111111   0011001100110011001100110011001100110011001100110011
//  + 1023  -1023 1.0011001100110011001100110011001100110011001100110011
//  + 0           1.0011001100110011001100110011001100110011001100110011
//
//d=1.400000000000000
//hex=3F F6 66 66 66 66 66 66
//bin=0011111111110110011001100110011001100110011001100110011001100110
//bin=0 01111111111   0110011001100110011001100110011001100110011001100110
//  + 1023  -1023 1.0110011001100110011001100110011001100110011001100110
//  + 0           1.0110011001100110011001100110011001100110011001100110
//
//d=1.600000000000000
//hex=3F F9 99 99 99 99 99 99
//bin=0011111111111001100110011001100110011001100110011001100110011001
//bin=0 01111111111   1001100110011001100110011001100110011001100110011001
//  + 1023  -1023 1.1001100110011001100110011001100110011001100110011001
//  + 0           1.1001100110011001100110011001100110011001100110011001
//
//d=1.800000000000000
//hex=3F FC CC CC CC CC CC CC
//bin=0011111111111100110011001100110011001100110011001100110011001100
//bin=0 01111111111   1100110011001100110011001100110011001100110011001100
//  + 1023  -1023 1.1100110011001100110011001100110011001100110011001100
//  + 0           1.1100110011001100110011001100110011001100110011001100
//
//d=2.000000000000000
//hex=3F FF FF FF FF FF FF FF
//bin=0011111111111111111111111111111111111111111111111111111111111111
//bin=0 01111111111   1111111111111111111111111111111111111111111111111111
//  + 1023  -1023 1.1111111111111111111111111111111111111111111111111111
//  + 0           1.1111111111111111111111111111111111111111111111111111
//
//d=2.200000000000000
//hex=40 01 99 99 99 99 99 99
//bin=0100000000000001100110011001100110011001100110011001100110011001
//bin=0 10000000000   0001100110011001100110011001100110011001100110011001
//  + 1024  -1023 1.0001100110011001100110011001100110011001100110011001
//  + 1           1.0001100110011001100110011001100110011001100110011001
//------------
//
//d=2.000000000000000
//hex=40 00 00 00 00 00 00 00
//bin=0100000000000000000000000000000000000000000000000000000000000000
//bin=0 10000000000   0000000000000000000000000000000000000000000000000000
//  + 1024  -1023 1.0000000000000000000000000000000000000000000000000000
//  + 1           1.0000000000000000000000000000000000000000000000000000
//

赵4老师 2014-11-25