Question

我正在尝试在没有FPU的体系结构上实现CORDIC算法，以单精度近似正弦函数。我将实现中的结果与标准C数学函数中的结果进行比较。我尝试了两种实现方式：1）直接使用浮点运算，2）将输入转换为定点并使用基于整数的运算。我比较从sinf（），sin（）和sin（）转换为float所获得的结果。比较是基于将结果的十六进制表示形式与数学函数的期望值进行比较。

在（1）中，实现使用双精度类型，然后将结果强制转换为浮点型。无论使用CORDIC进行多少次迭代，我的计算值始终至少有一个十六进制数字。

在（2）中，最初将输入映射到32位整数。该误差与（1）相同。只有将定点大小增加到64位（迭代次数增加到64位）后，精度才能提高。但是，仍然存在一些算法不精确的输入范围。如果我将不动点的大小增加到128位（迭代次数增加到128位），则可能足以获取精确的值，但这是完全不切实际的。

（1）中的算法是https://www.jjj.de/fxt/fxtbook.pdf本书的修改版本

#include <math.h>
#include <stdio.h>
const double cordic_1K = 0.6072529350088812561694467525049282631123908521500897724;
double *cordic_ctab;

void make_cordic_ctab(ulong na)
{
    double s = 1.0;
    for (ulong k=0; k<na; ++k)
    {
        cordic_ctab[k] = atan(s);
        s *= 0.5;
    }
}


void cordic(int theta, double* s, double* c, int n)
{
    double x, y, z, v;
    double tx, ty, tz;
    double d;

    x = cordic_1K;
    y = 0;
    z = theta;
    v = 1.0;

    for (int k = 0; k < n; ++k) {
        d = (z >= 0 ? +1 : -1);
        tx = x - d * v * y;
        ty = y + d * v * x;
        tz = z - d * cordic_ctab[k];
        x = tx;
        y = ty;
        z = tz;
        v *= 0.5;
    }
    *c = x;
    *s = y;
}

（2）a中的算法是在http://www.dcs.gla.ac.uk/~jhw/cordic/

上找到的修改版本。

#include <math.h>
#include <stdio.h>
#define cordic_1K 0x26dd3b6a10d79600
#define CORDIC_NTAB 64

void cordic(long theta, long *s, long *c, int n) {
  long d, tx, ty, tz;
  long x = cordic_1K, y = 0, z = theta;
  n = (n > CORDIC_NTAB) ? CORDIC_NTAB : n;

  for (int k = 0; k < n; ++k) {
    d = z >= 0 ? 0 : -1;
    tx = x - (((y >> k) ^ d) - d);
    ty = y + (((x >> k) ^ d) - d);
    tz = z - ((cordic_ctab[k] ^ d) - d);
    x = tx;
    y = ty;
    z = tz;
  }

  *c = x;
  *s = y;
}

CORDIC表类似地以bit = 64生成。

（1）的测试如下：

int main(int argc, char **argv) {
  float angle;
  long s, c;
  int failed = 0;

  cordic_ctab = (double*)malloc(sizeof(double) * 64);
  make_cordic_ctab(64);

  for (int i = 0; i < step; i++) {
    angle = (i / step) * M_PI / 4;

    cordic(angle, &s, &c, 64);
    float result = s;
    float expected = sinf(angle);

    if (angle < pow(2, -27))
      result = angle;

    if (memcmp(&result, &expected, sizeof(float)) != 0) {
      failed += 1;
      printf("%e : %e\n", result, expected);
      printf("0x%x : 0x%x\n", *((unsigned int *)&result),
             *((unsigned int *)&expected));
      printf("\n");

    }
  }
  printf("failed:%d\n", failed);
}

（2）的测试如下：

int main(int argc, char **argv) {
  float angle;
  long s, c;
  int failed = 0;
  double mul = 4611686018427387904.000000;
  double step = 1000000000.0;

  for (int i = 0; i < step; i++) {
    angle = (i / step) * M_PI / 4;

    cordic((angle * mul), &s, &c, 64);
    float result = s / mul;
    float expected = sinf(angle);

    if (angle < pow(2, -27))
      result = angle;

    if (memcmp(&result, &expected, sizeof(float)) != 0) {
      failed += 1;
      printf("%e : %e\n", result, expected);
      printf("0x%x : 0x%x\n", *((unsigned int *)&result),
             *((unsigned int *)&expected));
      printf("\n");

    }
  }
  printf("failed:%d\n", failed);
}

对于CORDIC，有没有我不考虑的东西吗？是否有可能完全不适合使用CORDIC并应考虑其他方法？

Answer 1

我试了一下，但是正如评论中提到的那样，您不能指望精确的位匹配，因为数学测角学通常基于Chebyshev多项式。另外，您还没有定义position: absolute常量。经过一番搜索后，我设法在 C ++ / VCL中做到了这一点：

cordic_1K

您可以忽略诸如//--------------------------------------------------------------------------- // IEEE 754 single masks const DWORD _f32_sig =0x80000000; // sign const DWORD _f32_exp =0x7F800000; // exponent const DWORD _f32_exp_sig=0x40000000; // exponent sign const DWORD _f32_exp_bia=0x3F800000; // exponent bias const DWORD _f32_exp_lsb=0x00800000; // exponent LSB const DWORD _f32_exp_pos= 23; // exponent LSB bit position const DWORD _f32_man =0x007FFFFF; // mantisa const DWORD _f32_man_msb=0x00400000; // mantisa MSB const DWORD _f32_man_bits= 23; // mantisa bits const float _f32_lsb = 3.4e-38; // abs min number //--------------------------------------------------------------------------- float CORDIC32_atan[_f32_man_bits+1]; void f32_sincos(float &s,float &c,float a) { int k; float x,y=0.0,v=1.0,d,tx,ty,ta; x=0.6072529350088812561694; // cordic_1K for (k=0;k<=_f32_man_bits;k++) { d =(a>=0.0?+1.0:-1.0); tx=x-d*v*y; ty=y+d*v*x; ta=a-d*CORDIC32_atan[k]; x=tx; y=ty; a=ta; v*=0.5; } c=x; s=y; } //--------------------------------------------------------------------------- double CORDIC64_atan[_f32_man_bits+1]; void f64_sincos(float &s,float &c,double a) { int k; double x,y=0.0,v=1.0,d,tx,ty,ta; x=0.6072529350088812561694; // cordic_1K for (k=0;k<=_f32_man_bits;k++) { d =(a>=0.0?+1.0:-1.0); tx=x-d*v*y; ty=y+d*v*x; ta=a-d*CORDIC64_atan[k]; x=tx; y=ty; a=ta; v*=0.5; } c=x; s=y; } //--------------------------------------------------------------------------- //--- Builder: -------------------------------------------------------------- //--------------------------------------------------------------------------- __fastcall TForm1::TForm1(TComponent* Owner):TForm(Owner) { int i; float s0,c0,s1,c1,s2,c2,d32,d64,D32,D64,x; AnsiString txt=""; // init CORDIC tables for (x=1.0,i=0;i<=_f32_man_bits;i++,x*=0.5) { CORDIC32_atan[i]=atan(x); CORDIC64_atan[i]=atan(double(x)); } // 32 bit D32=0.0; D64=0.0; for (x=-0.5*M_PI;x<=+0.5*M_PI;x+=0.025) { s0=sin(x); c0=cos(x); f32_sincos(s1,c1,x); d32=fabs(s1-s0); if (D32<d32) D32=d32; f64_sincos(s2,c2,x); d64=fabs(s2-s0); if (D64<d64) D64=d64; if (d32+d64>1e-16) { txt+=AnsiString().sprintf("sin(%2.5f) == %2.5f != %2.5f != %2.5f | %.10f %.10f\r\n",x,s0,s1,s2,d32,d64); f32_sincos(s0,c0,x); // debug breakpoint f64_sincos(s2,c2,x); } } txt=AnsiString().sprintf("max err: %.10f %.10f\r\n",D32,D64)+txt; mm_log->Lines->Add(txt); } //-------------------------------------------------------------------------之类的 VCL 东西（或将其移植到您的环境中），仅用于打印结果...

代码为我提供了以下输出：

AnsiString

如您所见，64位表与数学max err: 0.0000002384 0.0000001192 sin(-1.54580) == -0.99969 != -0.99969 != -0.99969 | 0.0000000596 0.0000000000 sin(-1.52080) == -0.99875 != -0.99875 != -0.99875 | 0.0000001192 0.0000000000 sin(-1.49580) == -0.99719 != -0.99719 != -0.99719 | 0.0000001192 0.0000000000 sin(-1.44580) == -0.99220 != -0.99220 != -0.99220 | 0.0000000596 0.0000000000 sin(-1.42080) == -0.98877 != -0.98877 != -0.98877 | 0.0000000596 0.0000000596 sin(-1.39580) == -0.98473 != -0.98473 != -0.98473 | 0.0000001192 0.0000000596 sin(-1.37080) == -0.98007 != -0.98007 != -0.98007 | 0.0000000000 0.0000000596 sin(-1.34580) == -0.97479 != -0.97479 != -0.97479 | 0.0000000596 0.0000000000 sin(-1.27080) == -0.95534 != -0.95534 != -0.95534 | 0.0000001192 0.0000000000 sin(-1.24580) == -0.94765 != -0.94765 != -0.94765 | 0.0000000596 0.0000000596 sin(-1.22080) == -0.93937 != -0.93937 != -0.93937 | 0.0000000596 0.0000000000 sin(-1.19580) == -0.93051 != -0.93051 != -0.93051 | 0.0000000596 0.0000000596 sin(-1.17080) == -0.92106 != -0.92106 != -0.92106 | 0.0000000596 0.0000000000 sin(-1.14580) == -0.91104 != -0.91104 != -0.91104 | 0.0000001192 0.0000000596 sin(-1.12080) == -0.90045 != -0.90045 != -0.90045 | 0.0000001192 0.0000000000 sin(-1.07080) == -0.87758 != -0.87758 != -0.87758 | 0.0000001788 0.0000000596 sin(-1.04580) == -0.86532 != -0.86532 != -0.86532 | 0.0000001788 0.0000000596 sin(-1.02080) == -0.85252 != -0.85252 != -0.85252 | 0.0000001192 0.0000000596 sin(-0.99580) == -0.83919 != -0.83919 != -0.83919 | 0.0000000000 0.0000000596 sin(-0.97080) == -0.82534 != -0.82534 != -0.82534 | 0.0000001192 0.0000000596 sin(-0.94580) == -0.81096 != -0.81096 != -0.81096 | 0.0000000596 0.0000000596 sin(-0.92080) == -0.79608 != -0.79608 != -0.79608 | 0.0000000000 0.0000000596 sin(-0.89580) == -0.78071 != -0.78071 != -0.78071 | 0.0000001788 0.0000000596 sin(-0.87080) == -0.76484 != -0.76484 != -0.76484 | 0.0000000596 0.0000000000 sin(-0.84580) == -0.74850 != -0.74850 != -0.74850 | 0.0000000596 0.0000000000 sin(-0.82080) == -0.73169 != -0.73169 != -0.73169 | 0.0000001192 0.0000000000 sin(-0.79580) == -0.71442 != -0.71442 != -0.71442 | 0.0000000596 0.0000000000 sin(-0.77080) == -0.69671 != -0.69671 != -0.69671 | 0.0000000596 0.0000000596 sin(-0.74580) == -0.67856 != -0.67856 != -0.67856 | 0.0000000000 0.0000000596 sin(-0.72080) == -0.65998 != -0.65998 != -0.65998 | 0.0000001192 0.0000000596 sin(-0.69580) == -0.64100 != -0.64100 != -0.64100 | 0.0000000596 0.0000000000 sin(-0.67080) == -0.62161 != -0.62161 != -0.62161 | 0.0000001788 0.0000000596 sin(-0.64580) == -0.60184 != -0.60184 != -0.60184 | 0.0000000596 0.0000001192 sin(-0.62080) == -0.58168 != -0.58168 != -0.58168 | 0.0000000596 0.0000001192 sin(-0.59580) == -0.56117 != -0.56117 != -0.56117 | 0.0000000596 0.0000000596 sin(-0.57080) == -0.54030 != -0.54030 != -0.54030 | 0.0000001788 0.0000001192 sin(-0.54580) == -0.51910 != -0.51910 != -0.51910 | 0.0000001788 0.0000001192 sin(-0.52080) == -0.49757 != -0.49757 != -0.49757 | 0.0000000596 0.0000000894 sin(-0.49580) == -0.47573 != -0.47573 != -0.47573 | 0.0000000894 0.0000000596 sin(-0.47080) == -0.45360 != -0.45360 != -0.45360 | 0.0000000894 0.0000000298 sin(-0.44580) == -0.43118 != -0.43118 != -0.43118 | 0.0000000298 0.0000000298 sin(-0.42080) == -0.40849 != -0.40849 != -0.40849 | 0.0000000894 0.0000000596 sin(-0.39580) == -0.38554 != -0.38554 != -0.38554 | 0.0000001192 0.0000000596 sin(-0.37080) == -0.36236 != -0.36236 != -0.36236 | 0.0000000298 0.0000000000 sin(-0.34580) == -0.33895 != -0.33895 != -0.33895 | 0.0000000000 0.0000000298 sin(-0.32080) == -0.31532 != -0.31532 != -0.31532 | 0.0000000596 0.0000000000 sin(-0.29580) == -0.29150 != -0.29150 != -0.29150 | 0.0000000596 0.0000000596 sin(-0.27080) == -0.26750 != -0.26750 != -0.26750 | 0.0000000894 0.0000001192 sin(-0.24580) == -0.24333 != -0.24333 != -0.24333 | 0.0000000894 0.0000001192 sin(-0.22080) == -0.21901 != -0.21901 != -0.21901 | 0.0000000745 0.0000000894 sin(-0.19580) == -0.19455 != -0.19455 != -0.19455 | 0.0000000894 0.0000000596 sin(-0.17080) == -0.16997 != -0.16997 != -0.16997 | 0.0000001043 0.0000000894 sin(-0.14580) == -0.14528 != -0.14528 != -0.14528 | 0.0000000894 0.0000000894 sin(-0.12080) == -0.12050 != -0.12050 != -0.12050 | 0.0000000596 0.0000000671 sin(-0.09580) == -0.09565 != -0.09565 != -0.09565 | 0.0000000522 0.0000000522 sin(-0.07080) == -0.07074 != -0.07074 != -0.07074 | 0.0000000075 0.0000000224 sin(-0.04580) == -0.04578 != -0.04578 != -0.04578 | 0.0000000447 0.0000000335 sin(-0.02080) == -0.02080 != -0.02080 != -0.02080 | 0.0000000596 0.0000000577 sin(0.00420) == 0.00420 != 0.00420 != 0.00420 | 0.0000000545 0.0000000549 sin(0.02920) == 0.02920 != 0.02920 != 0.02920 | 0.0000000447 0.0000000410 sin(0.05420) == 0.05418 != 0.05418 != 0.05418 | 0.0000000149 0.0000000186 sin(0.07920) == 0.07912 != 0.07912 != 0.07912 | 0.0000000224 0.0000000373 sin(0.10420) == 0.10401 != 0.10401 != 0.10401 | 0.0000000820 0.0000000745 sin(0.12920) == 0.12884 != 0.12884 != 0.12884 | 0.0000001043 0.0000000894 sin(0.15420) == 0.15359 != 0.15359 != 0.15359 | 0.0000001043 0.0000001043 sin(0.17920) == 0.17825 != 0.17825 != 0.17825 | 0.0000000447 0.0000000745 sin(0.20420) == 0.20279 != 0.20279 != 0.20279 | 0.0000000596 0.0000000745 sin(0.22920) == 0.22720 != 0.22720 != 0.22720 | 0.0000001043 0.0000001043 sin(0.25420) == 0.25147 != 0.25147 != 0.25147 | 0.0000001192 0.0000000894 sin(0.27920) == 0.27559 != 0.27559 != 0.27559 | 0.0000000000 0.0000000596 sin(0.30420) == 0.29953 != 0.29953 != 0.29953 | 0.0000000596 0.0000000298 sin(0.32920) == 0.32329 != 0.32329 != 0.32329 | 0.0000000596 0.0000000596 sin(0.35420) == 0.34684 != 0.34684 != 0.34684 | 0.0000000298 0.0000000298 sin(0.37920) == 0.37018 != 0.37018 != 0.37018 | 0.0000000298 0.0000000298 sin(0.40420) == 0.39329 != 0.39329 != 0.39329 | 0.0000001788 0.0000000596 sin(0.42920) == 0.41615 != 0.41615 != 0.41615 | 0.0000000894 0.0000000894 sin(0.45420) == 0.43875 != 0.43875 != 0.43875 | 0.0000000298 0.0000000000 sin(0.47920) == 0.46107 != 0.46107 != 0.46107 | 0.0000000596 0.0000000298 sin(0.50420) == 0.48311 != 0.48311 != 0.48311 | 0.0000000596 0.0000000596 sin(0.52920) == 0.50485 != 0.50485 != 0.50485 | 0.0000001788 0.0000000596 sin(0.55420) == 0.52627 != 0.52627 != 0.52627 | 0.0000002384 0.0000001192 sin(0.57920) == 0.54736 != 0.54736 != 0.54736 | 0.0000001192 0.0000000596 sin(0.60420) == 0.56811 != 0.56811 != 0.56811 | 0.0000000596 0.0000000596 sin(0.62920) == 0.58850 != 0.58850 != 0.58850 | 0.0000000596 0.0000001192 sin(0.65420) == 0.60853 != 0.60853 != 0.60853 | 0.0000001192 0.0000001192 sin(0.67920) == 0.62817 != 0.62817 != 0.62817 | 0.0000000596 0.0000001192 sin(0.70420) == 0.64743 != 0.64743 != 0.64743 | 0.0000000596 0.0000000000 sin(0.72920) == 0.66628 != 0.66628 != 0.66628 | 0.0000000596 0.0000000000 sin(0.75420) == 0.68471 != 0.68471 != 0.68471 | 0.0000000596 0.0000000000 sin(0.77920) == 0.70271 != 0.70271 != 0.70271 | 0.0000000596 0.0000000000 sin(0.82920) == 0.73739 != 0.73739 != 0.73739 | 0.0000000596 0.0000000000 sin(0.85420) == 0.75405 != 0.75405 != 0.75405 | 0.0000001192 0.0000000000 sin(0.87920) == 0.77023 != 0.77023 != 0.77023 | 0.0000001192 0.0000000000 sin(0.90420) == 0.78593 != 0.78593 != 0.78593 | 0.0000000596 0.0000000596 sin(0.92920) == 0.80114 != 0.80114 != 0.80114 | 0.0000000596 0.0000001192 sin(0.95420) == 0.81585 != 0.81585 != 0.81585 | 0.0000001788 0.0000000596 sin(0.97920) == 0.83005 != 0.83005 != 0.83005 | 0.0000000000 0.0000000596 sin(1.00420) == 0.84373 != 0.84373 != 0.84373 | 0.0000001788 0.0000000000 sin(1.02920) == 0.85689 != 0.85689 != 0.85689 | 0.0000000596 0.0000000000 sin(1.05420) == 0.86951 != 0.86951 != 0.86951 | 0.0000001192 0.0000000000 sin(1.12920) == 0.90407 != 0.90407 != 0.90407 | 0.0000000596 0.0000000000 sin(1.15420) == 0.91447 != 0.91447 != 0.91447 | 0.0000000596 0.0000000596 sin(1.17920) == 0.92430 != 0.92430 != 0.92430 | 0.0000001788 0.0000000596 sin(1.20420) == 0.93355 != 0.93355 != 0.93355 | 0.0000000596 0.0000000000 sin(1.25420) == 0.95030 != 0.95030 != 0.95030 | 0.0000000596 0.0000000596 sin(1.27920) == 0.95779 != 0.95779 != 0.95779 | 0.0000001192 0.0000000596 sin(1.30420) == 0.96467 != 0.96467 != 0.96467 | 0.0000000596 0.0000000000 sin(1.35420) == 0.97664 != 0.97664 != 0.97663 | 0.0000000596 0.0000000596 sin(1.45420) == 0.99321 != 0.99321 != 0.99321 | 0.0000000596 0.0000000000 sin(1.47920) == 0.99581 != 0.99581 != 0.99581 | 0.0000001192 0.0000000000 sin(1.50420) == 0.99778 != 0.99778 != 0.99778 | 0.0000000596 0.0000000000 sin(1.52920) == 0.99914 != 0.99914 != 0.99914 | 0.0000000596 0.0000000000 sin(1.55420) == 0.99986 != 0.99986 != 0.99986 | 0.0000000596 0.0000000000的匹配更好，对于32位和sin（2 ^而言，错误高达4 ulp（2 ^ -24）。 -24）用于64位表。由于32位浮点数的尾数为2 ulp位，结果对应于2个最低有效位，因此其最后一个十六进制数字不匹配...

PS 23+1表为：

atan

使用CORDIC进行正弦计算会得出不精确的值

1 个答案: