当另一个公式似乎更有意义时,为什么基于表格的sin近似文献总是使用这个公式？

Question

关于sin用表格计算基本函数的文献参考公式:

sin(x) = sin(Cn) * cos(h) + cos(Cn) * sin(h)

其中x = Cn + h,Cn是针对其恒定sin(Cn)和cos(Cn)已被预先计算并在表中可用的,并且,如果以下半乳糖的方法,Cn已被选择为使得两个sin(Cn)和cos(Cn)密切由浮点数近似.数量h接近0.0.此公式的参考示例是本文(第7页).

我不明白为什么这是有道理的:cos(h)然而,它被计算,对于某些值,至少0.5 ULP可能是错误的h,并且因为它接近1.0,这似乎对结果的准确性有极大的影响.sin(x)以这种方式计算.

我不明白为什么不使用下面的公式:

sin(x) = sin(Cn) + (sin(Cn) * (cos(h) - 1.0) + cos(Cn) * sin(h))

然后两个量(cos(h) - 1.0),并sin(h)可以用,很容易做出准确的,因为它们产生接近零的结果多项式来近似.为价值观sin(Cn) * (cos(h) - 1.0), cos(Cn) * sin(h)并为他们的总和仍然很小,其绝对精度,该总和表示,因此,加入这个量的少量ULPS表达sin(Cn)几乎是正确舍入.

我错过了一些让早期,流行,更简单的公式表现得更好的东西吗？作者是否理所当然地认为读者会理解第一个公式实际上是作为第二个公式实现的？

编辑:示例

用于计算单精度的单精度表sinf(),cosf()可能包含单精度中的以下点:

         f             |        cos f          |       sin f      
-----------------------+-----------------------+---------------------
0.017967 0x1.2660bcp-6 |    0x1.ffead8p-1      |    0x1.265caep-6
                       |    (actual value:)    |    (actual value:)
                       | ~0x1.ffead8000715dp-1 | ~0x1.265cae000e6f9p-6

以下函数是专用的单精度函数0.017967:

float sinf_trad(float x)
{
  float h = x - 0x1.2660bcp-6f;

  return 0x1.265caep-6f * cos_0(h) + 0x1.ffead8p-1f * sin_0(h);
}

float sinf_new(float x)
{
  float h = x - 0x1.2660bcp-6f;

  return 0x1.265caep-6f + (0x1.265caep-6f * cosm1_0(h) + 0x1.ffead8p-1f * sin_0(h));
}

在0.01f和0.025f之间测试这些功能似乎表明新配方提供了更精确的结果:

$ gcc -std=c99 test.c && ./a.out 
relative error, traditional: 2.169624e-07, new: 1.288049e-07
sum of squares of absolute error, traditional: 6.616202e-12, new: 2.522784e-12

我采取了几个快捷方式,所以请看完整的程序.

Answer 1

下面的实现部分回答了这个问题,因为它是一个单精度的正弦实现,使用问题中建议的公式,精确到0.53 ULP超过[0 ... 1.57],准确到0.5 ULP为99.98%它在这个范围内的争论.

具体来说,我得到输出:

error 285758762/536870912 ULP sin(2.11219326e-01) ref:2.09652290e-01 new:2.09652275e-01 
differences: 176880 / 1070134723

意味着错误永远不会超过ULP的285/536(约0.53 ULP),而176880是错误高于0.5 ULP的参数数量,总共1070134723个参数.

使用普通sin(Cn) * cos(h) + cos(Cn) * sin(h)公式并且只进行单精度计算似乎不可能实现这种结果.该问题中引用的文章暗示"为c0*h扩展精度评估术语",以实现整体准确性.

#include <inttypes.h>
#include <stdint.h>
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <stdlib.h>

float c_cos_sin[][3] = {
  //  0x0.000000000p+0 /* 0.000000 */, 0x1.000000p+0, 0x0.000000p+0,
  //  0x0.00fb76590p+2 /* 0.015348 */, 0x1.fff090p-1, 0x1.f6e7a4p-7,
  //  0x0.01fd02f80p+2 /* 0.031068 */, 0x1.ffc0c0p-1, 0x1.fcee02p-6,
  //  0x0.0302f6280p+2 /* 0.047056 */, 0x1.ff6eeap-1, 0x1.8156aap-5,
  //  0x0.04029a400p+2 /* 0.062659 */, 0x1.fefec8p-1, 0x1.007b94p-4,
  //  0x0.0500a9d80p+2 /* 0.078165 */, 0x1.fe6fcap-1, 0x1.3fd706p-4,
  //  0x0.060215b80p+2 /* 0.093877 */, 0x1.fdbedcp-1, 0x1.7ff4e8p-4,
  //  0x0.070225580p+2 /* 0.109506 */, 0x1.fceee8p-1, 0x1.bfa3fcp-4,
  //  0x0.080460e00p+2 /* 0.125267 */, 0x1.fbfcf6p-1, 0x1.ffc0f6p-4,
  //  0x0.08fed4a00p+2 /* 0.140554 */, 0x1.faf372p-1, 0x1.1ee830p-3,
  //  0x0.0a0054100p+2 /* 0.156270 */, 0x1.f9c2d8p-1, 0x1.3ebd74p-3,
  //  0x0.0afc8eb00p+2 /* 0.171665 */, 0x1.f87978p-1, 0x1.5dd872p-3,
  0x0.0bff5db00p+2 /* 0.187461 */, 0x1.f707b0p-1, 0x1.7dad14p-3,
  0x0.0cfe70200p+2 /* 0.203030 */, 0x1.f57bcep-1, 0x1.9cf438p-3,
  0x0.0e024ef00p+2 /* 0.218891 */, 0x1.f3c87ap-1, 0x1.bcb7a0p-3,
  0x0.0efeab400p+2 /* 0.234294 */, 0x1.f202ecp-1, 0x1.db74a8p-3,
  0x0.10003da00p+2 /* 0.250015 */, 0x1.f014d0p-1, 0x1.fab664p-3,
  0x0.110242c00p+2 /* 0.265763 */, 0x1.ee0660p-1, 0x1.0cf2f4p-2,
  0x0.12055d400p+2 /* 0.281577 */, 0x1.ebd62ap-1, 0x1.1c8a4ap-2,
  0x0.13025de00p+2 /* 0.297019 */, 0x1.e994c2p-1, 0x1.2bb212p-2,
  0x0.13fc96600p+2 /* 0.312292 */, 0x1.e73c4ep-1, 0x1.3a9d34p-2,
  0x0.15014c400p+2 /* 0.328204 */, 0x1.e4abbcp-1, 0x1.4a1472p-2,
  0x0.15fe27a00p+2 /* 0.343637 */, 0x1.e210eep-1, 0x1.58fffep-2,
  0x0.1703b1200p+2 /* 0.359600 */, 0x1.df4050p-1, 0x1.685884p-2,
  0x0.180296e00p+2 /* 0.375158 */, 0x1.dc63e8p-1, 0x1.7736b2p-2,
  0x0.18fc8a600p+2 /* 0.390414 */, 0x1.d9790cp-1, 0x1.85b472p-2,
  0x0.19ffac000p+2 /* 0.406230 */, 0x1.d654fap-1, 0x1.94a1ecp-2,
  0x0.1aff07c00p+2 /* 0.421816 */, 0x1.d31f26p-1, 0x1.a33e6ap-2,
  0x0.1c0162800p+2 /* 0.437585 */, 0x1.cfc21ep-1, 0x1.b1ec42p-2,
  0x0.1cfe63200p+2 /* 0.453027 */, 0x1.cc5a50p-1, 0x1.c0317ep-2,
  0x0.1e0153a00p+2 /* 0.468831 */, 0x1.c8c0f4p-1, 0x1.ceb01ep-2,
  0x0.1efe6d800p+2 /* 0.484279 */, 0x1.c52024p-1, 0x1.dcbe7ep-2,
  0x0.1ffde5600p+2 /* 0.499872 */, 0x1.c15a92p-1, 0x1.ead0fcp-2,
  0x0.20fa9ac00p+2 /* 0.515296 */, 0x1.bd83eap-1, 0x1.f89e82p-2,
  0x0.220491000p+2 /* 0.531529 */, 0x1.b95c6cp-1, 0x1.038212p-1,
  0x0.22ff9c800p+2 /* 0.546851 */, 0x1.b55542p-1, 0x1.0a3d7ap-1,
  0x0.23faafc00p+2 /* 0.562176 */, 0x1.b133aep-1, 0x1.10e916p-1,
  0x0.250a2cc00p+2 /* 0.578746 */, 0x1.ac9ed2p-1, 0x1.180d0ep-1,
  0x0.25fee2800p+2 /* 0.593682 */, 0x1.a863d2p-1, 0x1.1e6bdep-1,
  0x0.2700b4000p+2 /* 0.609418 */, 0x1.a3d498p-1, 0x1.251056p-1,
  0x0.28025e000p+2 /* 0.625144 */, 0x1.9f2b7ap-1, 0x1.2ba13ap-1,
  0x0.28f975400p+2 /* 0.640226 */, 0x1.9a9aa0p-1, 0x1.31db54p-1,
  0x0.29fc6dc00p+2 /* 0.656032 */, 0x1.95b7ecp-1, 0x1.384ef4p-1,
  0x0.2afc27c00p+2 /* 0.671640 */, 0x1.90cb6cp-1, 0x1.3e9a4ap-1,
  0x0.2c0659c00p+2 /* 0.687888 */, 0x1.8b90c6p-1, 0x1.45127ap-1,
  0x0.2d017dc00p+2 /* 0.703216 */, 0x1.868952p-1, 0x1.4b18dep-1,
  0x0.2e04f3c00p+2 /* 0.719052 */, 0x1.813e8cp-1, 0x1.513d70p-1,
  0x0.2efcb8800p+2 /* 0.734175 */, 0x1.7c19bcp-1, 0x1.5706f0p-1,
  0x0.300642800p+2 /* 0.750382 */, 0x1.767dc8p-1, 0x1.5d2464p-1,
  0x0.30ff5cc00p+2 /* 0.765586 */, 0x1.7123d0p-1, 0x1.62cb9cp-1,
  0x0.3204f6c00p+2 /* 0.781553 */, 0x1.6b6d98p-1, 0x1.68a4d6p-1,
  0x0.3303af000p+2 /* 0.797100 */, 0x1.65c70cp-1, 0x1.6e4010p-1,
  0x0.34002f400p+2 /* 0.812511 */, 0x1.601740p-1, 0x1.73b86cp-1,
  0x0.35080ac00p+2 /* 0.828616 */, 0x1.5a0f1cp-1, 0x1.79579cp-1,
  0x0.35fda7800p+2 /* 0.843607 */, 0x1.545d16p-1, 0x1.7e7cc6p-1,
  0x0.37040f800p+2 /* 0.859623 */, 0x1.4e31bep-1, 0x1.83e3aep-1,
  0x0.3800eac00p+2 /* 0.875056 */, 0x1.482b1cp-1, 0x1.89002ap-1,
  0x0.390737c00p+2 /* 0.891066 */, 0x1.41d5b8p-1, 0x1.8e3432p-1,
  0x0.39fce7800p+2 /* 0.906061 */, 0x1.3bd3dep-1, 0x1.92fc2ap-1,
  0x0.3b0596c00p+2 /* 0.922216 */, 0x1.3546c4p-1, 0x1.9808d0p-1,
  0x0.3bf971c00p+2 /* 0.937100 */, 0x1.2f2b58p-1, 0x1.9c979ep-1,
  0x0.3d0275800p+2 /* 0.953275 */, 0x1.2874c8p-1, 0x1.a17120p-1,
  0x0.3e02c4400p+2 /* 0.968919 */, 0x1.21e3cap-1, 0x1.a60740p-1,
  0x0.3ef759000p+2 /* 0.983847 */, 0x1.1b8ec4p-1, 0x1.aa4f02p-1,
  0x0.3ff90a800p+2 /* 0.999575 */, 0x1.14d158p-1, 0x1.aeb732p-1,
  0x0.40f703800p+2 /* 1.015077 */, 0x1.0e1baep-1, 0x1.b2f468p-1,
  0x0.420693000p+2 /* 1.031651 */, 0x1.06dcb2p-1, 0x1.b75f2ap-1,
  0x0.4300fb800p+2 /* 1.046935 */, 0x1.001dcep-1, 0x1.bb5678p-1,
  0x0.440282800p+2 /* 1.062653 */, 0x1.f23bb2p-2, 0x1.bf4efcp-1,
  0x0.44fb18000p+2 /* 1.077826 */, 0x1.e49a58p-2, 0x1.c3095ep-1,
  0x0.45fe26000p+2 /* 1.093637 */, 0x1.d647a4p-2, 0x1.c6cfaap-1,
  0x0.4700de800p+2 /* 1.109428 */, 0x1.c7dba0p-2, 0x1.ca77aap-1,
  0x0.47fd2d800p+2 /* 1.124828 */, 0x1.b9af14p-2, 0x1.cdec48p-1,
  0x0.48fd3c000p+2 /* 1.140456 */, 0x1.ab3138p-2, 0x1.d1515ep-1,
  0x0.49f66d000p+2 /* 1.155666 */, 0x1.9cfd2cp-2, 0x1.d48338p-1,
  0x0.4b05ec000p+2 /* 1.172236 */, 0x1.8d67dcp-2, 0x1.d7deb0p-1,
  0x0.4bfebf800p+2 /* 1.187424 */, 0x1.7f0718p-2, 0x1.dad544p-1,
  0x0.4cfa07800p+2 /* 1.202761 */, 0x1.706b10p-2, 0x1.ddb6e0p-1,
  0x0.4e0324800p+2 /* 1.218942 */, 0x1.60e920p-2, 0x1.e0a1e6p-1,
  0x0.4efdb7800p+2 /* 1.234236 */, 0x1.522b24p-2, 0x1.e34658p-1,
  0x0.4ffb51000p+2 /* 1.249714 */, 0x1.432afap-2, 0x1.e5d57ep-1,
  0x0.50fb6d000p+2 /* 1.265346 */, 0x1.33f0b2p-2, 0x1.e84ce2p-1,
  0x0.5200bc000p+2 /* 1.281295 */, 0x1.24536ep-2, 0x1.eab19cp-1,
  0x0.52fbc0800p+2 /* 1.296616 */, 0x1.1541a6p-2, 0x1.ece01ep-1,
  0x0.54066d800p+2 /* 1.312892 */, 0x1.052cfep-2, 0x1.ef1104p-1,
  0x0.550424000p+2 /* 1.328378 */, 0x1.eb9ff8p-3, 0x1.f1077cp-1,
  0x0.55f93c800p+2 /* 1.343337 */, 0x1.cdd470p-3, 0x1.f2cfeap-1,
  0x0.56fa9d000p+2 /* 1.359046 */, 0x1.ae6e42p-3, 0x1.f49074p-1,
  0x0.57ff02000p+2 /* 1.374939 */, 0x1.8e8e2cp-3, 0x1.f63612p-1,
  0x0.58f813000p+2 /* 1.390141 */, 0x1.6ff8f0p-3, 0x1.f7aaf6p-1,
  0x0.5a0036800p+2 /* 1.406263 */, 0x1.4f722cp-3, 0x1.f915dcp-1,
  0x0.5b005b800p+2 /* 1.421897 */, 0x1.2fd214p-3, 0x1.fa55aep-1,
  0x0.5bfe01800p+2 /* 1.437378 */, 0x1.106e24p-3, 0x1.fb732ap-1,
  0x0.5cf89f000p+2 /* 1.452675 */, 0x1.e2b3c0p-4, 0x1.fc6ea8p-1,
  0x0.5e00f4800p+2 /* 1.468808 */, 0x1.a104e8p-4, 0x1.fd56eap-1,
  0x0.5f0527000p+2 /* 1.484689 */, 0x1.60420cp-4, 0x1.fe1a64p-1,
  0x0.5fffc8000p+2 /* 1.499987 */, 0x1.21cb4cp-4, 0x1.feb78ap-1,
  0x0.610318000p+2 /* 1.515814 */, 0x1.c23092p-5, 0x1.ff39eep-1,
  0x0.62032d800p+2 /* 1.531444 */, 0x1.424a9cp-5, 0x1.ff9a86p-1,
  0x0.62fd46000p+2 /* 1.546709 */, 0x1.8a9d8cp-6, 0x1.ffd9fap-1,
  0x0.63fbc1800p+2 /* 1.562241 */, 0x1.1856c2p-7, 0x1.fffb34p-1,
  0x0.65011f000p+2 /* 1.578193 */, -0x1.e4c59ap-8, 0x1.fffc6ap-1,
};

/*@ requires 0 <= x <= 1.6 ; */
float my_sinf(float x)
{
  const float offs = 0x0.0b8p+2f;
  if (x < offs)
    {
      float xx = x * x;
      /* Remez-optimized polynomial for relative accuracy on -0.164 .. 0.164,
         Not the full -0.18 .. 0.18 where it is used, which makes it worse
         on -0.164 .. 0.164. But even optimized without regard for 0.164 .. 0.18
         It is better than the table entry + correction there so we use it there
      */
      return x + x * xx * (-0.16666660487324f + xx * 8.3259065018069e-3f);
    }
  int i = (x - offs) * 64.0f;
  float *p = c_cos_sin[i];
  float F = p[0];
  float C = p[1];
  float S = p[2];
  float h = x - F;
#if 0
  float s = S * (cosl(h) - 1.0) + C * sinl(h); // ext-double computation
#endif
#if 1
  // Two Remez-optimized polynomials for absolute accuracy on -0.008 .. 0.008
  float s =  h * (C + h * (-0.4999976959797f * S + h * -0.166666183241f * C));
#endif
  return S + s; 
}

unsigned int m, c, t;
uint64_t max_ulp;

int main(){
  for (float f = 0.0f; f < 1.57f; f = nextafterf(f, 3.0f))
    {
      double rd = sin(f);
      float r = rd;
      float n = my_sinf(f);
      t++;
      if (r != n)
        {
          c++;
          uint64_t in, ir;
          double nd = n;
          memcpy(&in, &nd, 8);
          memcpy(&ir, &rd, 8);
          uint64_t ulp = in > ir ? in - ir : ir - in;
          if (ulp > max_ulp)
            printf("error %" PRIu64 "/536870912 ULP sin(%.8e) ref:%.8e new:%.8e \n", 
                   ulp, f, r, n);
          if (ulp > max_ulp)
              max_ulp = ulp;
        }
    }
  printf("differences: %u / %u\n", c, t);
}

Answer 2

好吧，这个公式是一个开始。然后可以根据上下文进行其他转换。我同意，如果该公式sin(x) = sin(Cn) * cos(h) + cos(Cn) * sin(h)应用于目标精度，则 的舍入误差sin(Cn) * cos(h)最多为结果的 1/2 ulp，如果目标是获得准确的结果，则这是不好的。然而，某些术语有时可以通过使用伪扩展来更精确地表达。例如，一个数字可以用一对 ( a , b ) 表示，其中b远小于a，其值被视为a + b。在这种情况下， cos( h ) 可以由一对 (1, h ')表示，并且计算将等同于您的建议。

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或者，一旦给出了计算 cos( h ) 和 sin( h )的公式，就可以详细说明实现。请参阅 Stehl\xc3\xa9 中的第 3.1 节和您引用的 Zimmermann 论文：它们定义了 C ^* ( h )\xc2\xa0=\xc2\xa0C( h )\xc2\xa0\xe2\x88\x92\xc2\xa01 ，并且在最终公式中使用 C ^{* ，这基本上就是您的建议。}

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注意：我确信使用上述公式是最好的选择。人们可以从 开始sin(x)\xc2\xa0=\xc2\xa0sin(Cn)\xc2\xa0+\xc2\xa0error_term，并以其他方式计算误差项。

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