计算初等函数的文献sin带表是指公式:
sin(x) = sin(Cn) * cos(h) + cos(Cn) * sin(h)
哪里
x = Cn + h , Cn是一个常数，其中 sin(Cn)和 cos(Cn)已预先计算并在表中可用，并且，如果遵循 Gal 的方法，Cn已被选中，以便 sin(Cn)和 cos(Cn)由浮点数近似。数量h靠近0.0 .引用这个公式的一个例子是这个 article (第 7 页)。我不明白为什么这是有道理的:
cos(h) ，无论它是如何计算的，对于 h 的某些值，可能至少会出错 0.5 ULP ，并且因为它接近 1.0 ，这似乎对结果的准确性产生了巨大影响 sin(x)当以这种方式计算时。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))
然后是两个数量
(cos(h) - 1.0)和 sin(h)可以用多项式来近似，这些多项式很容易精确，因为它们产生的结果接近于零。 sin(Cn) * (cos(h) - 1.0) 的值, cos(Cn) * sin(h)并且因为它们的总和仍然很小，并且其绝对精度以总和所代表的小数量的 ULP 表示，因此将此数量添加到 sin(Cn)几乎正确四舍五入。我是否遗漏了一些使更早、更流行、更简单的公式也表现良好的东西？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()和 cosf()可能包含以下单精度点: 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
The following functions are specialized single-precision functions to use around 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 之间测试这些函数似乎表明新公式给出了更精确的结果:
$ gcc -std=c99 test.c && ./a.out
相对误差，传统:2.169624e-07，新:1.288049e-07
绝对误差平方和，传统:6.616202e-12，新:2.522784e-12
我走了几个捷径所以请看the complete program .
最佳答案
下面的实现部分回答了这个问题，因为它是正弦的单精度实现，使用问题中建议的公式，在 [0 … 1.57] 上精确到 0.53 ULP，对于 99.98% 精确到 0.5 ULP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
Meaning the error was never more than 285/536 of an ULP (about 0.53 ULP), and 176880 is the number of arguments where the error was above 0.5 ULP, out of a total of 1070134723 arguments.
It does not seem possible to achieve this sort of result with the plain sin(Cn) * cos(h) + cos(Cn) * sin(h) formula and only single-precision computations. The article cited in the question alludes to “the term c0*h being evaluated in an extended precision” in order to achieve overall accuracy.
#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);
}
关于floating-point - 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？，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/23703408/