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我已经实现了在 C 中以单精度（32 位浮点）计算的三角函数（sin、cos、arctan）的一些近似值。它们精确到大约 +/- 2 ulp。

我的目标设备不支持任何<cmath>或<math.h>方法。它不提供 FMA，而是提供 MAC ALU。ALU 和 LU 以 32 位格式计算。

我的 arctan 近似实际上是N.juffa 近似的修改版本，它在整个范围内近似 arctan。正弦和余弦函数在 [-pi,pi] 范围内精确到 2 ulp。

我现在的目标是为正弦和余弦提供更大的输入范围（尽可能大，理想情况下为 [FLT_MIN,FLT_MAX]），这导致我减少参数。

我目前正在阅读不同的论文，例如 KCNg 的 A RGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit或关于这种新的参数缩减算法的论文，但我无法从中得出实现。

我还想提两个涉及相关问题的 stackoverflow 问题：有一种基于我链接的第一篇论文的matlab 和 c++ 方法。它实际上使用 matlab、cmath 方法，并将输入限制为 [0,20.000]。另一个已经在评论中提到了。它是一种在 C 中实现 sin 和 cos 的方法，使用我无法使用的各种 c 库。由于这两个帖子已经有好几年了，可能会有一些新的发现。

似乎在这种情况下主要使用的算法是将 2/pi 的数量精确地存储到所需的位数，以便能够准确地计算模计算并同时避免取消。我的设备不提供大型 DMEM，这意味着无法使用具有数百位的大型查找表。这个过程实际上在本参考资料的第 70 页上进行了描述，顺便说一下，它提供了很多关于浮点数学的有用信息。

所以我的问题是：是否有另一种有效的方法来减少正弦和余弦获得单精度避免大 LUT 的参数？上面提到的论文实际上侧重于双精度，最多使用 1000 位，这不适合我的用例。

我实际上没有在 C 中找到任何实现，也没有找到针对单精度计算的实现，我将不胜感激任何类型的提示/链接/示例...

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1 回答 1

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以下代码基于先前的答案，在该答案中，我演示了如何通过使用 Cody-Waite 方法对小参数使用拆分常数的 Cody-Waite 方法和对大参数使用 Payne-Hanek 方法对三角函数执行相当准确的参数缩减震级。有关 Payne-Hanek 算法的详细信息，请参见那里，有关 Cody-Waite 算法的详细信息，请参见我以前的答案。

这里我做了必要的调整以适应提问者平台的限制，不支持 64 位类型，不支持融合乘加，并且辅助函数math.h不可用。我假设float映射到 IEEE-754binary32格式，并且有一种方法可以将这样的 32 位浮点数重新解释为 32 位无符号整数，反之亦然。我已经通过标准的可移植习语实现了这种重新解释，即使用memcpy()，但可以选择适合未指定目标平台的其他方法，例如内联汇编、特定于机器的内在函数或易失性联合。

由于这段代码基本上是我以前的代码移植到一个更严格的环境，它可能缺乏专门针对该环境的从头设计的优雅。我基本上已经用一些位旋转替换了frexp()辅助函数，math.h用 32 位整数对模拟了 64 位整数计算，用 32 位定点计算替换了双精度计算（这比我预期的要好得多)，并将所有 FMA 替换为未融合的等效项。

重新处理参数减少的 Cody-Waite 部分需要做很多工作。显然，如果没有可用的 FMA，我们需要确保常数 π/2 的组成部分（除了最低有效位）中有足够数量的尾随零位，以确保乘积是准确的。我花了几个小时在实验上费解了一个特定的拆分，它提供了准确的结果，但也将切换点推到了 Payne-Hanek 方法的尽可能高。

指定时USE_FMA = 1，使用高质量数学库编译的测试应用程序的输出应类似于以下内容：

Testing sinf ...  PASSED. max ulp err = 1.493253  diffsum = 337633490
Testing cosf ...  PASSED. max ulp err = 1.495098  diffsum = 342020968

随着USE_FMA = 0准确度的变化略有恶化：

Testing sinf ...  PASSED. max ulp err = 1.498012  diffsum = 359702532
Testing cosf ...  PASSED. max ulp err = 1.504061  diffsum = 364682650

diffsum输出是整体准确度的粗略指标，这里显示大约 90% 的输入会产生正确舍入的单精度响应。

请注意，使用编译器提供的最严格的浮点设置和对 IEEE-754 的最高遵守程度来编译代码非常重要。对于我用来开发和测试此代码的 Intel 编译器，可以通过使用/fp:strict. 此外，用于参考的数学库的质量对于准确评估此单精度代码的 ulp 误差至关重要。英特尔编译器附带一个数学库，可提供双精度初等数学函数，在 HA（高精度）变体中误差略高于 0.5 ulp。使用多精度参考库可能更可取，但在这里会减慢我的速度。

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>   // for memcpy()
#include <math.h>     // for test purposes, and when PORTABLE=1 or USE_FMA=1

#define USE_FMA   (0) // use fmaf() calls for arithmetic
#define PORTABLE  (0) // allow helper functions from math.h
#define HAVE_U64  (0) // 64-bit integer type available
#define CW_STAGES (3) // number of stages in Cody-Waite reduction when USE_FMA=0

#if USE_FMA
#define SIN_RED_SWITCHOVER  (117435.992f)
#define COS_RED_SWITCHOVER  (71476.0625f)
#define MAX_DIFF            (1)
#else // USE_FMA
#if CW_STAGES == 2
#define SIN_RED_SWITCHOVER  (3.921875f)
#define COS_RED_SWITCHOVER  (3.921875f)
#elif CW_STAGES == 3
#define SIN_RED_SWITCHOVER  (201.15625f)
#define COS_RED_SWITCHOVER  (142.90625f)
#endif // CW_STAGES
#define MAX_DIFF            (2)
#endif // USE_FMA

/* re-interpret the bit pattern of an IEEE-754 float as a uint32 */
uint32_t float_as_uint32 (float a)
{
    uint32_t r;
    memcpy (&r, &a, sizeof r);
    return r;
}

/* re-interpret the bit pattern of a uint32 as an IEEE-754 float */
float uint32_as_float (uint32_t a)
{
    float r;
    memcpy (&r, &a, sizeof r);
    return r;
}

/* Compute the upper 32 bits of the product of two unsigned 32-bit integers */
#if HAVE_U64
uint32_t umul32_hi (uint32_t a, uint32_t b)
{
    return (uint32_t)(((uint64_t)a * b) >> 32);
}
#else // HAVE_U64
/* Henry S. Warren, "Hacker's Delight, 2nd ed.", Addison-Wesley 2012. Fig. 8-2 */
uint32_t umul32_hi (uint32_t a, uint32_t b)
{
    uint16_t a_lo = (uint16_t)a;
    uint16_t a_hi = a >> 16;
    uint16_t b_lo = (uint16_t)b;
    uint16_t b_hi = b >> 16;
    uint32_t p0 = (uint32_t)a_lo * b_lo;
    uint32_t p1 = (uint32_t)a_lo * b_hi;
    uint32_t p2 = (uint32_t)a_hi * b_lo;
    uint32_t p3 = (uint32_t)a_hi * b_hi;
    uint32_t t = (p0 >> 16) + p1;
    return (t >> 16) + (((uint32_t)(uint16_t)t + p2) >> 16) + p3;
}
#endif // HAVE_U64

/* 190 bits of 2/PI for Payne-Hanek style argument reduction. */
const uint32_t two_over_pi_f [] = 
{
    0x28be60db,
    0x9391054a,
    0x7f09d5f4,
    0x7d4d3770,
    0x36d8a566,
    0x4f10e410
};

/* Reduce a trig function argument using the slow Payne-Hanek method */
float trig_red_slowpath_f (float a, int *quadrant)
{
    uint32_t ia, hi, mid, lo, tmp, i, l, h, plo, phi;
    int32_t e, q;
    float r;

#if PORTABLE
    ia = (uint32_t)(fabsf (frexpf (a, &e)) * 0x1.0p32f); // 4.29496730e+9
#else // PORTABLE
    ia = ((float_as_uint32 (a) & 0x007fffff) << 8) | 0x80000000;
    e = ((float_as_uint32 (a) >> 23) & 0xff) - 126;
#endif // PORTABLE
    
    /* compute product x * 2/pi in 2.62 fixed-point format */
    i = (uint32_t)e >> 5;
    e = (uint32_t)e & 31;

    hi  = i ? two_over_pi_f [i-1] : 0;
    mid = two_over_pi_f [i+0];
    lo  = two_over_pi_f [i+1];
    tmp = two_over_pi_f [i+2];
 
    if (e) {
        hi  = (hi  << e) | (mid >> (32 - e));
        mid = (mid << e) | (lo  >> (32 - e));
        lo  = (lo  << e) | (tmp >> (32 - e));
    }

    /* compute 64-bit product phi:plo */
    phi = 0;
    l = ia * lo;
    h = umul32_hi (ia, lo);
    plo = phi + l;
    phi = h + (plo < l);
    l = ia * mid;
    h = umul32_hi (ia, mid);
    plo = phi + l;
    phi = h + (plo < l);
    l = ia * hi;
    phi = phi + l;

    /* split fixed-point result into integer and fraction portions */
    q = phi >> 30;               // integral portion = quadrant<1:0>
    phi = phi & 0x3fffffff;      // fraction
    if (phi & 0x20000000) {      // fraction >= 0.5
        phi = phi - 0x40000000;  // fraction - 1.0
        q = q + 1;
    }

    /* compute remainder of x / (pi/2) */
#if USE_FMA
    float phif, plof, chif, clof, thif, tlof;
    phif = 0x1.0p27f * (float)(int32_t)(phi & 0xffffffe0);
    plof = (float)((plo >> 5) | (phi << (32-5)));
    thif = phif + plof;
    plof = (phif - thif) + plof;
    phif = thif;
    chif =  0x1.921fb6p-57f; // (1.5707963267948966 * 0x1.0p-57)_hi
    clof = -0x1.777a5cp-82f; // (1.5707963267948966 * 0x1.0p-57)_lo
    thif = phif * chif;
    tlof = fmaf (phif, chif, -thif);
    tlof = fmaf (phif, clof, tlof);
    tlof = fmaf (plof, chif, tlof);
    r = thif + tlof;
#else // USE_FMA
    /* record sign of fraction */
    uint32_t s = phi & 0x80000000;
    
    /* take absolute value of fraction */
    if ((int32_t)phi < 0) {
        phi = ~phi;
        plo = 0 - plo;
        phi += (plo == 0);
    }
    
    /* normalize fraction */
    e = 0;
    while ((int32_t)phi > 0) {
        phi = (phi << 1) | (plo >> 31);
        plo = plo << 1;
        e--;
    }
    
    /* multiply 32 high-order bits of fraction with pi/2 */
    phi = umul32_hi (phi, 0xc90fdaa2); // (uint32_t)rint(PI/2 * 2**31)
    
    /* normalize product */
    if ((int32_t)phi > 0) {
        phi = phi << 1;
        e--;
    }

    /* round and convert to floating point */
    uint32_t ri = s + ((e + 128) << 23) + (phi >> 8) + ((phi & 0xff) > 0x7e);
    r = uint32_as_float (ri);
#endif // USE_FMA
    if (a < 0.0f) {
        r = -r;
        q = -q;
    }

    *quadrant = q;
    return r;
}

/* Argument reduction for trigonometric functions that reduces the argument
   to the interval [-PI/4, +PI/4] and also returns the quadrant. It returns 
   -0.0f for an input of -0.0f 
*/
float trig_red_f (float a, float switch_over, int *q)
{    
    float j, r;

    if (fabsf (a) > switch_over) {
        /* Payne-Hanek style reduction. M. Payne and R. Hanek, "Radian reduction
           for trigonometric functions". SIGNUM Newsletter, 18:19-24, 1983
        */
        r = trig_red_slowpath_f (a, q);
    } else {
        /* Cody-Waite style reduction. W. J. Cody and W. Waite, "Software Manual
           for the Elementary Functions", Prentice-Hall 1980
        */
#if USE_FMA
        j = fmaf (a, 0x1.45f306p-1f, 0x1.8p+23f) - 0x1.8p+23f; // 6.36619747e-1, 1.25829120e+7
        r = fmaf (j, -0x1.921fb0p+00f, a); // -1.57079601e+00 // pio2_high
        r = fmaf (j, -0x1.5110b4p-22f, r); // -3.13916473e-07 // pio2_mid
        r = fmaf (j, -0x1.846988p-48f, r); // -5.39030253e-15 // pio2_low
#else // USE_FMA
        j = (a * 0x1.45f306p-1f + 0x1.8p+23f) - 0x1.8p+23f; // 6.36619747e-1, 1.25829120e+7
#if CW_STAGES == 2
        r = a - j * 0x1.921fb4p+0f;  // pio2_high
        r = r - j * 0x1.4442d2p-24f; // pio2_low
#elif CW_STAGES == 3
        r = a - j * 0x1.921f00p+00f; // 1.57078552e+00 // pio2_high
        r = r - j * 0x1.6a8880p-17f; // 1.08043314e-05 // pio2_mid
        r = r - j * 0x1.68c234p-39f; // 2.56334407e-12 // pio2_low
#endif // CW_STAGES
#endif // USE_FMA
        *q = (int)j;
    }
    return r;
}

/* Approximate sine on [-PI/4,+PI/4]. Maximum ulp error with USE_FMA = 0.64196
   Returns -0.0f for an argument of -0.0f
   Polynomial approximation based on T. Myklebust, "Computing accurate 
   Horner form approximations to special functions in finite precision
   arithmetic", http://arxiv.org/abs/1508.03211, retrieved on 8/29/2016
*/
float sinf_poly (float a, float s)
{
    float r, t;
#if USE_FMA
    r =              0x1.80a000p-19f;  //  2.86567956e-6
    r = fmaf (r, s, -0x1.a0690cp-13f); // -1.98559923e-4
    r = fmaf (r, s,  0x1.111182p-07f); //  8.33338592e-3
    r = fmaf (r, s, -0x1.555556p-03f); // -1.66666672e-1
    t = fmaf (a, s, 0.0f); // ensure -0 is passed through
    r = fmaf (r, t, a);
#else // USE_FMA
    r =         0x1.80a000p-19f; //  2.86567956e-6
    r = r * s - 0x1.a0690cp-13f; // -1.98559923e-4
    r = r * s + 0x1.111182p-07f; //  8.33338592e-3
    r = r * s - 0x1.555556p-03f; // -1.66666672e-1
    t = a * s + 0.0f; // ensure -0 is passed through
    r = r * t + a;
#endif // USE_FMA
    return r;
}

/* Approximate cosine on [-PI/4,+PI/4]. Maximum ulp error with USE_FMA = 0.87444 */
float cosf_poly (float s)
{
    float r;
#if USE_FMA
    r =              0x1.9a8000p-16f;  //  2.44677067e-5
    r = fmaf (r, s, -0x1.6c0efap-10f); // -1.38877297e-3
    r = fmaf (r, s,  0x1.555550p-05f); //  4.16666567e-2
    r = fmaf (r, s, -0x1.000000p-01f); // -5.00000000e-1
    r = fmaf (r, s,  0x1.000000p+00f); //  1.00000000e+0
#else // USE_FMA
    r =         0x1.9a8000p-16f; //  2.44677067e-5
    r = r * s - 0x1.6c0efap-10f; // -1.38877297e-3
    r = r * s + 0x1.555550p-05f; //  4.16666567e-2
    r = r * s - 0x1.000000p-01f; // -5.00000000e-1
    r = r * s + 0x1.000000p+00f; //  1.00000000e+0
#endif // USE_FMA
    return r;
}

/* Map sine or cosine value based on quadrant */
float sinf_cosf_core (float a, int i)
{
    float r, s;

    s = a * a;
    r = (i & 1) ? cosf_poly (s) : sinf_poly (a, s);
    if (i & 2) {
        r = 0.0f - r; // don't change "sign" of NaNs
    }
    return r;
}

/* maximum ulp error with USE_FMA = 1: 1.495098  */
float my_sinf (float a)
{
    float r;
    int i;

    a = a * 0.0f + a; // inf -> NaN
    r = trig_red_f (a, SIN_RED_SWITCHOVER, &i);
    r = sinf_cosf_core (r, i);
    return r;
}

/* maximum ulp error with USE_FMA = 1: 1.493253 */
float my_cosf (float a)
{
    float r;
    int i;

    a = a * 0.0f + a; // inf -> NaN
    r = trig_red_f (a, COS_RED_SWITCHOVER, &i);
    r = sinf_cosf_core (r, i + 1);
    return r;
}

/* re-interpret bit pattern of an IEEE-754 double as a uint64 */
uint64_t double_as_uint64 (double a)
{
    uint64_t r;
    memcpy (&r, &a, sizeof r);
    return r;
}

double floatUlpErr (float res, double ref)
{
    uint64_t i, j, err, refi;
    int expoRef;
    
    /* ulp error cannot be computed if either operand is NaN, infinity, zero */
    if (isnan (res) || isnan (ref) || isinf (res) || isinf (ref) ||
        (res == 0.0f) || (ref == 0.0f)) {
        return 0.0;
    }
    /* Convert the float result to an "extended float". This is like a float
       with 56 instead of 24 effective mantissa bits.
    */
    i = ((uint64_t)float_as_uint32(res)) << 32;
    /* Convert the double reference to an "extended float". If the reference is
       >= 2^129, we need to clamp to the maximum "extended float". If reference
       is < 2^-126, we need to denormalize because of the float types's limited
       exponent range.
    */
    refi = double_as_uint64(ref);
    expoRef = (int)(((refi >> 52) & 0x7ff) - 1023);
    if (expoRef >= 129) {
        j = 0x7fffffffffffffffULL;
    } else if (expoRef < -126) {
        j = ((refi << 11) | 0x8000000000000000ULL) >> 8;
        j = j >> (-(expoRef + 126));
    } else {
        j = ((refi << 11) & 0x7fffffffffffffffULL) >> 8;
        j = j | ((uint64_t)(expoRef + 127) << 55);
    }
    j = j | (refi & 0x8000000000000000ULL);
    err = (i < j) ? (j - i) : (i - j);
    return err / 4294967296.0;
}

int main (void) 
{
    float arg, res, reff;
    uint32_t argi, resi, refi;
    int64_t diff, diffsum;
    double ref, ulp, maxulp;

    printf ("Testing sinf ...  ");
    diffsum = 0;
    maxulp = 0;
    argi = 0;
    do {
        arg = uint32_as_float (argi);
        res = my_sinf (arg);
        ref = sin ((double)arg);
        reff = (float)ref;
        resi = float_as_uint32 (res);
        refi = float_as_uint32 (reff);
        ulp = floatUlpErr (res, ref);
        if (ulp > maxulp) {
            maxulp = ulp;
        }
        diff = (resi > refi) ? (resi - refi) : (refi - resi);
        if (diff > MAX_DIFF) {
            printf ("\nerror @ %08x (% 15.8e): res=%08x (% 15.8e)  ref=%08x (%15.8e)\n", argi, arg, resi, res, refi, reff);
            return EXIT_FAILURE;
        }
        diffsum = diffsum + diff;
        argi++;
    } while (argi);
    printf ("PASSED. max ulp err = %.6f  diffsum = %lld\n", maxulp, diffsum);

    printf ("Testing cosf ...  ");
    diffsum = 0;
    maxulp = 0;
    argi = 0;
    do {
        arg = uint32_as_float (argi);
        res = my_cosf (arg);
        ref = cos ((double)arg);
        reff = (float)ref;
        resi = float_as_uint32 (res);
        refi = float_as_uint32 (reff);
        ulp = floatUlpErr (res, ref);
        if (ulp > maxulp) {
            maxulp = ulp;
        }
        diff = (resi > refi) ? (resi - refi) : (refi - resi);
        if (diff > MAX_DIFF) {
            printf ("\nerror @ %08x (% 15.8e): res=%08x (% 15.8e)  ref=%08x (%15.8e)\n", argi, arg, resi, res, refi, reff);
            return EXIT_FAILURE;
        }
        diffsum = diffsum + diff;
        argi++;
    } while (argi);
    diffsum = diffsum + diff;
    printf ("PASSED. max ulp err = %.6f  diffsum = %lld\n", maxulp, diffsum);
    return EXIT_SUCCESS;
}
于 2020-09-25T12:02:05.247 回答