[英]Single precision argument reduction for trigonometric functions in C
我已經為 C 中的單精度(32 位浮點)計算的三角函數(sin、cos、arctan)實現了一些近似值。它們的精度約為 +/- 2 ulp。
我的目標設備不支持任何<cmath>
或<math.h>
方法。 它不提供FMA,而是提供MAC ALU。 ALU 和 LU 以 32 位格式計算。
我的 arctan 近似實際上是N.juffa 近似的修改版本,它在整個范圍內近似於 arctan。 正弦和余弦 function 在 [-pi,pi] 范圍內精確到 2 ulp。
我現在的目標是為正弦和余弦提供更大的輸入范圍(盡可能大,最好是 [FLT_MIN,FLT_MAX]),這使我減少了自變量。
我目前正在閱讀不同的論文,例如 K.C.Ng 的 A RGUMENT REDUCTION FOR HUGE ARGUMENTS:Good to the Last Bit或關於這種新參數縮減算法的論文,但我無法從中得出實現。
另外,我想提兩個涉及相關問題的計算器問題:有一種方法 matlab 和 c++是基於我鏈接的第一篇論文。 它實際上使用 matlab,cmath 方法並將輸入限制為 [0,20.000]。 另一個已經在評論中提到了。 這是一種在 C 中實現 sin 和 cos 的方法,使用了我無法使用的各種 c 庫。 由於這兩個帖子已經有好幾年了,因此可能會有一些新發現。
似乎在這種情況下主要使用的算法是將 2/pi 的數量精確地存儲到所需的位數,以便能夠准確地計算模計算並同時避免取消。 我的設備不提供大型 DMEM,這意味着具有數百位的大型查找表是不可能的。 該過程實際上在 該參考文獻的第 70 頁上進行了描述,順便說一下,它提供了很多有關浮點數學的有用信息。
所以我的問題是:是否有另一種有效的方法來減少 arguments 的正弦和余弦獲得單精度避免大型 LUT? 上面提到的論文實際上專注於雙精度並且最多使用 1000 位數字,這不適合我的用例。
我實際上沒有在 C 中找到任何實現,也沒有找到針對單精度計算的實現,我將不勝感激任何類型的提示/鏈接/示例...
下面的代碼基於之前的答案,其中我演示了如何通過使用 Cody-Waite 方法對小參數進行拆分常量,以及對大參數使用 Payne-Hanek 方法,對三角函數執行相當准確的參數減少震級。 有關 Payne-Hanek 算法的詳細信息,請參閱此處,有關 Cody-Waite 算法的詳細信息,請參閱我之前的答案。
在這里,我進行了必要的調整以適應提問者平台的限制,因為不支持 64 位類型,不支持融合乘加,並且math.h
中的輔助函數不可用。 我假設float
映射到 IEEE-754 binary32
格式,並且有一種方法可以將這種 32 位浮點數重新解釋為 32 位無符號整數,反之亦然。 我已經通過標准的可移植習慣用法實現了這種重新解釋,即通過使用memcpy()
,但可以選擇適合未指定目標平台的其他方法,例如內聯匯編、特定於機器的內在函數或易失性聯合。
由於這段代碼基本上是我以前的代碼到一個更嚴格的環境的移植,它可能缺乏專門針對該環境的從頭設計的優雅。 我基本上已經用一些位frexp()
替換了math.h
的frexp()
輔助函數,用 32 位整數對模擬了 64 位整數計算,用 32 位定點計算替換了雙精度計算(有效)比我預期的要好得多),並用未融合的等效項替換了所有 FMA。
重新處理參數減少的 Cody-Waite 部分需要大量的工作。 顯然,如果沒有可用的 FMA,我們需要確保常數 π/2 的組成部分中有足夠數量的尾隨零位(最低有效位除外),以確保乘積是准確的。 我花了幾個小時實驗性地弄清了一個特定的拆分,它提供了准確的結果,但也盡可能地將切換點推到了 Payne-Hanek 方法。
當指定USE_FMA = 1
,測試應用程序的輸出在使用高質量數學庫編譯時應類似於以下內容:
Testing sinf ... PASSED. max ulp err = 1.493253 diffsum = 337633490
Testing cosf ... PASSED. max ulp err = 1.495098 diffsum = 342020968
當USE_FMA = 0
,精度會稍微變差:
Testing sinf ... PASSED. max ulp err = 1.498012 diffsum = 359702532
Testing cosf ... PASSED. max ulp err = 1.504061 diffsum = 364682650
diffsum
輸出是總體准確度的粗略指標,此處顯示所有輸入的大約 90% 導致正確舍入的單精度響應。
請注意,使用編譯器提供的最嚴格的浮點設置和最高程度的 IEEE-754 來編譯代碼非常重要。 對於我用來開發和測試此代碼的英特爾編譯器,可以通過編譯/fp:strict
來實現。 此外,用於參考的數學庫的質量對於准確評估此單精度代碼的 ulp 誤差至關重要。 英特爾編譯器附帶一個數學庫,該庫提供雙精度基本數學函數,HA(高精度)變體中的誤差略高於 0.5 ulp。 使用多精度參考庫可能更可取,但在這里會減慢我的速度。
#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;
}
數學論壇上有一個帖子,其中用戶J. M. 不是數學家介紹了改進的 Taylor/Padé 想法來近似 [-pi,pi] 范圍內的 cos 和 sin 函數。 這是轉換為 C++ 的正弦版本。此近似值不如庫 std::sin() function 快,但可能值得檢查 SSE/AVX/FMA 實現是否對速度有足夠幫助。
我沒有針對庫 sin() 或 cos() function 測試 ULP 錯誤,但通過Julia Function 精度測試工具,它看起來像是一個很好的近似方法(將以下代碼添加到屬於 Julia 測試套件的 runtest.jl 模塊):
function test_sine(x::AbstractFloat)
f=0.5
z=x*0.5
k=0
while (abs(z)>f)
z*=0.5
k=k+1
end
z2=z^2;
r=z*(1+(z2/105-1)*((z/3)^2))/
(1+(z2/7-4)*((z/3)^2));
while(k > 0)
r = (2*r)/(1-r*r);
k=k-1
end
return (2*r)/(1+r*r)
end
function test_cosine(x::AbstractFloat)
f=0.5
z=x*0.5
k=0
while (abs(z)>f)
z*=0.5
k=k+1
end
z2=z^2;
r=z*(1+(z2/105-1)*((z/3)^2))/
(1+(z2/7-4)*((z/3)^2));
while (k > 0)
r = (2*r)/(1-r*r);
k=k-1
end
return (1-r*r)/(1+r*r)
end
pii = 3.141592653589793238462643383279502884
MAX_SIN(n::Val{pii}, ::Type{Float16}) = 3.1415926535897932f0
MAX_SIN(n::Val{pii}, ::Type{Float32}) = 3.1415926535897932f0
#MAX_SIN(n::Val{pii}, ::Type{Float64}) = 3.141592653589793238462643383279502884
MIN_SIN(n::Val{pii}, ::Type{Float16}) = -3.1415926535897932f0
MIN_SIN(n::Val{pii}, ::Type{Float32}) = -3.1415926535897932f0
#MIN_SIN(n::Val{pii}, ::Type{Float64}) = -3.141592653589793238462643383279502884
for (func, base) in (sin=>Val(pii), test_sine=>Val(pii), cos=>Val(pii), test_cosine=>Val(pii))
for T in (Float16, Float32)
xx = range(MIN_SIN(base,T), MAX_SIN(base,T), length = 10^6);
test_acc(func, xx)
end
end
[-pi,pi] 范圍內的近似值和 sin() 和 cos() 的結果:
Tol debug failed 0.0% of the time.
sin
ULP max 0.5008857846260071 at x = 2.203355
ULP mean 0.24990503381476237
Test Summary: | Pass Total
Float32 sin | 1 1
Tol debug failed 0.0% of the time.
sin
ULP max 0.5008857846260071 at x = 2.203355
ULP mean 0.24990503381476237
Test Summary: | Pass Total
Float32 sin | 1 1
Tol debug failed 0.0% of the time.
test_sine
ULP max 0.001272978144697845 at x = 2.899093
ULP mean 1.179825295005716e-8
Test Summary: | Pass Total
Float32 test_sine | 1 1
Tol debug failed 0.0% of the time.
test_sine
ULP max 0.001272978144697845 at x = 2.899093
ULP mean 1.179825295005716e-8
Test Summary: | Pass Total
Float32 test_sine | 1 1
Tol debug failed 0.0% of the time.
cos
ULP max 0.5008531212806702 at x = 0.45568538
ULP mean 0.2499933592458589
Test Summary: | Pass Total
Float32 cos | 1 1
Tol debug failed 0.0% of the time.
cos
ULP max 0.5008531212806702 at x = 0.45568538
ULP mean 0.2499933592458589
Test Summary: | Pass Total
Float32 cos | 1 1
Tol debug failed 0.0% of the time.
test_cosine
ULP max 0.0011584102176129818 at x = 1.4495481
ULP mean 1.6793535615395134e-8
Test Summary: | Pass Total
Float32 test_cosine | 1 1
Tol debug failed 0.0% of the time.
test_cosine
ULP max 0.0011584102176129818 at x = 1.4495481
ULP mean 1.6793535615395134e-8
Test Summary: | Pass Total
Float32 test_cosine | 1 1
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