如何在 GNU C 库中测试三角函数? [英] How are the trigonometric functions tested in the GNU C Library?
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问题描述
我正在尝试找出如何最好地为科学和/或数学函数编写单元测试.我在 GNU C 库的源代码中搜索了 sin() 和 cos() 函数的单元测试，并发现了 atest-sincos.c 源文件，转载如下.(可以找到 这里)
I'm trying to figure out how to best write unit tests for scientific and/or mathematical functions. I searched the source code for the GNU C Library for unit tests for the sin() and cos() functions and came across the atest-sincos.c source file, reproduced below. (It can be found here)
有人可以带我浏览这个文件并给出一个粗略的想法这里正在测试什么吗?我看到看起来非常像用于数值求解微分方程的 Runge-Kutta 算法，也可能与列表值进行比较，但我不太确定.非常欢迎这里的任何指导.
Can someone walk me through this file and give a rough idea what is being tested here? I see what looks very much like the Runge-Kutta algorithm for numerically solving differential equations, and also possibly a comparison with tabulated values, but I'm not quite sure. Any guidance here would be very welcome.
/* Copyright (C) 1997-2016 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Geoffrey Keating <Geoff.Keating@anu.edu.au>, 1997.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <stdio.h>
#include <math.h>
#include <gmp.h>
#include <string.h>
#include <limits.h>
#include <assert.h>
#define PRINT_ERRORS 0
#define N 0
#define N2 20
#define FRAC (32 * 4)
#define mpbpl (CHAR_BIT * sizeof (mp_limb_t))
#define SZ (FRAC / mpbpl + 1)
typedef mp_limb_t mp1[SZ], mp2[SZ * 2];
/* These strings have exactly 100 hex digits in them. */
static const char sin1[101] =
"d76aa47848677020c6e9e909c50f3c3289e511132f518b4def"
"b6ca5fd6c649bdfb0bd9ff1edcd4577655b5826a3d3b50c264";
static const char cos1[101] =
"8a51407da8345c91c2466d976871bd29a2373a894f96c3b7f2"
"300240b760e6fa96a94430a52d0e9e43f3450e3b8ff99bc934";
static const char hexdig[] = "0123456789abcdef";
static void
print_mpn_hex (const mp_limb_t *x, unsigned size)
{
char value[size + 1];
unsigned i;
const unsigned final = (size * 4 > SZ * mpbpl) ? SZ * mpbpl / 4 : size;
memset (value, '0', size);
for (i = 0; i < final ; i++)
value[size-1-i] = hexdig[x[i * 4 / mpbpl] >> (i * 4) % mpbpl & 0xf];
value[size] = '\0';
fputs (value, stdout);
}
static void
sincosx_mpn (mp1 si, mp1 co, mp1 xx, mp1 ix)
{
int i;
mp2 s[4], c[4];
mp1 tmp, x;
if (ix == NULL)
{
memset (si, 0, sizeof (mp1));
memset (co, 0, sizeof (mp1));
co[SZ-1] = 1;
memcpy (x, xx, sizeof (mp1));
}
else
mpn_sub_n (x, xx, ix, SZ);
for (i = 0; i < 1 << N; i++)
{
#define add_shift_mulh(d,x,s1,s2,sh,n) \
do { \
if (s2 != NULL) { \
if (sh > 0) { \
assert (sh < mpbpl); \
mpn_lshift (tmp, s1, SZ, sh); \
if (n) \
mpn_sub_n (tmp,tmp,s2+FRAC/mpbpl,SZ); \
else \
mpn_add_n (tmp,tmp,s2+FRAC/mpbpl,SZ); \
} else { \
if (n) \
mpn_sub_n (tmp,s1,s2+FRAC/mpbpl,SZ); \
else \
mpn_add_n (tmp,s1,s2+FRAC/mpbpl,SZ); \
} \
mpn_mul_n(d,tmp,x,SZ); \
} else \
mpn_mul_n(d,s1,x,SZ); \
assert(N+sh < mpbpl); \
if (N+sh > 0) mpn_rshift(d,d,2*SZ,N+sh); \
} while(0)
#define summ(d,ss,s,n) \
do { \
mpn_add_n(tmp,s[1]+FRAC/mpbpl,s[2]+FRAC/mpbpl,SZ); \
mpn_lshift(tmp,tmp,SZ,1); \
mpn_add_n(tmp,tmp,s[0]+FRAC/mpbpl,SZ); \
mpn_add_n(tmp,tmp,s[3]+FRAC/mpbpl,SZ); \
mpn_divmod_1(tmp,tmp,SZ,6); \
if (n) \
mpn_sub_n (d,ss,tmp,SZ); \
else \
mpn_add_n (d,ss,tmp,SZ); \
} while (0)
add_shift_mulh (s[0], x, co, NULL, 0, 0); /* s0 = h * c; */
add_shift_mulh (c[0], x, si, NULL, 0, 0); /* c0 = h * s; */
add_shift_mulh (s[1], x, co, c[0], 1, 1); /* s1 = h * (c - c0/2); */
add_shift_mulh (c[1], x, si, s[0], 1, 0); /* c1 = h * (s + s0/2); */
add_shift_mulh (s[2], x, co, c[1], 1, 1); /* s2 = h * (c - c1/2); */
add_shift_mulh (c[2], x, si, s[1], 1, 0); /* c2 = h * (s + s1/2); */
add_shift_mulh (s[3], x, co, c[2], 0, 1); /* s3 = h * (c - c2); */
add_shift_mulh (c[3], x, si, s[2], 0, 0); /* c3 = h * (s + s2); */
summ (si, si, s, 0); /* s = s + (s0+2*s1+2*s2+s3)/6; */
summ (co, co, c, 1); /* c = c - (c0+2*c1+2*c2+c3)/6; */
}
#undef add_shift_mulh
#undef summ
}
static int
mpn_bitsize (const mp_limb_t *SRC_PTR, mp_size_t SIZE)
{
int i, j;
for (i = SIZE - 1; i > 0; i--)
if (SRC_PTR[i] != 0)
break;
for (j = mpbpl - 1; j >= 0; j--)
if ((SRC_PTR[i] & (mp_limb_t)1 << j) != 0)
break;
return i * mpbpl + j;
}
static int
do_test (void)
{
mp1 si, co, x, ox, xt, s2, c2, s3, c3;
int i;
int sin_errors = 0, cos_errors = 0;
int sin_failures = 0, cos_failures = 0;
mp1 sin_maxerror, cos_maxerror;
int sin_maxerror_s = 0, cos_maxerror_s = 0;
const double sf = pow (2, mpbpl);
/* assert(mpbpl == mp_bits_per_limb); */
assert(FRAC / mpbpl * mpbpl == FRAC);
memset (sin_maxerror, 0, sizeof (mp1));
memset (cos_maxerror, 0, sizeof (mp1));
memset (xt, 0, sizeof (mp1));
xt[(FRAC - N2) / mpbpl] = (mp_limb_t)1 << (FRAC - N2) % mpbpl;
for (i = 0; i < 1 << N2; i++)
{
int s2s, s3s, c2s, c3s, j;
double ds2,dc2;
mpn_mul_1 (x, xt, SZ, i);
sincosx_mpn (si, co, x, i == 0 ? NULL : ox);
memcpy (ox, x, sizeof (mp1));
ds2 = sin (i / (double) (1 << N2));
dc2 = cos (i / (double) (1 << N2));
for (j = SZ-1; j >= 0; j--)
{
s2[j] = (mp_limb_t) ds2;
ds2 = (ds2 - s2[j]) * sf;
c2[j] = (mp_limb_t) dc2;
dc2 = (dc2 - c2[j]) * sf;
}
if (mpn_cmp (si, s2, SZ) >= 0)
mpn_sub_n (s3, si, s2, SZ);
else
mpn_sub_n (s3, s2, si, SZ);
if (mpn_cmp (co, c2, SZ) >= 0)
mpn_sub_n (c3, co, c2, SZ);
else
mpn_sub_n (c3, c2, co, SZ);
s2s = mpn_bitsize (s2, SZ);
s3s = mpn_bitsize (s3, SZ);
c2s = mpn_bitsize (c2, SZ);
c3s = mpn_bitsize (c3, SZ);
if ((s3s >= 0 && s2s - s3s < 54)
|| (c3s >= 0 && c2s - c3s < 54)
|| 0)
{
#if PRINT_ERRORS
printf ("%06x ", i * (0x100000 / (1 << N2)));
print_mpn_hex(si, (FRAC / 4) + 1);
putchar (' ');
print_mpn_hex (co, (FRAC / 4) + 1);
putchar ('\n');
fputs (" ", stdout);
print_mpn_hex (s2, (FRAC / 4) + 1);
putchar (' ');
print_mpn_hex (c2, (FRAC / 4) + 1);
putchar ('\n');
printf (" %c%c ",
s3s >= 0 && s2s-s3s < 54 ? s2s - s3s == 53 ? 'e' : 'F' : 'P',
c3s >= 0 && c2s-c3s < 54 ? c2s - c3s == 53 ? 'e' : 'F' : 'P');
print_mpn_hex (s3, (FRAC / 4) + 1);
putchar (' ');
print_mpn_hex (c3, (FRAC / 4) + 1);
putchar ('\n');
#endif
sin_errors += s2s - s3s == 53;
cos_errors += c2s - c3s == 53;
sin_failures += s2s - s3s < 53;
cos_failures += c2s - c3s < 53;
}
if (s3s >= sin_maxerror_s
&& mpn_cmp (s3, sin_maxerror, SZ) > 0)
{
memcpy (sin_maxerror, s3, sizeof (mp1));
sin_maxerror_s = s3s;
}
if (c3s >= cos_maxerror_s
&& mpn_cmp (c3, cos_maxerror, SZ) > 0)
{
memcpy (cos_maxerror, c3, sizeof (mp1));
cos_maxerror_s = c3s;
}
}
/* Check Range-Kutta against precomputed values of sin(1) and cos(1). */
memset (x, 0, sizeof (mp1));
x[FRAC / mpbpl] = (mp_limb_t)1 << FRAC % mpbpl;
sincosx_mpn (si, co, x, ox);
memset (s2, 0, sizeof (mp1));
memset (c2, 0, sizeof (mp1));
for (i = 0; i < 100 && i < FRAC / 4; i++)
{
s2[(FRAC - i * 4 - 4) / mpbpl] |= ((mp_limb_t) (strchr (hexdig, sin1[i])
- hexdig)
<< (FRAC - i * 4 - 4) % mpbpl);
c2[(FRAC - i * 4 - 4) / mpbpl] |= ((mp_limb_t) (strchr (hexdig, cos1[i])
- hexdig)
<< (FRAC - i * 4 - 4) % mpbpl);
}
if (mpn_cmp (si, s2, SZ) >= 0)
mpn_sub_n (s3, si, s2, SZ);
else
mpn_sub_n (s3, s2, si, SZ);
if (mpn_cmp (co, c2, SZ) >= 0)
mpn_sub_n (c3, co, c2, SZ);
else
mpn_sub_n (c3, c2, co, SZ);
printf ("sin:\n");
printf ("%d failures; %d errors; error rate %0.2f%%\n",
sin_failures, sin_errors, sin_errors * 100.0 / (double) (1 << N2));
fputs ("maximum error: ", stdout);
print_mpn_hex (sin_maxerror, (FRAC / 4) + 1);
fputs ("\nerror in sin(1): ", stdout);
print_mpn_hex (s3, (FRAC / 4) + 1);
fputs ("\n\ncos:\n", stdout);
printf ("%d failures; %d errors; error rate %0.2f%%\n",
cos_failures, cos_errors, cos_errors * 100.0 / (double) (1 << N2));
fputs ("maximum error: ", stdout);
print_mpn_hex (cos_maxerror, (FRAC / 4) + 1);
fputs ("\nerror in cos(1): ", stdout);
print_mpn_hex (c3, (FRAC / 4) + 1);
putchar ('\n');
return (sin_failures == 0 && cos_failures == 0) ? 0 : 1;
}
#define TIMEOUT 600
#define TEST_FUNCTION do_test ()
#include "../test-skeleton.c"
推荐答案
是的，处理器.将 glibc 计算的 sin 和 cos 值与 sin'=cos, cos'=-sin 的 Runge-Kutta 解进行比较，该解在基于 mpn大整数.
Yes, the processor resp. glibc computed values of sin and cos are compared to the Runge-Kutta solution for sin'=cos, cos'=-sin computed in multi-precision fixed-point floats modeled on mpn big integers.
如果我没看错，对于 4 阶 RK4，步长可能有点过大，但安全总比抱歉好.
If I read it right, the step size might be overkill for 4th order RK4, but better safe than sorry.
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