问题
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)
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"
回答1:
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.
If I read it right, the step size might be overkill for 4th order RK4, but better safe than sorry.
来源：https://stackoverflow.com/questions/38213584/how-are-the-trigonometric-functions-tested-in-the-gnu-c-library