The following code is based on a previous answer in which I demonstrated how to perform a fairly accurate argument reduction for trigonometric functions by using the Cody-Waite method of split constants for arguments small in magnitude, and the Payne-Hanek method for arguments large in magnitude. For details on the Payne-Hanek algorithm see there, for details on the Cody-Waite algorithm see this previous answer of mine.
Here I have made adjustments necessary to adjust to the restrictions of the asker's platform, in that no 64-bit types are supported, fused multiply-add is not supported, and helper functions from math.h are not available. I am assuming that float maps to IEEE-754 binary32 format, and that there is a way to re-interpret such a 32-bit float as a 32-bit unsigned integer and vice versa. I have implemented this re-interpretation via the standard portable idiom, that is, by using memcpy(), but other methods may be chosen appropriate for the unspecified target platform, such as inline assembly, machine-specific intrinsics, or volatile unions.
Since this code is basically a port of my previous code to a more restrictive environment, it lacks perhaps the elegance of a de novo design specifically targeted at that environment. I have basically replaced the frexp() helper function from math.h with some bit twiddling, emulated 64-bit integer computation with pairs of 32-bit integers, replaced the double-precision computation with 32-bit fixed-point computation (which worked much better than I had anticipated), and replaced all FMAs with the unfused equivalent.
Re-working the Cody-Waite portion of the argument reduction took quite a bit of work. Clearly, without FMA available, we need to ensure a sufficient number of trailing zero bits in the constituent parts of the constant π/2 (except the least significant one) to make sure the products are exact. I spent several hours experimentally puzzling out a particular split that delivers accurate results but also pushes the switchover point to the Payne-Hanek method as high as possible.
When USE_FMA = 1 is specified, the output of the test app, when compiled with a high-quality math library, should look similar to this:
Testing sinf ... PASSED. max ulp err = 1.493253 diffsum = 337633490
Testing cosf ... PASSED. max ulp err = 1.495098 diffsum = 342020968
With USE_FMA = 0 the accuracy changes slightly for the worse:
Testing sinf ... PASSED. max ulp err = 1.498012 diffsum = 359702532
Testing cosf ... PASSED. max ulp err = 1.504061 diffsum = 364682650
The diffsum output is a rough indicator of overall accuracy, here showing that about 90% of all inputs result in a correctly rounded single-precision response.
Note that it is important to compile the code with the strictest floating-point settings and highest degree of adherence to IEEE-754 the compiler offers. For the Intel compiler that I used to develop and test this code, that can be achieved by compiling with /fp:strict. Also, the quality of the math library used for reference is crucial for accurate assessment of the ulp error of this single-precision code. The Intel compiler comes with a math library that provides double-precision elementary math functions with just slightly over 0.5 ulp error in the HA (high accuracy) variant. Use of a multi-precision reference library may be preferable but would have slowed me down too much here.
#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)) * 4.29496730e+9f); // 0x1.0p32
#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 = 1.34217728e+8f * (float)(int32_t)(phi & 0xffffffe0); // 0x1.0p27
plof = (float)((plo >> 5) | (phi << (32-5)));
thif = phif + plof;
plof = (phif - thif) + plof;
phif = thif;
chif = 1.08995894e-17f; // 0x1.921fb6p-57 // (1.5707963267948966 * 0x1.0p-57)_hi
clof = -3.03308686e-25f; // -0x1.777a5cp-82 // (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, 6.36619747e-1f, 1.2582912e+7f); // 0x1.45f306p-1, 0x1.8p+23
j = j - 1.25829120e+7f; // 0x1.8p+23
r = fmaf (j, -1.57079601e+00f, a); // -0x1.921fb0p+00 // pio2_high
r = fmaf (j, -3.13916473e-07f, r); // -0x1.5110b4p-22 // pio2_mid
r = fmaf (j, -5.39030253e-15f, r); // -0x1.846988p-48 // pio2_low
#else // USE_FMA
j = (a * 6.36619747e-1f + 1.2582912e+7f); // 0x1.45f306p-1, 0x1.8p+23
j = j - 1.25829120e+7f; // 0x1.8p+23
#if CW_STAGES == 2
r = a - j * 1.57079625e+00f; // 0x1.921fb4p+0 // pio2_high
r = r - j * 7.54979013e-08f; // 0x1.4442d2p-24 // pio2_low
#elif CW_STAGES == 3
r = a - j * 1.57078552e+00f; // 0x1.921f00p+00 // pio2_high
r = r - j * 1.08043314e-05f; // 0x1.6a8880p-17 // pio2_mid
r = r - j * 2.56334407e-12f; // 0x1.68c234p-39 // 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 = 2.86567956e-6f; // 0x1.80a000p-19
r = fmaf (r, s, -1.98559923e-4f); // -0x1.a0690cp-13
r = fmaf (r, s, 8.33338592e-3f); // 0x1.111182p-07
r = fmaf (r, s, -1.66666672e-1f); // -0x1.555556p-03
t = fmaf (a, s, 0.0f); // ensure -0 is passed through
r = fmaf (r, t, a);
#else // USE_FMA
r = 2.86567956e-6f; // 0x1.80a000p-19
r = r * s - 1.98559923e-4f; // -0x1.a0690cp-13
r = r * s + 8.33338592e-3f; // 0x1.111182p-07
r = r * s - 1.66666672e-1f; // -0x1.555556p-03
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 = 2.44677067e-5f; // 0x1.9a8000p-16
r = fmaf (r, s, -1.38877297e-3f); // -0x1.6c0efap-10
r = fmaf (r, s, 4.16666567e-2f); // 0x1.555550p-05
r = fmaf (r, s, -5.00000000e-1f); // -0x1.000000p-01
r = fmaf (r, s, 1.00000000e+0f); // 0x1.000000p+00
#else // USE_FMA
r = 2.44677067e-5f; // 0x1.9a8000p-16
r = r * s - 1.38877297e-3f; // -0x1.6c0efap-10
r = r * s + 4.16666567e-2f; // 0x1.555550p-05
r = r * s - 5.00000000e-1f; // -0x1.000000p-01
r = r * s + 1.00000000e+0f; // 0x1.000000p+00
#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;
}
