ref: f2e8e641bf8ee06fa5128c522b151a7f4d9d003b
dir: /lib/math/ancillary/generate-arctan-tuples-for-GB91.c/
/* cc -o generate-arctan-tuples-for-GB91 generate-arctan-tuples-for-GB91.c -lmpfr # -fno-strict-aliasing */
/* cc -static -std=c99 -D_POSIX_C_SOURCE=999999999 -fno-strict-aliasing -O2 -o generate-arctan-tuples-for-GB91 generate-arctan-tuples-for-GB91.c -lmpfr -lgmp */
#include <errno.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>
#include <unistd.h>
#include <mpfr.h>
/*
[GB91] treat arctan by a table + minimax method: the range [0,1]
is partitioned up, and on each partition arctan is approximated
by a minimax polynomial pi(x - xi) =~= arctan(x - xi).
The twist of [GB91], that of Highly Accurate Tables, here applies
to the first two coefficients of the minimax polynomial: by
adjusting xi and computing different polynomials, we obtain
coefficients cij for pi such that ci0 and ci1, in perfect accuracy,
have a bunch of zeroes in the binary expansion after the 53rd
bit. This gives the stored cij a bit more accuracy for free.
Note that there's a sign flip somewhere: so the even-degree
elements need to be negated for use in atan-impl.myr.
*/
/* Something something -fno-strict-aliasing */
#define FLT64_TO_UINT64(f) (*((uint64_t *) ((char *) &f)))
#define UINT64_TO_FLT64(u) (*((double *) ((char *) &u)))
#define xmin(a, b) ((a) < (b) ? (a) : (b))
#define xmax(a, b) ((a) > (b) ? (a) : (b))
#define EXP_OF_FLT64(f) (((FLT64_TO_UINT64(f)) >> 52) & 0x7ff)
typedef int (*mpfr_fn)(mpfr_ptr, mpfr_srcptr, mpfr_rnd_t);
#define N 5
static int leeway_of(mpfr_t temp, mpfr_t f)
{
double d1 = mpfr_get_d(f, MPFR_RNDN);
double d2 = 0.0;
mpfr_set_d(temp, d1, MPFR_RNDN);
mpfr_sub(temp, f, temp, MPFR_RNDN);
d2 = mpfr_get_d(temp, MPFR_RNDN);
return EXP_OF_FLT64(d1) - 52 - EXP_OF_FLT64(d2);
}
static void determinant_poorly(mpfr_t det, mpfr_t A[][N + 2])
{
int sgn = 1;
int sigma[N + 2];
mpfr_t prod;
for (int j = 0; j < (N + 2); ++j) {
sigma[j] = j;
}
mpfr_set_si(det, 0, MPFR_RNDN);
mpfr_init2(prod, 200);
while (1) {
/* ∏ a_{j, sigma[j]} */
mpfr_set_si(prod, sgn, MPFR_RNDN);
for (int j = 0; j < (N + 2); ++j) {
mpfr_mul(prod, prod, A[j][sigma[j]], MPFR_RNDN);
}
mpfr_add(det, det, prod, MPFR_RNDN);
/* increment sigma: algorithm K/L/something... */
int k = N + 1;
int j = N + 1;
int t;
while (k > 0 &&
sigma[k - 1] >= sigma[k]) {
k--;
}
if (!k) {
break;
}
while (sigma[j] <= sigma[k - 1]) {
j--;
}
if (k - 1 != j) {
t = sigma[k - 1];
sigma[k - 1] = sigma[j];
sigma[j] = t;
sgn *= -1;
}
for (int l = N + 1; l > k; --l, ++k) {
t = sigma[l];
sigma[l] = sigma[k];
sigma[k] = t;
sgn *= -1;
}
}
}
static void invert_poorly(mpfr_t A[][N + 2], mpfr_t Ainv[][N + 2])
{
mpfr_t Mij[N + 2][N + 2];
mpfr_t det;
mpfr_t Mijdet;
mpfr_init2(det, 200);
mpfr_init2(Mijdet, 200);
determinant_poorly(det, A);
for (int i = 0; i < N + 2; ++i) {
for (int j = 0; j < N + 2; ++j) {
mpfr_init2(Mij[i][j], 200);
if (i == (N + 1) &&
j == (N + 1)) {
mpfr_set_si(Mij[i][j], 1, MPFR_RNDN);
} else if (i == (N + 1) ||
j == (N + 1)) {
mpfr_set_si(Mij[i][j], 0, MPFR_RNDN);
}
}
}
/* Construct transpose adjugate poorly */
for (int i = 0; i < N + 2; ++i) {
for (int j = 0; j < N + 2; ++j) {
/* Copy over A, sans i, j, to Mij */
for (int ii = 0; ii < N + 2; ii++) {
if (ii == i) {
continue;
}
int ri = ii > i ? ii - 1 : ii;
for (int jj = 0; jj < N + 2; jj++) {
if (jj == j) {
continue;
}
int rj = jj > j ? jj - 1 : jj;
mpfr_set(Mij[ri][rj], A[ii][jj],
MPFR_RNDN);
}
}
/* Ainv[j][i] = | Mij | / det */
determinant_poorly(Mijdet, Mij);
mpfr_div(Ainv[j][i], Mijdet, det, MPFR_RNDN);
if ((i + j) % 2) {
mpfr_mul_si(Ainv[j][i], Ainv[j][i], -1,
MPFR_RNDN);
}
}
}
}
static int find_tuple(int ii, int min_leeway)
{
int64_t r = 0;
double xi_orig_d = ii / 256.0;
uint64_t xi_orig = FLT64_TO_UINT64(xi_orig_d);
double range_a = -1 / 512.0;
double range_b = 1 / 512.0;
uint64_t xi;
double xi_d;
mpfr_t xi_m;
int best_lee = 0;
long int best_r = 0;
mpfr_t t[10];
mpfr_t cn[N + 2];
mpfr_t bi[N + 2];
mpfr_t best_bi[N + 2];
mpfr_t best_xi;
mpfr_t xij[N + 2][N + 2];
mpfr_t xijinv[N + 2][N + 2];
mpfr_t fxi[N + 2];
double t_d = 0.0;
uint64_t t_u = 0;
long ec = 1;
long start = time(0);
long end = start;
mpfr_init2(xi_m, 200);
mpfr_init2(best_xi, 200);
for (int i = 0; i < 10; ++i) {
mpfr_init2(t[i], 200);
}
mpfr_set_d(t[1], range_a, MPFR_RNDN);
mpfr_set_d(t[2], range_b, MPFR_RNDN);
mpfr_add(t[3], t[2], t[1], MPFR_RNDN);
mpfr_sub(t[4], t[2], t[1], MPFR_RNDN);
mpfr_div_si(t[3], t[3], 2, MPFR_RNDN);
mpfr_div_si(t[4], t[4], 2, MPFR_RNDN);
/* Calculate Chebyshev nodes for the range */
for (int i = 0; i < (N + 2); ++i) {
mpfr_init2(cn[i], 200);
mpfr_init2(bi[i], 200);
mpfr_init2(best_bi[i], 200);
mpfr_set_si(best_bi[i], 0, MPFR_RNDN);
mpfr_init2(fxi[i], 200);
mpfr_set_si(cn[i], 2 * i - 1, MPFR_RNDN);
mpfr_div_si(cn[i], cn[i], 2 * (N + 2), MPFR_RNDN);
mpfr_cos(cn[i], cn[i], MPFR_RNDN);
mpfr_mul(cn[i], cn[i], t[4], MPFR_RNDN);
mpfr_add(cn[i], cn[i], t[3], MPFR_RNDN);
}
/*
Set up M×M (M = N+2) matrix for one step of Remez
algorithm: the cnI^Js in
b0 + b1·cn1 + ⋯ + bN·cn1^n + (-1)^1·E = f(cn1)
b0 + b1·cn2 + ⋯ + bN·cn2^n + (-1)^2·E = f(cn2)
⋮ ⋮ ⋱ ⋮ ⋮ ⋮
b0 + b1·cnM + ⋯ + bN·cnM^n + (-1)^M·E = f(cnM)
*/
for (int i = 0; i < (N + 2); ++i) {
mpfr_set_si(t[1], 1, MPFR_RNDN);
for (int j = 0; j < (N + 1); ++j) {
mpfr_init2(xij[i][j], 200);
mpfr_init2(xijinv[i][j], 200);
mpfr_set(xij[i][j], t[1], MPFR_RNDN);
mpfr_mul(t[1], t[1], cn[i], MPFR_RNDN);
}
mpfr_init2(xij[i][N + 1], 200);
mpfr_init2(xijinv[i][N + 1], 200);
mpfr_set_si(xij[i][N + 1], ec, MPFR_RNDN);
ec *= -1;
}
/* Compute (xij)^(-1) */
invert_poorly(xij, xijinv);
while (r < (1 << 28)) {
xi = xi_orig + r;
xi_d = UINT64_TO_FLT64(xi);
mpfr_set_d(xi_m, xi_d, MPFR_RNDN);
/* compute f(cn[i]) = atan(cn[i] - xi) */
for (int i = 0; i < (N + 2); ++i) {
mpfr_sub(fxi[i], cn[i], xi_m, MPFR_RNDN);
mpfr_atan(fxi[i], fxi[i], MPFR_RNDN);
}
/* Now solve the linear system above for bi */
for (int i = 0; i < (N + 2); ++i) {
mpfr_set_si(bi[i], 0, MPFR_RNDN);
for (int j = 0; j < (N + 2); ++j) {
mpfr_mul(t[i], xijinv[i][j], fxi[j], MPFR_RNDN);
mpfr_add(bi[i], bi[i], t[i], MPFR_RNDN);
}
}
/*
If the error term isn't within close to 0, we
should, by all rights, try a few more iterations
of Remez. But that's incredibly slow, and we're
in a tight loop, so let's just bail.
*/
double e = mpfr_get_d(bi[N + 1], MPFR_RNDN);
if (FLT64_TO_UINT64(e) & 0x7fffffffffffffff > 0x08) {
goto next_r;
}
/* Test if b[0] and b[1] are very precise */
int leeA = leeway_of(t[0], bi[0]);
int leeB = 0;
int lee = 0;
if (leeA <= min_leeway) {
goto next_r;
}
leeB = leeway_of(t[0], bi[1]);
if (leeB + 4 <= min_leeway) {
goto next_r;
}
lee = xmin(leeA, leeB + 4);
if (lee <= best_lee) {
goto next_r;
}
best_lee = lee;
best_r = r;
mpfr_set(best_xi, xi_m, MPFR_RNDN);
for (int i = 0; i < (N + 2); ++i) {
mpfr_set(best_bi[i], bi[i], MPFR_RNDN);
}
next_r:
/* increment r */
if (r <= 0) {
r = 1 - r;
} else {
r = -r;
}
}
end = time(0);
if (best_lee < min_leeway) {
return -1;
}
/* Recall the N+1 entry in output is the error, which we don't care about */
t_d = mpfr_get_d(best_xi, MPFR_RNDN);
t_u = FLT64_TO_UINT64(t_d);
printf("(%#018lx, ", t_u);
for (int i = 0; i < N; ++i) {
t_d = mpfr_get_d(best_bi[i], MPFR_RNDN);
printf("%#018lx, ", FLT64_TO_UINT64(t_d));
}
t_d = mpfr_get_d(best_bi[N], MPFR_RNDN);
t_u = FLT64_TO_UINT64(t_d);
printf("%#018lx), ", t_u);
printf("/* i = %03d, l = %02d, r = %010ld, t = %ld */\n", ii, best_lee,
best_r, end - start);
return 0;
}
static void usage(void)
{
printf("generate-arctan-tuples-for-GB91\n");
printf(" [-i start_idx]\n");
printf(" [-j end_idx]\n");
}
int main(int argc, char **argv)
{
int c = 0;
long i_start_arg = 1;
long i_end_arg = 256;
int i_start = 1;
int i_end = 256;
for (int k = 0; k < argc; ++k) {
printf("%s ", argv[k]);
}
printf("\n");
while ((c = getopt(argc, argv, "i:j:")) != -1) {
switch (c) {
case 'i':
errno = 0;
i_start_arg = strtoll(optarg, 0, 0);
if (errno) {
fprintf(stderr, "bad start index %s\n", optarg);
return 1;
}
break;
case 'j':
errno = 0;
i_end_arg = strtoll(optarg, 0, 0);
if (errno) {
fprintf(stderr, "bad end index %s\n", optarg);
return 1;
}
break;
default:
usage();
break;
}
}
if (i_start_arg <= 0 ||
i_start_arg > 256) {
printf("truncating start to (0, %d]\n", 256);
i_start_arg = xmin(xmax(i_start_arg, 1), 256);
}
if (i_end_arg <= 0 ||
i_end_arg > 256) {
printf("truncating end to (0, %d]\n", 256);
i_end_arg = xmin(xmax(i_end_arg, 1), 256);
}
i_start = i_start_arg;
i_end = i_end_arg;
for (int i = i_start; i <= i_end; ++i) {
if (find_tuple(i, 1) < 0) {
printf("CANNOT FIND SUITABLE CANDIDATE FOR i = %03d\n",
i);
}
}
return 0;
}