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pi_fftcs.cpp
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/*
Based on "Calculation of PI(= 3.14159...) using FFT and AGM" by T.Ooura, Nov. 1999.
https://github.com/Fibonacci43/SuperPI
Modified for Arduino by Lucas Saavedra Vaz, 2024.
*/
#include <Arduino.h>
#include <assert.h>
#include <math.h>
#include <limits.h>
#include <float.h>
#include <stdio.h>
#include <stdlib.h>
#include "fftsg_h.h"
#include "pi_fftcs.h"
void pi_calc(int nfft) {
int log2_nfft, radix, log10_radix, n, npow, nprc;
#if PRINT_DIGITS
int j = 0, k = 0, l = 0;
#endif
double err;
int *a, *b, *c, *e, *i1, *i2;
double *d1, *d2, *d3;
char *dgt;
uint32_t start_time;
double elap_time, loop_time;
log_d("Calculation of PI using FFT and AGM, %s", PI_FFTC_VER);
// DGTINT is defined as short int, so it should be 2 bytes
assert(sizeof(DGTINT) == 2);
log_d("initializing...");
nfft /= 4;
start_time = millis();
for (log2_nfft = 1; (1 << log2_nfft) < nfft; log2_nfft++);
nfft = 1 << log2_nfft;
n = nfft + 2;
a = (int *)malloc(2 * sizeof(int) + n * sizeof(DGTINT));
b = (int *)malloc(2 * sizeof(int) + n * sizeof(DGTINT));
c = (int *)malloc(2 * sizeof(int) + n * sizeof(DGTINT));
e = (int *)malloc(2 * sizeof(int) + n * sizeof(DGTINT));
i1 = (int *)malloc(2 * sizeof(int) + n * sizeof(DGTINT));
i2 = (int *)malloc(2 * sizeof(int) + n * sizeof(DGTINT));
d1 = (double *)malloc((nfft + 2) * sizeof(double));
d2 = (double *)malloc((nfft + 2) * sizeof(double));
d3 = (double *)malloc((nfft + 2) * sizeof(double));
if (d3 == NULL) {
printf("Allocation Failure!\n");
exit(1);
}
/* ---- radix test ---- */
log10_radix = 1;
radix = 10;
err = mp_mul_radix_test(n, radix, nfft, d1);
err += DBL_EPSILON * (n * radix * radix / 4);
while (100 * err < DBL_ERROR_MARGIN && radix <= DGTINT_MAX / 20) {
err *= 100;
log10_radix++;
radix *= 10;
}
log_d("nfft= %d, radix= %d, error_margin= %g", nfft, radix, err);
log_d("calculating %d digits of PI...", log10_radix * (n - 2));
/*
* ---- a formula based on the AGM (Arithmetic-Geometric Mean) ----
* c = sqrt(0.125);
* a = 1 + 3 * c;
* b = sqrt(a);
* e = b - 0.625;
* b = 2 * b;
* c = e - c;
* a = a + e;
* npow = 4;
* do {
* npow = 2 * npow;
* e = (a + b) / 2;
* b = sqrt(a * b);
* e = e - b;
* b = 2 * b;
* c = c - e;
* a = e + b;
* } while (e > SQRT_SQRT_EPSILON);
* e = e * e / 4;
* a = a + b;
* pi = (a * a - e - e / 2) / (a * c - e) / npow;
* ---- modification ----
* This is a modified version of Gauss-Legendre formula
* (by T.Ooura). It is faster than original version.
* ---- reference ----
* 1. E.Salamin,
* Computation of PI Using Arithmetic-Geometric Mean,
* Mathematics of Computation, Vol.30 1976.
* 2. R.P.Brent,
* Fast Multiple-Precision Evaluation of Elementary Functions,
* J. ACM 23 1976.
* 3. D.Takahasi, Y.Kanada,
* Calculation of PI to 51.5 Billion Decimal Digits on
* Distributed Memoriy Parallel Processors,
* Transactions of Information Processing Society of Japan,
* Vol.39 No.7 1998.
* 4. T.Ooura,
* Improvement of the PI Calculation Algorithm and
* Implementation of Fast Multiple-Precision Computation,
* Information Processing Society of Japan SIG Notes,
* 98-HPC-74, 1998.
*/
/* ---- c = 1 / sqrt(8) ---- */
mp_invisqrt(n, radix, 8, c, i1, i2, nfft, d1, d2);
/* ---- a = 1 + 3 * c ---- */
mp_imul(n, radix, c, 3, e);
mp_sscanf(n, log10_radix, (char *)"1", a);
mp_add(n, radix, a, e, a);
/* ---- b = sqrt(a) ---- */
mp_sqrt(n, radix, a, b, i1, i2, nfft, d1, d2);
/* ---- e = b - 0.625 ---- */
mp_sscanf(n, log10_radix, (char *)"0.625", e);
mp_sub(n, radix, b, e, e);
/* ---- b = 2 * b ---- */
mp_add(n, radix, b, b, b);
/* ---- c = e - c ---- */
mp_sub(n, radix, e, c, c);
/* ---- a = a + e ---- */
mp_add(n, radix, a, e, a);
log_d("AGM iteration");
npow = 4;
elap_time = ((double)(millis() - start_time)) / 1000;
do {
uint32_t start_loop_time = millis();
npow *= 2;
/* ---- e = (a + b) / 2 ---- */
mp_add(n, radix, a, b, e);
mp_idiv_2(n, radix, e, e);
/* ---- b = sqrt(a * b) ---- */
mp_mul(n, radix, a, b, a, i1, nfft, d1, d2, d3);
mp_sqrt(n, radix, a, b, i1, i2, nfft, d1, d2);
/* ---- e = e - b ---- */
mp_sub(n, radix, e, b, e);
/* ---- b = 2 * b ---- */
mp_add(n, radix, b, b, b);
/* ---- c = c - e ---- */
mp_sub(n, radix, c, e, c);
/* ---- a = e + b ---- */
mp_add(n, radix, e, b, a);
/* ---- convergence check ---- */
nprc = -e[1];
if (e[0] == 0) {
nprc = n;
}
loop_time = ((double)(millis() - start_loop_time)) / 1000;
elap_time += loop_time;
log_d("precision= %d: %0.2f sec", 4 * nprc * log10_radix, loop_time);
} while (4 * nprc <= n);
start_time = millis();
/* ---- e = e * e / 4 (half precision) ---- */
mp_idiv_2(n, radix, e, e);
mp_squh(n, radix, e, e, nfft, d1);
/* ---- a = a + b ---- */
mp_add(n, radix, a, b, a);
/* ---- a = (a * a - e - e / 2) / (a * c - e) / npow ---- */
mp_mulhf(n, radix, a, c, c, i1, nfft, d1, d2);
mp_sub(n, radix, c, e, c);
mp_inv(n, radix, c, b, i1, i2, nfft, d2, d3);
mp_squhf_use_infft(n, radix, d1, a, a, i1, nfft, d2);
mp_sub(n, radix, a, e, a);
mp_idiv_2(n, radix, e, e);
mp_sub(n, radix, a, e, a);
mp_mul(n, radix, a, b, a, i1, nfft, d1, d2, d3);
mp_idiv(n, radix, a, npow, a);
/* ---- output ---- */
dgt = (char *)d1;
mp_sprintf(n - 1, log10_radix, a, dgt);
elap_time = ((double)(millis() - start_time)) / 1000;
#if PRINT_DIGITS
do {
if (!isdigit(*dgt)) {
if (isalpha(*dgt) != 0) {
fputc('\n', stdout);
fputc('\n', stdout);
}
fputc(*dgt, stdout);
fputc('\n', stdout);
fputc('\n', stdout);
j = 0;
k = 0;
l = 0;
continue;
}
fputc(*dgt, stdout);
if (++j >= DGT_PACK) {
j = 0;
if (++k >= DGT_PACK_LINE) {
k = 0;
fputc('\n', stdout);
if (++l >= DGT_LINE_BLOCK) {
l = 0;
fputc('\n', stdout);
}
} else {
fputc(' ', stdout);
}
}
} while (*dgt++ && *dgt != 'e');
fputc('\n', stdout);
fprintf(stdout, "%s\n", dgt);
#endif
free(d3);
free(d2);
free(d1);
free(i2);
free(i1);
free(e);
free(c);
free(b);
free(a);
/* ---- difftime ---- */
log_d("%0.2f sec. (real time)", elap_time);
}
/* -------- multiple precision routines -------- */
/* -------- mp_load routines -------- */
void mp_load_0(int n, int radix, int out[]) {
int j;
DGTINT *outr;
outr = ((DGTINT *)&out[2]) - 2;
out[0] = 0;
out[1] = 0;
for (j = 2; j <= n + 1; j++) {
outr[j] = 0;
}
}
void mp_load_1(int n, int radix, int out[]) {
int j;
DGTINT *outr;
outr = ((DGTINT *)&out[2]) - 2;
out[0] = 1;
out[1] = 0;
outr[2] = 1;
for (j = 3; j <= n + 1; j++) {
outr[j] = 0;
}
}
void mp_round(int n, int radix, int m, int inout[]) {
int j, x;
DGTINT *inoutr;
inoutr = ((DGTINT *)&inout[2]) - 2;
if (m < n) {
for (j = n + 1; j > m + 2; j--) {
inoutr[j] = 0;
}
x = 2 * inoutr[m + 2];
inoutr[m + 2] = 0;
if (x >= radix) {
for (j = m + 1; j >= 2; j--) {
x = inoutr[j] + 1;
if (x < radix) {
inoutr[j] = (DGTINT)x;
break;
}
inoutr[j] = 0;
}
if (x >= radix) {
inoutr[2] = 1;
inout[1]++;
}
}
}
}
/* -------- mp_add routines -------- */
int mp_cmp(int n, int radix, int in1[], int in2[]) {
int mp_unsgn_cmp(int n, int in1[], int in2[]);
if (in1[0] > in2[0]) {
return 1;
} else if (in1[0] < in2[0]) {
return -1;
}
return in1[0] * mp_unsgn_cmp(n, &in1[1], &in2[1]);
}
void mp_add(int n, int radix, int in1[], int in2[], int out[]) {
int mp_unsgn_cmp(int n, int in1[], int in2[]);
int mp_unexp_add(int n, int radix, int expdif, DGTINT in1[], DGTINT in2[], DGTINT out[]);
int mp_unexp_sub(int n, int radix, int expdif, DGTINT in1[], DGTINT in2[], DGTINT out[]);
int outsgn, outexp, expdif;
expdif = in1[1] - in2[1];
outexp = in1[1];
if (expdif < 0) {
outexp = in2[1];
}
outsgn = in1[0] * in2[0];
if (outsgn >= 0) {
if (outsgn > 0) {
outsgn = in1[0];
} else {
outsgn = in1[0] + in2[0];
outexp = in1[1] + in2[1];
expdif = 0;
}
if (expdif >= 0) {
outexp += mp_unexp_add(n, radix, expdif, (DGTINT *)&in1[2], (DGTINT *)&in2[2], (DGTINT *)&out[2]);
} else {
outexp += mp_unexp_add(n, radix, -expdif, (DGTINT *)&in2[2], (DGTINT *)&in1[2], (DGTINT *)&out[2]);
}
} else {
outsgn = mp_unsgn_cmp(n, &in1[1], &in2[1]);
if (outsgn >= 0) {
expdif = mp_unexp_sub(n, radix, expdif, (DGTINT *)&in1[2], (DGTINT *)&in2[2], (DGTINT *)&out[2]);
} else {
expdif = mp_unexp_sub(n, radix, -expdif, (DGTINT *)&in2[2], (DGTINT *)&in1[2], (DGTINT *)&out[2]);
}
outexp -= expdif;
outsgn *= in1[0];
if (expdif == n) {
outsgn = 0;
}
}
if (outsgn == 0) {
outexp = 0;
}
out[0] = outsgn;
out[1] = outexp;
}
void mp_sub(int n, int radix, int in1[], int in2[], int out[]) {
int mp_unsgn_cmp(int n, int in1[], int in2[]);
int mp_unexp_add(int n, int radix, int expdif, DGTINT in1[], DGTINT in2[], DGTINT out[]);
int mp_unexp_sub(int n, int radix, int expdif, DGTINT in1[], DGTINT in2[], DGTINT out[]);
int outsgn, outexp, expdif;
expdif = in1[1] - in2[1];
outexp = in1[1];
if (expdif < 0) {
outexp = in2[1];
}
outsgn = in1[0] * in2[0];
if (outsgn <= 0) {
if (outsgn < 0) {
outsgn = in1[0];
} else {
outsgn = in1[0] - in2[0];
outexp = in1[1] + in2[1];
expdif = 0;
}
if (expdif >= 0) {
outexp += mp_unexp_add(n, radix, expdif, (DGTINT *)&in1[2], (DGTINT *)&in2[2], (DGTINT *)&out[2]);
} else {
outexp += mp_unexp_add(n, radix, -expdif, (DGTINT *)&in2[2], (DGTINT *)&in1[2], (DGTINT *)&out[2]);
}
} else {
outsgn = mp_unsgn_cmp(n, &in1[1], &in2[1]);
if (outsgn >= 0) {
expdif = mp_unexp_sub(n, radix, expdif, (DGTINT *)&in1[2], (DGTINT *)&in2[2], (DGTINT *)&out[2]);
} else {
expdif = mp_unexp_sub(n, radix, -expdif, (DGTINT *)&in2[2], (DGTINT *)&in1[2], (DGTINT *)&out[2]);
}
outexp -= expdif;
outsgn *= in1[0];
if (expdif == n) {
outsgn = 0;
}
}
if (outsgn == 0) {
outexp = 0;
}
out[0] = outsgn;
out[1] = outexp;
}
/* -------- mp_add child routines -------- */
int mp_unsgn_cmp(int n, int in1[], int in2[]) {
int j, cmp;
DGTINT *in1r, *in2r;
in1r = ((DGTINT *)&in1[1]) - 1;
in2r = ((DGTINT *)&in2[1]) - 1;
cmp = in1[0] - in2[0];
for (j = 1; j <= n && cmp == 0; j++) {
cmp = in1r[j] - in2r[j];
}
if (cmp > 0) {
cmp = 1;
} else if (cmp < 0) {
cmp = -1;
}
return cmp;
}
int mp_unexp_add(int n, int radix, int expdif, DGTINT in1[], DGTINT in2[], DGTINT out[]) {
int j, x, carry;
carry = 0;
if (expdif == 0 && in1[0] + in2[0] >= radix) {
x = in1[n - 1] + in2[n - 1];
carry = x >= radix ? -1 : 0;
for (j = n - 1; j > 0; j--) {
x = in1[j - 1] + in2[j - 1] - carry;
carry = x >= radix ? -1 : 0;
out[j] = (DGTINT)(x - (radix & carry));
}
out[0] = (DGTINT)-carry;
} else {
if (expdif > n) {
expdif = n;
}
for (j = n - 1; j >= expdif; j--) {
x = in1[j] + in2[j - expdif] - carry;
carry = x >= radix ? -1 : 0;
out[j] = (DGTINT)(x - (radix & carry));
}
for (j = expdif - 1; j >= 0; j--) {
x = in1[j] - carry;
carry = x >= radix ? -1 : 0;
out[j] = (DGTINT)(x - (radix & carry));
}
if (carry != 0) {
for (j = n - 1; j > 0; j--) {
out[j] = out[j - 1];
}
out[0] = (DGTINT)-carry;
}
}
return -carry;
}
int mp_unexp_sub(int n, int radix, int expdif, DGTINT in1[], DGTINT in2[], DGTINT out[]) {
int j, x, borrow, ncancel;
if (expdif > n) {
expdif = n;
}
borrow = 0;
for (j = n - 1; j >= expdif; j--) {
x = in1[j] - in2[j - expdif] + borrow;
borrow = x < 0 ? -1 : 0;
out[j] = (DGTINT)(x + (radix & borrow));
}
for (j = expdif - 1; j >= 0; j--) {
x = in1[j] + borrow;
borrow = x < 0 ? -1 : 0;
out[j] = (DGTINT)(x + (radix & borrow));
}
ncancel = 0;
for (j = 0; j < n && out[j] == 0; j++) {
ncancel = j + 1;
}
if (ncancel > 0 && ncancel < n) {
for (j = 0; j < n - ncancel; j++) {
out[j] = out[j + ncancel];
}
for (j = n - ncancel; j < n; j++) {
out[j] = 0;
}
}
return ncancel;
}
/* -------- mp_imul routines -------- */
void mp_imul(int n, int radix, int in1[], int in2, int out[]) {
void mp_unsgn_imul(int n, double dradix, int in1[], double din2, int out[]);
if (in2 > 0) {
out[0] = in1[0];
} else if (in2 < 0) {
out[0] = -in1[0];
in2 = -in2;
} else {
out[0] = 0;
}
mp_unsgn_imul(n, radix, &in1[1], in2, &out[1]);
if (out[0] == 0) {
out[1] = 0;
}
}
int mp_idiv(int n, int radix, int in1[], int in2, int out[]) {
void mp_load_0(int n, int radix, int out[]);
void mp_unsgn_idiv(int n, double dradix, int in1[], double din2, int out[]);
if (in2 == 0) {
return -1;
}
if (in2 > 0) {
out[0] = in1[0];
} else {
out[0] = -in1[0];
in2 = -in2;
}
if (in1[0] == 0) {
mp_load_0(n, radix, out);
return 0;
}
mp_unsgn_idiv(n, radix, &in1[1], in2, &out[1]);
return 0;
}
void mp_idiv_2(int n, int radix, int in[], int out[]) {
int j, ix, carry, shift;
DGTINT *inr, *outr;
inr = ((DGTINT *)&in[2]) - 2;
outr = ((DGTINT *)&out[2]) - 2;
out[0] = in[0];
shift = 0;
if (inr[2] == 1) {
shift = 1;
}
out[1] = in[1] - shift;
carry = -shift;
for (j = 2; j <= n + 1 - shift; j++) {
ix = inr[j + shift] + (radix & carry);
carry = -(ix & 1);
outr[j] = (DGTINT)(ix >> 1);
}
if (shift > 0) {
outr[n + 1] = (DGTINT)((radix & carry) >> 1);
}
}
/* -------- mp_imul child routines -------- */
void mp_unsgn_imul(int n, double dradix, int in1[], double din2, int out[]) {
int j, carry, shift;
double x, d1_radix;
DGTINT *in1r, *outr;
in1r = ((DGTINT *)&in1[1]) - 1;
outr = ((DGTINT *)&out[1]) - 1;
d1_radix = 1.0 / dradix;
carry = 0;
for (j = n; j >= 1; j--) {
x = din2 * in1r[j] + carry + 0.5;
carry = (int)(d1_radix * x);
outr[j] = (DGTINT)(x - dradix * carry);
}
shift = 0;
x = carry + 0.5;
while (x > 1) {
x *= d1_radix;
shift++;
}
out[0] = in1[0] + shift;
if (shift > 0) {
while (shift > n) {
carry = (int)(d1_radix * carry + 0.5);
shift--;
}
for (j = n; j >= shift + 1; j--) {
outr[j] = outr[j - shift];
}
for (j = shift; j >= 1; j--) {
x = carry + 0.5;
carry = (int)(d1_radix * x);
outr[j] = (DGTINT)(x - dradix * carry);
}
}
}
void mp_unsgn_idiv(int n, double dradix, int in1[], double din2, int out[]) {
int j, ix, carry, shift;
double x, d1_in2;
DGTINT *in1r, *outr;
in1r = ((DGTINT *)&in1[1]) - 1;
outr = ((DGTINT *)&out[1]) - 1;
d1_in2 = 1.0 / din2;
shift = 0;
x = 0;
do {
shift++;
x *= dradix;
if (shift <= n) {
x += in1r[shift];
}
} while (x < din2 - 0.5);
x += 0.5;
ix = (int)(d1_in2 * x);
carry = (int)(x - din2 * ix);
outr[1] = (DGTINT)ix;
shift--;
out[0] = in1[0] - shift;
if (shift >= n) {
shift = n - 1;
}
for (j = 2; j <= n - shift; j++) {
x = in1r[j + shift] + dradix * carry + 0.5;
ix = (int)(d1_in2 * x);
carry = (int)(x - din2 * ix);
outr[j] = (DGTINT)ix;
}
for (j = n - shift + 1; j <= n; j++) {
x = dradix * carry + 0.5;
ix = (int)(d1_in2 * x);
carry = (int)(x - din2 * ix);
outr[j] = (DGTINT)ix;
}
}
/* -------- mp_mul routines -------- */
double mp_mul_radix_test(int n, int radix, int nfft, double tmpfft[]) {
void mp_mul_csqu(int nfft, double d1[]);
double mp_mul_d2i_test(int radix, int nfft, double din[]);
int j, ndata, radix_2;
ndata = (nfft >> 1) + 1;
if (ndata > n) {
ndata = n;
}
tmpfft[nfft + 1] = radix - 1;
for (j = nfft; j > ndata; j--) {
tmpfft[j] = 0;
}
radix_2 = (radix + 1) / 2;
for (j = ndata; j > 2; j--) {
tmpfft[j] = radix_2;
}
tmpfft[2] = radix;
tmpfft[1] = radix - 1;
tmpfft[0] = 0;
mp_mul_csqu(nfft, tmpfft);
return 2 * mp_mul_d2i_test(radix, nfft, tmpfft);
}
void mp_mul(int n, int radix, int in1[], int in2[], int out[], int tmp[], int nfft, double tmp1fft[], double tmp2fft[], double tmp3fft[]) {
void mp_add(int n, int radix, int in1[], int in2[], int out[]);
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul_nt_out(int nfft, double d1[], double d2[]);
void mp_mul_cmul_nt_d2(int nfft, double d1[], double d2[]);
void mp_mul_cmul_nt_d1_add(int nfft, double d1[], double d2[], double d3[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
int n_h, shift;
DGTINT *in1r, *in2r;
in1r = ((DGTINT *)&in1[2]) - 2;
in2r = ((DGTINT *)&in2[2]) - 2;
shift = (nfft >> 1) + 1;
while (n > shift) {
if (in1r[shift + 2] + in2r[shift + 2] != 0) {
break;
}
shift++;
}
n_h = n / 2 + 1;
if (n_h < n - shift) {
n_h = n - shift;
}
/* ---- tmp3fft = (upper) in1 * (lower) in2 ---- */
mp_mul_i2d(n, radix, nfft, 0, in1, tmp1fft);
mp_mul_i2d(n, radix, nfft, shift, in2, tmp3fft);
mp_mul_cmul_nt_out(nfft, tmp1fft, tmp3fft);
/* ---- tmp = (upper) in1 * (upper) in2 ---- */
mp_mul_i2d(n, radix, nfft, 0, in2, tmp2fft);
mp_mul_cmul_nt_d2(nfft, tmp2fft, tmp1fft);
mp_mul_d2i(n, radix, nfft, tmp1fft, tmp);
/* ---- tmp3fft += (upper) in2 * (lower) in1 ---- */
mp_mul_i2d(n, radix, nfft, shift, in1, tmp1fft);
mp_mul_cmul_nt_d1_add(nfft, tmp2fft, tmp1fft, tmp3fft);
/* ---- out = tmp + tmp3fft ---- */
mp_mul_d2i(n_h, radix, nfft, tmp3fft, out);
mp_add(n, radix, out, tmp, out);
}
void mp_squ(int n, int radix, int in[], int out[], int tmp[], int nfft, double tmp1fft[], double tmp2fft[]) {
void mp_add(int n, int radix, int in1[], int in2[], int out[]);
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul(int nfft, double d1[], double d2[]);
void mp_mul_csqu_nt_d1(int nfft, double d1[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
int n_h, shift;
DGTINT *inr;
inr = ((DGTINT *)&in[2]) - 2;
shift = (nfft >> 1) + 1;
while (n > shift) {
if (inr[shift + 2] != 0) {
break;
}
shift++;
}
n_h = n / 2 + 1;
if (n_h < n - shift) {
n_h = n - shift;
}
/* ---- tmp = 2 * (upper) in * (lower) in ---- */
mp_mul_i2d(n, radix, nfft, 0, in, tmp1fft);
mp_mul_i2d(n, radix, nfft, shift, in, tmp2fft);
mp_mul_cmul(nfft, tmp1fft, tmp2fft);
mp_mul_d2i(n_h, radix, nfft, tmp2fft, tmp);
mp_add(n_h, radix, tmp, tmp, tmp);
/* ---- out = tmp + ((upper) in)^2 ---- */
mp_mul_csqu_nt_d1(nfft, tmp1fft);
mp_mul_d2i(n, radix, nfft, tmp1fft, out);
mp_add(n, radix, out, tmp, out);
}
void mp_mulhf(int n, int radix, int in1[], int in2[], int out[], int tmp[], int nfft, double in1fft[], double tmpfft[]) {
void mp_add(int n, int radix, int in1[], int in2[], int out[]);
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul(int nfft, double d1[], double d2[]);
void mp_mul_cmul_nt_d1(int nfft, double d1[], double d2[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
int n_h, shift;
DGTINT *in2r;
in2r = ((DGTINT *)&in2[2]) - 2;
shift = (nfft >> 1) + 1;
while (n > shift) {
if (in2r[shift + 2] != 0) {
break;
}
shift++;
}
n_h = n / 2 + 1;
if (n_h < n - shift) {
n_h = n - shift;
}
/* ---- tmp = (upper) in1 * (upper) in2 ---- */
mp_mul_i2d(n, radix, nfft, 0, in1, in1fft);
mp_mul_i2d(n, radix, nfft, 0, in2, tmpfft);
mp_mul_cmul(nfft, in1fft, tmpfft);
mp_mul_d2i(n, radix, nfft, tmpfft, tmp);
/* ---- out = tmp + (upper) in1 * (lower) in2 ---- */
mp_mul_i2d(n, radix, nfft, shift, in2, tmpfft);
mp_mul_cmul_nt_d1(nfft, in1fft, tmpfft);
mp_mul_d2i(n_h, radix, nfft, tmpfft, out);
mp_add(n, radix, out, tmp, out);
}
void mp_mulhf_use_in1fft(int n, int radix, double in1fft[], int in2[], int out[], int tmp[], int nfft, double tmpfft[]) {
void mp_add(int n, int radix, int in1[], int in2[], int out[]);
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul_nt_d1(int nfft, double d1[], double d2[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
int n_h, shift;
DGTINT *in2r;
in2r = ((DGTINT *)&in2[2]) - 2;
shift = (nfft >> 1) + 1;
while (n > shift) {
if (in2r[shift + 2] != 0) {
break;
}
shift++;
}
n_h = n / 2 + 1;
if (n_h < n - shift) {
n_h = n - shift;
}
/* ---- tmp = (upper) in1fft * (upper) in2 ---- */
mp_mul_i2d(n, radix, nfft, 0, in2, tmpfft);
mp_mul_cmul_nt_d1(nfft, in1fft, tmpfft);
mp_mul_d2i(n, radix, nfft, tmpfft, tmp);
/* ---- out = tmp + (upper) in1 * (lower) in2 ---- */
mp_mul_i2d(n, radix, nfft, shift, in2, tmpfft);
mp_mul_cmul_nt_d1(nfft, in1fft, tmpfft);
mp_mul_d2i(n_h, radix, nfft, tmpfft, out);
mp_add(n, radix, out, tmp, out);
}
void mp_squhf_use_infft(int n, int radix, double infft[], int in[], int out[], int tmp[], int nfft, double tmpfft[]) {
void mp_add(int n, int radix, int in1[], int in2[], int out[]);
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul_nt_d1(int nfft, double d1[], double d2[]);
void mp_mul_csqu_nt_d1(int nfft, double d1[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
int n_h, shift;
DGTINT *inr;
inr = ((DGTINT *)&in[2]) - 2;
shift = (nfft >> 1) + 1;
while (n > shift) {
if (inr[shift + 2] != 0) {
break;
}
shift++;
}
n_h = n / 2 + 1;
if (n_h < n - shift) {
n_h = n - shift;
}
/* ---- tmp = (upper) infft * (lower) in ---- */
mp_mul_i2d(n, radix, nfft, shift, in, tmpfft);
mp_mul_cmul_nt_d1(nfft, infft, tmpfft);
mp_mul_d2i(n_h, radix, nfft, tmpfft, tmp);
/* ---- out = tmp + ((upper) infft)^2 ---- */
mp_mul_csqu_nt_d1(nfft, infft);
mp_mul_d2i(n, radix, nfft, infft, out);
mp_add(n, radix, out, tmp, out);
}
void mp_mulh(int n, int radix, int in1[], int in2[], int out[], int nfft, double in1fft[], double outfft[]) {
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul(int nfft, double d1[], double d2[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
mp_mul_i2d(n, radix, nfft, 0, in1, in1fft);
mp_mul_i2d(n, radix, nfft, 0, in2, outfft);
mp_mul_cmul(nfft, in1fft, outfft);
mp_mul_d2i(n, radix, nfft, outfft, out);
}
void mp_mulh_use_in1fft(int n, int radix, double in1fft[], int shift, int in2[], int out[], int nfft, double outfft[]) {
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_cmul_nt_d1(int nfft, double d1[], double d2[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
DGTINT *in2r;
in2r = ((DGTINT *)&in2[2]) - 2;
while (n > shift) {
if (in2r[shift + 2] != 0) {
break;
}
shift++;
}
mp_mul_i2d(n, radix, nfft, shift, in2, outfft);
mp_mul_cmul_nt_d1(nfft, in1fft, outfft);
mp_mul_d2i(n, radix, nfft, outfft, out);
}
void mp_squh(int n, int radix, int in[], int out[], int nfft, double outfft[]) {
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_csqu(int nfft, double d1[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
mp_mul_i2d(n, radix, nfft, 0, in, outfft);
mp_mul_csqu(nfft, outfft);
mp_mul_d2i(n, radix, nfft, outfft, out);
}
void mp_squh_save_infft(int n, int radix, int in[], int out[], int nfft, double infft[], double outfft[]) {
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]);
void mp_mul_csqu_save_d1(int nfft, double d1[], double d2[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
mp_mul_i2d(n, radix, nfft, 0, in, infft);
mp_mul_csqu_save_d1(nfft, infft, outfft);
mp_mul_d2i(n, radix, nfft, outfft, out);
}
void mp_squh_use_in1fft(int n, int radix, double inoutfft[], int out[], int nfft) {
void mp_mul_csqu_nt_d1(int nfft, double d1[]);
void mp_mul_d2i(int n, int radix, int nfft, double din[], int out[]);
mp_mul_csqu_nt_d1(nfft, inoutfft);
mp_mul_d2i(n, radix, nfft, inoutfft, out);
}
/* -------- mp_mul child routines -------- */
void mp_mul_i2d(int n, int radix, int nfft, int shift, int in[], double dout[]) {
int j, x, carry, ndata, radix_2, topdgt;
DGTINT *inr;
inr = ((DGTINT *)&in[2]) - 2;
ndata = 0;
topdgt = 0;
if (n > shift) {
topdgt = inr[shift + 2];
ndata = (nfft >> 1) + 1;
if (ndata > n - shift) {
ndata = n - shift;
}
}
dout[nfft + 1] = in[0] * topdgt;
for (j = nfft; j > ndata; j--) {
dout[j] = 0;
}
/* ---- abs(dout[j]) <= radix/2 (to keep FFT precision) ---- */
if (ndata > 1) {
radix_2 = radix / 2;
carry = 0;
for (j = ndata + 1; j > 3; j--) {
x = inr[j + shift] - carry;
carry = x >= radix_2 ? -1 : 0;
dout[j - 1] = x - (radix & carry);
}
dout[2] = inr[shift + 3] - carry;
}
dout[1] = topdgt;
dout[0] = in[1] - shift;
}
void mp_mul_cmul(int nfft, double d1[], double d2[]) {
void cdft(int n, int isgn, double *a);
void mp_mul_rcmul(int n, double *a, double *b);
double xr, xi;
cdft(nfft, 1, &d1[1]);
cdft(nfft, 1, &d2[1]);
d2[0] += d1[0];
xr = d1[1] * d2[1] + d1[2] * d2[2];
xi = d1[1] * d2[2] + d1[2] * d2[1];
d2[1] = xr;
d2[2] = xi;
if (nfft > 2) {
mp_mul_rcmul(nfft, &d1[1], &d2[1]);
}
d2[nfft + 1] *= d1[nfft + 1];
cdft(nfft, -1, &d2[1]);
}
void mp_mul_cmul_nt_d1(int nfft, double d1[], double d2[]) {
void cdft(int n, int isgn, double *a);
void mp_mul_rcmul_nt_in1(int n, double *a, double *b);
double xr, xi;
cdft(nfft, 1, &d2[1]);
d2[0] += d1[0];
xr = d1[1] * d2[1] + d1[2] * d2[2];
xi = d1[1] * d2[2] + d1[2] * d2[1];
d2[1] = xr;
d2[2] = xi;
if (nfft > 2) {
mp_mul_rcmul_nt_in1(nfft, &d1[1], &d2[1]);
}
d2[nfft + 1] *= d1[nfft + 1];
cdft(nfft, -1, &d2[1]);
}
void mp_mul_cmul_nt_d2(int nfft, double d1[], double d2[]) {
void cdft(int n, int isgn, double *a);
void mp_mul_rcmul_nt_in2(int n, double *a, double *b);
double xr, xi;
cdft(nfft, 1, &d1[1]);
d2[0] += d1[0];
xr = d1[1] * d2[1] + d1[2] * d2[2];
xi = d1[1] * d2[2] + d1[2] * d2[1];
d2[1] = xr;
d2[2] = xi;
if (nfft > 2) {
mp_mul_rcmul_nt_in2(nfft, &d1[1], &d2[1]);
}
d2[nfft + 1] *= d1[nfft + 1];
cdft(nfft, -1, &d2[1]);
}
void mp_mul_cmul_nt_out(int nfft, double d1[], double d2[]) {
void cdft(int n, int isgn, double *a);
void mp_mul_rcmul_nt_out(int n, double *a, double *b);
double xr, xi;
cdft(nfft, 1, &d1[1]);
cdft(nfft, 1, &d2[1]);
d2[0] += d1[0];
xr = d1[1] * d2[1] + d1[2] * d2[2];
xi = d1[1] * d2[2] + d1[2] * d2[1];
d2[1] = xr;
d2[2] = xi;
if (nfft > 2) {
mp_mul_rcmul_nt_out(nfft, &d1[1], &d2[1]);
}
d2[nfft + 1] *= d1[nfft + 1];
}
void mp_mul_cmul_nt_d1_add(int nfft, double d1[], double d2[], double d3[]) {
void cdft(int n, int isgn, double *a);
void mp_mul_rcmul_nt_in1_add(int n, double *a, double *b, double *badd);
double xr, xi;
cdft(nfft, 1, &d2[1]);
xr = d1[1] * d2[1] + d1[2] * d2[2];
xi = d1[1] * d2[2] + d1[2] * d2[1];
d3[1] += xr;
d3[2] += xi;
if (nfft > 2) {
mp_mul_rcmul_nt_in1_add(nfft, &d1[1], &d2[1], &d3[1]);
}
d3[nfft + 1] += d1[nfft + 1] * d2[nfft + 1];
cdft(nfft, -1, &d3[1]);
}
void mp_mul_csqu(int nfft, double d1[]) {
void cdft(int n, int isgn, double *a);
void mp_mul_rcsqu(int n, double *a);
double xr, xi;
cdft(nfft, 1, &d1[1]);
d1[0] *= 2;
xr = d1[1] * d1[1] + d1[2] * d1[2];
xi = 2 * d1[1] * d1[2];
d1[1] = xr;
d1[2] = xi;
if (nfft > 2) {