furnace/extern/fftw/libbench2/mp.c

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#include "config.h"
#include "libbench2/bench.h"
#include <math.h>
#define DG unsigned short
#define ACC unsigned long
#define REAL bench_real
#define BITS_IN_REAL 53 /* mantissa */
#define SHFT 16
#define RADIX 65536L
#define IRADIX (1.0 / RADIX)
#define LO(x) ((x) & (RADIX - 1))
#define HI(x) ((x) >> SHFT)
#define HI_SIGNED(x) \
((((x) + (ACC)(RADIX >> 1) * RADIX) >> SHFT) - (RADIX >> 1))
#define ZEROEXP (-32768)
#define LEN 10
typedef struct {
short sign;
short expt;
DG d[LEN];
} N[1];
#define EXA a->expt
#define EXB b->expt
#define EXC c->expt
#define AD a->d
#define BD b->d
#define SGNA a->sign
#define SGNB b->sign
static const N zero = {{ 1, ZEROEXP, {0} }};
static void cpy(const N a, N b)
{
*b = *a;
}
static void fromreal(REAL x, N a)
{
int i, e;
cpy(zero, a);
if (x == 0.0) return;
if (x >= 0) { SGNA = 1; }
else { SGNA = -1; x = -x; }
e = 0;
while (x >= 1.0) { x *= IRADIX; ++e; }
while (x < IRADIX) { x *= RADIX; --e; }
EXA = e;
for (i = LEN - 1; i >= 0 && x != 0.0; --i) {
REAL y;
x *= RADIX;
y = (REAL) ((int) x);
AD[i] = (DG)y;
x -= y;
}
}
static void fromshort(int x, N a)
{
cpy(zero, a);
if (x < 0) { x = -x; SGNA = -1; }
else { SGNA = 1; }
EXA = 1;
AD[LEN - 1] = x;
}
static void pack(DG *d, int e, int s, int l, N a)
{
int i, j;
for (i = l - 1; i >= 0; --i, --e)
if (d[i] != 0)
break;
if (i < 0) {
/* number is zero */
cpy(zero, a);
} else {
EXA = e;
SGNA = s;
if (i >= LEN - 1) {
for (j = LEN - 1; j >= 0; --i, --j)
AD[j] = d[i];
} else {
for (j = LEN - 1; i >= 0; --i, --j)
AD[j] = d[i];
for ( ; j >= 0; --j)
AD[j] = 0;
}
}
}
/* compare absolute values */
static int abscmp(const N a, const N b)
{
int i;
if (EXA > EXB) return 1;
if (EXA < EXB) return -1;
for (i = LEN - 1; i >= 0; --i) {
if (AD[i] > BD[i])
return 1;
if (AD[i] < BD[i])
return -1;
}
return 0;
}
static int eq(const N a, const N b)
{
return (SGNA == SGNB) && (abscmp(a, b) == 0);
}
/* add magnitudes, for |a| >= |b| */
static void addmag0(int s, const N a, const N b, N c)
{
int ia, ib;
ACC r = 0;
DG d[LEN + 1];
for (ia = 0, ib = EXA - EXB; ib < LEN; ++ia, ++ib) {
r += (ACC)AD[ia] + (ACC)BD[ib];
d[ia] = LO(r);
r = HI(r);
}
for (; ia < LEN; ++ia) {
r += (ACC)AD[ia];
d[ia] = LO(r);
r = HI(r);
}
d[ia] = LO(r);
pack(d, EXA + 1, s * SGNA, LEN + 1, c);
}
static void addmag(int s, const N a, const N b, N c)
{
if (abscmp(a, b) > 0) addmag0(1, a, b, c); else addmag0(s, b, a, c);
}
/* subtract magnitudes, for |a| >= |b| */
static void submag0(int s, const N a, const N b, N c)
{
int ia, ib;
ACC r = 0;
DG d[LEN];
for (ia = 0, ib = EXA - EXB; ib < LEN; ++ia, ++ib) {
r += (ACC)AD[ia] - (ACC)BD[ib];
d[ia] = LO(r);
r = HI_SIGNED(r);
}
for (; ia < LEN; ++ia) {
r += (ACC)AD[ia];
d[ia] = LO(r);
r = HI_SIGNED(r);
}
pack(d, EXA, s * SGNA, LEN, c);
}
static void submag(int s, const N a, const N b, N c)
{
if (abscmp(a, b) > 0) submag0(1, a, b, c); else submag0(s, b, a, c);
}
/* c = a + b */
static void add(const N a, const N b, N c)
{
if (SGNA == SGNB) addmag(1, a, b, c); else submag(1, a, b, c);
}
static void sub(const N a, const N b, N c)
{
if (SGNA == SGNB) submag(-1, a, b, c); else addmag(-1, a, b, c);
}
static void mul(const N a, const N b, N c)
{
DG d[2 * LEN];
int i, j, k;
ACC r;
for (i = 0; i < LEN; ++i)
d[2 * i] = d[2 * i + 1] = 0;
for (i = 0; i < LEN; ++i) {
ACC ai = AD[i];
if (ai) {
r = 0;
for (j = 0, k = i; j < LEN; ++j, ++k) {
r += ai * (ACC)BD[j] + (ACC)d[k];
d[k] = LO(r);
r = HI(r);
}
d[k] = LO(r);
}
}
pack(d, EXA + EXB, SGNA * SGNB, 2 * LEN, c);
}
static REAL toreal(const N a)
{
REAL h, l, f;
int i, bits;
ACC r;
DG sticky;
if (EXA != ZEROEXP) {
f = IRADIX;
i = LEN;
bits = 0;
h = (r = AD[--i]) * f; f *= IRADIX;
for (bits = 0; r > 0; ++bits)
r >>= 1;
/* first digit */
while (bits + SHFT <= BITS_IN_REAL) {
h += AD[--i] * f; f *= IRADIX; bits += SHFT;
}
/* guard digit (leave one bit for sticky bit, hence `<' instead
of `<=') */
bits = 0; l = 0.0;
while (bits + SHFT < BITS_IN_REAL) {
l += AD[--i] * f; f *= IRADIX; bits += SHFT;
}
/* sticky bit */
sticky = 0;
while (i > 0)
sticky |= AD[--i];
if (sticky)
l += (RADIX / 2) * f;
h += l;
for (i = 0; i < EXA; ++i) h *= (REAL)RADIX;
for (i = 0; i > EXA; --i) h *= IRADIX;
if (SGNA == -1) h = -h;
return h;
} else {
return 0.0;
}
}
static void neg(N a)
{
SGNA = -SGNA;
}
static void inv(const N a, N x)
{
N w, z, one, two;
fromreal(1.0 / toreal(a), x); /* initial guess */
fromshort(1, one);
fromshort(2, two);
for (;;) {
/* Newton */
mul(a, x, w);
sub(two, w, z);
if (eq(one, z)) break;
mul(x, z, x);
}
}
/* 2 pi */
static const N n2pi = {{
1, 1,
{18450, 59017, 1760, 5212, 9779, 4518, 2886, 54545, 18558, 6}
}};
/* 1 / 31! */
static const N i31fac = {{
1, -7,
{28087, 45433, 51357, 24545, 14291, 3954, 57879, 8109, 38716, 41382}
}};
/* 1 / 32! */
static const N i32fac = {{
1, -7,
{52078, 60811, 3652, 39679, 37310, 47227, 28432, 57597, 13497, 1293}
}};
static void msin(const N a, N b)
{
N a2, g, k;
int i;
cpy(i31fac, g);
cpy(g, b);
mul(a, a, a2);
/* Taylor */
for (i = 31; i > 1; i -= 2) {
fromshort(i * (i - 1), k);
mul(k, g, g);
mul(a2, b, k);
sub(g, k, b);
}
mul(a, b, b);
}
static void mcos(const N a, N b)
{
N a2, g, k;
int i;
cpy(i32fac, g);
cpy(g, b);
mul(a, a, a2);
/* Taylor */
for (i = 32; i > 0; i -= 2) {
fromshort(i * (i - 1), k);
mul(k, g, g);
mul(a2, b, k);
sub(g, k, b);
}
}
static void by2pi(REAL m, REAL n, N a)
{
N b;
fromreal(n, b);
inv(b, a);
fromreal(m, b);
mul(a, b, a);
mul(n2pi, a, a);
}
static void sin2pi(REAL m, REAL n, N a);
static void cos2pi(REAL m, REAL n, N a)
{
N b;
if (m < 0) cos2pi(-m, n, a);
else if (m > n * 0.5) cos2pi(n - m, n, a);
else if (m > n * 0.25) {sin2pi(m - n * 0.25, n, a); neg(a);}
else if (m > n * 0.125) sin2pi(n * 0.25 - m, n, a);
else { by2pi(m, n, b); mcos(b, a); }
}
static void sin2pi(REAL m, REAL n, N a)
{
N b;
if (m < 0) {sin2pi(-m, n, a); neg(a);}
else if (m > n * 0.5) {sin2pi(n - m, n, a); neg(a);}
else if (m > n * 0.25) {cos2pi(m - n * 0.25, n, a);}
else if (m > n * 0.125) {cos2pi(n * 0.25 - m, n, a);}
else {by2pi(m, n, b); msin(b, a);}
}
/*----------------------------------------------------------------------*/
/* FFT stuff */
/* (r0 + i i0)(r1 + i i1) */
static void cmul(N r0, N i0, N r1, N i1, N r2, N i2)
{
N s, t, q;
mul(r0, r1, s);
mul(i0, i1, t);
sub(s, t, q);
mul(r0, i1, s);
mul(i0, r1, t);
add(s, t, i2);
cpy(q, r2);
}
/* (r0 - i i0)(r1 + i i1) */
static void cmulj(N r0, N i0, N r1, N i1, N r2, N i2)
{
N s, t, q;
mul(r0, r1, s);
mul(i0, i1, t);
add(s, t, q);
mul(r0, i1, s);
mul(i0, r1, t);
sub(s, t, i2);
cpy(q, r2);
}
static void mcexp(int m, int n, N r, N i)
{
static int cached_n = -1;
static N w[64][2];
int k, j;
if (n != cached_n) {
for (j = 1, k = 0; j < n; j += j, ++k) {
cos2pi(j, n, w[k][0]);
sin2pi(j, n, w[k][1]);
}
cached_n = n;
}
fromshort(1, r);
fromshort(0, i);
if (m > 0) {
for (k = 0; m; ++k, m >>= 1)
if (m & 1)
cmul(w[k][0], w[k][1], r, i, r, i);
} else {
m = -m;
for (k = 0; m; ++k, m >>= 1)
if (m & 1)
cmulj(w[k][0], w[k][1], r, i, r, i);
}
}
static void bitrev(int n, N *a)
{
int i, j, m;
for (i = j = 0; i < n - 1; ++i) {
if (i < j) {
N t;
cpy(a[2*i], t); cpy(a[2*j], a[2*i]); cpy(t, a[2*j]);
cpy(a[2*i+1], t); cpy(a[2*j+1], a[2*i+1]); cpy(t, a[2*j+1]);
}
/* bit reversed counter */
m = n; do { m >>= 1; j ^= m; } while (!(j & m));
}
}
static void fft0(int n, N *a, int sign)
{
int i, j, k;
bitrev(n, a);
for (i = 1; i < n; i = 2 * i) {
for (j = 0; j < i; ++j) {
N wr, wi;
mcexp(sign * (int)j, 2 * i, wr, wi);
for (k = j; k < n; k += 2 * i) {
N *a0 = a + 2 * k;
N *a1 = a0 + 2 * i;
N r0, i0, r1, i1, t0, t1, xr, xi;
cpy(a0[0], r0); cpy(a0[1], i0);
cpy(a1[0], r1); cpy(a1[1], i1);
mul(r1, wr, t0); mul(i1, wi, t1); sub(t0, t1, xr);
mul(r1, wi, t0); mul(i1, wr, t1); add(t0, t1, xi);
add(r0, xr, a0[0]); add(i0, xi, a0[1]);
sub(r0, xr, a1[0]); sub(i0, xi, a1[1]);
}
}
}
}
/* a[2*k]+i*a[2*k+1] = exp(2*pi*i*k^2/(2*n)) */
static void bluestein_sequence(int n, N *a)
{
int k, ksq, n2 = 2 * n;
ksq = 1; /* (-1)^2 */
for (k = 0; k < n; ++k) {
/* careful with overflow */
ksq = ksq + 2*k - 1; while (ksq > n2) ksq -= n2;
mcexp(ksq, n2, a[2*k], a[2*k+1]);
}
}
static int pow2_atleast(int x)
{
int h;
for (h = 1; h < x; h = 2 * h)
;
return h;
}
static N *cached_bluestein_w = 0;
static N *cached_bluestein_y = 0;
static int cached_bluestein_n = -1;
static void bluestein(int n, N *a)
{
int nb = pow2_atleast(2 * n);
N *b = (N *)bench_malloc(2 * nb * sizeof(N));
N *w = cached_bluestein_w;
N *y = cached_bluestein_y;
N nbinv;
int i;
fromreal(1.0 / nb, nbinv); /* exact because nb = 2^k */
if (cached_bluestein_n != n) {
if (w) bench_free(w);
if (y) bench_free(y);
w = (N *)bench_malloc(2 * n * sizeof(N));
y = (N *)bench_malloc(2 * nb * sizeof(N));
cached_bluestein_n = n;
cached_bluestein_w = w;
cached_bluestein_y = y;
bluestein_sequence(n, w);
for (i = 0; i < 2*nb; ++i) cpy(zero, y[i]);
for (i = 0; i < n; ++i) {
cpy(w[2*i], y[2*i]);
cpy(w[2*i+1], y[2*i+1]);
}
for (i = 1; i < n; ++i) {
cpy(w[2*i], y[2*(nb-i)]);
cpy(w[2*i+1], y[2*(nb-i)+1]);
}
fft0(nb, y, -1);
}
for (i = 0; i < 2*nb; ++i) cpy(zero, b[i]);
for (i = 0; i < n; ++i)
cmulj(w[2*i], w[2*i+1], a[2*i], a[2*i+1], b[2*i], b[2*i+1]);
/* scaled convolution b * y */
fft0(nb, b, -1);
for (i = 0; i < nb; ++i)
cmul(b[2*i], b[2*i+1], y[2*i], y[2*i+1], b[2*i], b[2*i+1]);
fft0(nb, b, 1);
for (i = 0; i < n; ++i) {
cmulj(w[2*i], w[2*i+1], b[2*i], b[2*i+1], a[2*i], a[2*i+1]);
mul(nbinv, a[2*i], a[2*i]);
mul(nbinv, a[2*i+1], a[2*i+1]);
}
bench_free(b);
}
static void swapri(int n, N *a)
{
int i;
for (i = 0; i < n; ++i) {
N t;
cpy(a[2 * i], t);
cpy(a[2 * i + 1], a[2 * i]);
cpy(t, a[2 * i + 1]);
}
}
static void fft1(int n, N *a, int sign)
{
if (power_of_two(n)) {
fft0(n, a, sign);
} else {
if (sign == 1) swapri(n, a);
bluestein(n, a);
if (sign == 1) swapri(n, a);
}
}
static void fromrealv(int n, bench_complex *a, N *b)
{
int i;
for (i = 0; i < n; ++i) {
fromreal(c_re(a[i]), b[2 * i]);
fromreal(c_im(a[i]), b[2 * i + 1]);
}
}
static void compare(int n, N *a, N *b, double *err)
{
int i;
double e1, e2, einf;
double n1, n2, ninf;
e1 = e2 = einf = 0.0;
n1 = n2 = ninf = 0.0;
# define DO(x1, x2, xinf, var) { \
double d = var; \
if (d < 0) d = -d; \
x1 += d; x2 += d * d; if (d > xinf) xinf = d; \
}
for (i = 0; i < 2 * n; ++i) {
N dd;
sub(a[i], b[i], dd);
DO(n1, n2, ninf, toreal(a[i]));
DO(e1, e2, einf, toreal(dd));
}
# undef DO
err[0] = e1 / n1;
err[1] = sqrt(e2 / n2);
err[2] = einf / ninf;
}
void fftaccuracy(int n, bench_complex *a, bench_complex *ffta,
int sign, double err[6])
{
N *b = (N *)bench_malloc(2 * n * sizeof(N));
N *fftb = (N *)bench_malloc(2 * n * sizeof(N));
N mn, ninv;
int i;
fromreal(n, mn); inv(mn, ninv);
/* forward error */
fromrealv(n, a, b); fromrealv(n, ffta, fftb);
fft1(n, b, sign);
compare(n, b, fftb, err);
/* backward error */
fromrealv(n, a, b); fromrealv(n, ffta, fftb);
for (i = 0; i < 2 * n; ++i) mul(fftb[i], ninv, fftb[i]);
fft1(n, fftb, -sign);
compare(n, b, fftb, err + 3);
bench_free(fftb);
bench_free(b);
}
void fftaccuracy_done(void)
{
if (cached_bluestein_w) bench_free(cached_bluestein_w);
if (cached_bluestein_y) bench_free(cached_bluestein_y);
cached_bluestein_w = 0;
cached_bluestein_y = 0;
cached_bluestein_n = -1;
}