642 lines
13 KiB
C
642 lines
13 KiB
C
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#include "config.h"
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#include "libbench2/bench.h"
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#include <math.h>
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#define DG unsigned short
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#define ACC unsigned long
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#define REAL bench_real
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#define BITS_IN_REAL 53 /* mantissa */
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#define SHFT 16
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#define RADIX 65536L
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#define IRADIX (1.0 / RADIX)
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#define LO(x) ((x) & (RADIX - 1))
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#define HI(x) ((x) >> SHFT)
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#define HI_SIGNED(x) \
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((((x) + (ACC)(RADIX >> 1) * RADIX) >> SHFT) - (RADIX >> 1))
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#define ZEROEXP (-32768)
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#define LEN 10
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typedef struct {
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short sign;
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short expt;
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DG d[LEN];
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} N[1];
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#define EXA a->expt
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#define EXB b->expt
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#define EXC c->expt
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#define AD a->d
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#define BD b->d
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#define SGNA a->sign
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#define SGNB b->sign
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static const N zero = {{ 1, ZEROEXP, {0} }};
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static void cpy(const N a, N b)
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{
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*b = *a;
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}
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static void fromreal(REAL x, N a)
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{
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int i, e;
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cpy(zero, a);
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if (x == 0.0) return;
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if (x >= 0) { SGNA = 1; }
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else { SGNA = -1; x = -x; }
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e = 0;
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while (x >= 1.0) { x *= IRADIX; ++e; }
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while (x < IRADIX) { x *= RADIX; --e; }
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EXA = e;
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for (i = LEN - 1; i >= 0 && x != 0.0; --i) {
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REAL y;
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x *= RADIX;
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y = (REAL) ((int) x);
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AD[i] = (DG)y;
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x -= y;
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}
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}
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static void fromshort(int x, N a)
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{
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cpy(zero, a);
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if (x < 0) { x = -x; SGNA = -1; }
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else { SGNA = 1; }
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EXA = 1;
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AD[LEN - 1] = x;
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}
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static void pack(DG *d, int e, int s, int l, N a)
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{
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int i, j;
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for (i = l - 1; i >= 0; --i, --e)
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if (d[i] != 0)
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break;
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if (i < 0) {
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/* number is zero */
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cpy(zero, a);
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} else {
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EXA = e;
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SGNA = s;
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if (i >= LEN - 1) {
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for (j = LEN - 1; j >= 0; --i, --j)
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AD[j] = d[i];
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} else {
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for (j = LEN - 1; i >= 0; --i, --j)
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AD[j] = d[i];
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for ( ; j >= 0; --j)
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AD[j] = 0;
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}
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}
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}
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/* compare absolute values */
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static int abscmp(const N a, const N b)
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{
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int i;
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if (EXA > EXB) return 1;
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if (EXA < EXB) return -1;
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for (i = LEN - 1; i >= 0; --i) {
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if (AD[i] > BD[i])
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return 1;
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if (AD[i] < BD[i])
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return -1;
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}
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return 0;
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}
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static int eq(const N a, const N b)
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{
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return (SGNA == SGNB) && (abscmp(a, b) == 0);
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}
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/* add magnitudes, for |a| >= |b| */
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static void addmag0(int s, const N a, const N b, N c)
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{
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int ia, ib;
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ACC r = 0;
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DG d[LEN + 1];
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for (ia = 0, ib = EXA - EXB; ib < LEN; ++ia, ++ib) {
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r += (ACC)AD[ia] + (ACC)BD[ib];
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d[ia] = LO(r);
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r = HI(r);
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}
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for (; ia < LEN; ++ia) {
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r += (ACC)AD[ia];
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d[ia] = LO(r);
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r = HI(r);
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}
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d[ia] = LO(r);
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pack(d, EXA + 1, s * SGNA, LEN + 1, c);
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}
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static void addmag(int s, const N a, const N b, N c)
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{
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if (abscmp(a, b) > 0) addmag0(1, a, b, c); else addmag0(s, b, a, c);
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}
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/* subtract magnitudes, for |a| >= |b| */
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static void submag0(int s, const N a, const N b, N c)
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{
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int ia, ib;
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ACC r = 0;
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DG d[LEN];
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for (ia = 0, ib = EXA - EXB; ib < LEN; ++ia, ++ib) {
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r += (ACC)AD[ia] - (ACC)BD[ib];
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d[ia] = LO(r);
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r = HI_SIGNED(r);
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}
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for (; ia < LEN; ++ia) {
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r += (ACC)AD[ia];
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d[ia] = LO(r);
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r = HI_SIGNED(r);
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}
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pack(d, EXA, s * SGNA, LEN, c);
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}
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static void submag(int s, const N a, const N b, N c)
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{
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if (abscmp(a, b) > 0) submag0(1, a, b, c); else submag0(s, b, a, c);
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}
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/* c = a + b */
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static void add(const N a, const N b, N c)
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{
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if (SGNA == SGNB) addmag(1, a, b, c); else submag(1, a, b, c);
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}
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static void sub(const N a, const N b, N c)
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{
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if (SGNA == SGNB) submag(-1, a, b, c); else addmag(-1, a, b, c);
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}
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static void mul(const N a, const N b, N c)
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{
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DG d[2 * LEN];
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int i, j, k;
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ACC r;
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for (i = 0; i < LEN; ++i)
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d[2 * i] = d[2 * i + 1] = 0;
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for (i = 0; i < LEN; ++i) {
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ACC ai = AD[i];
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if (ai) {
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r = 0;
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for (j = 0, k = i; j < LEN; ++j, ++k) {
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r += ai * (ACC)BD[j] + (ACC)d[k];
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d[k] = LO(r);
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r = HI(r);
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}
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d[k] = LO(r);
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}
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}
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pack(d, EXA + EXB, SGNA * SGNB, 2 * LEN, c);
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}
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static REAL toreal(const N a)
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{
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REAL h, l, f;
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int i, bits;
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ACC r;
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DG sticky;
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if (EXA != ZEROEXP) {
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f = IRADIX;
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i = LEN;
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bits = 0;
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h = (r = AD[--i]) * f; f *= IRADIX;
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for (bits = 0; r > 0; ++bits)
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r >>= 1;
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/* first digit */
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while (bits + SHFT <= BITS_IN_REAL) {
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h += AD[--i] * f; f *= IRADIX; bits += SHFT;
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}
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/* guard digit (leave one bit for sticky bit, hence `<' instead
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of `<=') */
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bits = 0; l = 0.0;
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while (bits + SHFT < BITS_IN_REAL) {
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l += AD[--i] * f; f *= IRADIX; bits += SHFT;
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}
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/* sticky bit */
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sticky = 0;
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while (i > 0)
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sticky |= AD[--i];
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if (sticky)
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l += (RADIX / 2) * f;
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h += l;
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for (i = 0; i < EXA; ++i) h *= (REAL)RADIX;
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for (i = 0; i > EXA; --i) h *= IRADIX;
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if (SGNA == -1) h = -h;
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return h;
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} else {
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return 0.0;
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}
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}
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static void neg(N a)
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{
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SGNA = -SGNA;
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}
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static void inv(const N a, N x)
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{
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N w, z, one, two;
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fromreal(1.0 / toreal(a), x); /* initial guess */
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fromshort(1, one);
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fromshort(2, two);
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for (;;) {
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/* Newton */
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mul(a, x, w);
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sub(two, w, z);
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if (eq(one, z)) break;
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mul(x, z, x);
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}
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}
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/* 2 pi */
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static const N n2pi = {{
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1, 1,
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{18450, 59017, 1760, 5212, 9779, 4518, 2886, 54545, 18558, 6}
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}};
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/* 1 / 31! */
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static const N i31fac = {{
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1, -7,
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{28087, 45433, 51357, 24545, 14291, 3954, 57879, 8109, 38716, 41382}
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}};
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/* 1 / 32! */
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static const N i32fac = {{
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1, -7,
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{52078, 60811, 3652, 39679, 37310, 47227, 28432, 57597, 13497, 1293}
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}};
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static void msin(const N a, N b)
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{
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N a2, g, k;
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int i;
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cpy(i31fac, g);
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cpy(g, b);
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mul(a, a, a2);
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/* Taylor */
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for (i = 31; i > 1; i -= 2) {
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fromshort(i * (i - 1), k);
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mul(k, g, g);
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mul(a2, b, k);
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sub(g, k, b);
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}
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mul(a, b, b);
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}
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static void mcos(const N a, N b)
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{
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N a2, g, k;
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int i;
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cpy(i32fac, g);
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cpy(g, b);
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mul(a, a, a2);
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/* Taylor */
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for (i = 32; i > 0; i -= 2) {
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fromshort(i * (i - 1), k);
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mul(k, g, g);
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mul(a2, b, k);
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sub(g, k, b);
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}
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}
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static void by2pi(REAL m, REAL n, N a)
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{
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N b;
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fromreal(n, b);
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inv(b, a);
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fromreal(m, b);
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mul(a, b, a);
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mul(n2pi, a, a);
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}
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static void sin2pi(REAL m, REAL n, N a);
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static void cos2pi(REAL m, REAL n, N a)
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{
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N b;
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if (m < 0) cos2pi(-m, n, a);
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else if (m > n * 0.5) cos2pi(n - m, n, a);
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else if (m > n * 0.25) {sin2pi(m - n * 0.25, n, a); neg(a);}
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else if (m > n * 0.125) sin2pi(n * 0.25 - m, n, a);
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else { by2pi(m, n, b); mcos(b, a); }
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}
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static void sin2pi(REAL m, REAL n, N a)
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{
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N b;
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if (m < 0) {sin2pi(-m, n, a); neg(a);}
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else if (m > n * 0.5) {sin2pi(n - m, n, a); neg(a);}
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else if (m > n * 0.25) {cos2pi(m - n * 0.25, n, a);}
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else if (m > n * 0.125) {cos2pi(n * 0.25 - m, n, a);}
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else {by2pi(m, n, b); msin(b, a);}
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}
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/*----------------------------------------------------------------------*/
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/* FFT stuff */
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/* (r0 + i i0)(r1 + i i1) */
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static void cmul(N r0, N i0, N r1, N i1, N r2, N i2)
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{
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N s, t, q;
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mul(r0, r1, s);
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mul(i0, i1, t);
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sub(s, t, q);
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mul(r0, i1, s);
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mul(i0, r1, t);
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add(s, t, i2);
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cpy(q, r2);
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}
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/* (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;
|
||
|
|
}
|