comparison src/fftw-3.3.8/libbench2/mp.c @ 82:d0c2a83c1364

Add FFTW 3.3.8 source, and a Linux build

+#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;
+}