cpow.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
#pragma weak cpow = __cpow
/* INDENT OFF */
/*
* dcomplex cpow(dcomplex z);
*
* z**w analytically equivalent to
*
* cpow(z,w) = cexp(w clog(z))
*
* Let z = x+iy, w = u+iv.
* Since
* _________
* / 2 2 -1 y
* log(x+iy) = log(\/ x + y ) + i tan (---)
* x
*
* 1 2 2 -1 y
* = --- log(x + y ) + i tan (---)
* 2 x
* u 2 2 -1 y
* (u+iv)* log(x+iy) = --- log(x + y ) - v tan (---) + (1)
* 2 x
*
* v 2 2 -1 y
* i * [ --- log(x + y ) + u tan (---) ] (2)
* 2 x
*
* = r + i q
*
* Therefore,
* w r+iq r
* z = e = e (cos(q)+i*sin(q))
* _______
* / 2 2
* r \/ x + y -v*atan2(y,x)
* Here e can be expressed as: u * e
*
* Special cases (in the order of appearance):
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. When v = 0, y = 0:
* If x is finite and negative, and u is finite, then
* x ** u = exp(u*pi i) * pow(|x|, u);
* otherwise,
* x ** u = pow(x, u);
* 4. When v = 0, x = 0 or |x| = |y| or x is inf or y is inf:
* (x + y i) ** u = r * exp(q i)
* where
* r = hypot(x,y) ** u
* q = u * atan2pi(y, x)
*
* 5. otherwise, z**w is NAN if any x, y, u, v is a Nan or inf
*
* Note: many results of special cases are obtained in terms of
* polar coordinate. In the conversion from polar to rectangle:
* r exp(q i) = r * cos(q) + r * sin(q) i,
* we regard r * 0 is 0 except when r is a NaN.
*/
/* INDENT ON */
#include "libm.h" /* atan2/exp/fabs/hypot/log/pow/scalbn */
/* atan2pi/exp2/sincos/sincospi/__k_clog_r/__k_atan2 */
#include "complex_wrapper.h"
extern void sincospi(double, double *, double *);
static const double
huge = 1e300,
tiny = 1e-300,
invln2 = 1.44269504088896338700e+00,
ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
one = 1.0,
zero = 0.0;
static const int hiinf = 0x7ff00000;
extern double atan2pi(double, double);
/*
* Assuming |t[0]| > |t[1]| and |t[2]| > |t[3]|, sum4fp subroutine
* compute t[0] + t[1] + t[2] + t[3] into two double fp numbers.
*/
static double
sum4fp(double ta[], double *w) {
double t1, t2, t3, t4, w1, w2, t;
t1 = ta[0]; t2 = ta[1]; t3 = ta[2]; t4 = ta[3];
/*
* Rearrange ti so that |t1| >= |t2| >= |t3| >= |t4|
*/
if (fabs(t4) > fabs(t1)) {
t = t1; t1 = t3; t3 = t;
t = t2; t2 = t4; t4 = t;
} else if (fabs(t3) > fabs(t1)) {
t = t1; t1 = t3;
if (fabs(t4) > fabs(t2)) {
t3 = t4; t4 = t2; t2 = t;
} else {
t3 = t2; t2 = t;
}
} else if (fabs(t3) > fabs(t2)) {
t = t2; t2 = t3;
if (fabs(t4) > fabs(t2)) {
t3 = t4; t4 = t;
} else
t3 = t;
}
/* summing r = t1 + t2 + t3 + t4 to w1 + w2 */
w1 = t3 + t4;
w2 = t4 - (w1 - t3);
t = t2 + w1;
w2 += w1 - (t - t2);
w1 = t + w2;
w2 += t - w1;
t = t1 + w1;
w2 += w1 - (t - t1);
w1 = t + w2;
*w = w2 - (w1 - t);
return (w1);
}
dcomplex
cpow(dcomplex z, dcomplex w) {
dcomplex ans;
double x, y, u, v, t, c, s, r, x2, y2;
double b[4], t1, t2, t3, t4, w1, w2, u1, v1, x1, y1;
int ix, iy, hx, lx, hy, ly, hv, hu, iu, iv, lu, lv;
int i, j, k;
x = D_RE(z);
y = D_IM(z);
u = D_RE(w);
v = D_IM(w);
hx = ((int *) &x)[HIWORD];
lx = ((int *) &x)[LOWORD];
hy = ((int *) &y)[HIWORD];
ly = ((int *) &y)[LOWORD];
hu = ((int *) &u)[HIWORD];
lu = ((int *) &u)[LOWORD];
hv = ((int *) &v)[HIWORD];
lv = ((int *) &v)[LOWORD];
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
iu = hu & 0x7fffffff;
iv = hv & 0x7fffffff;
j = 0;
if ((iv | lv) == 0) { /* z**(real) */
if (((hu - 0x3ff00000) | lu) == 0) { /* z ** 1 = z */
D_RE(ans) = x;
D_IM(ans) = y;
} else if ((iu | lu) == 0) { /* z ** 0 = 1 */
D_RE(ans) = one;
D_IM(ans) = zero;
} else if ((iy | ly) == 0) { /* (real)**(real) */
D_IM(ans) = zero;
if (hx < 0 && ix < hiinf && iu < hiinf) {
/* -x ** u is exp(i*pi*u)*pow(x,u) */
r = pow(-x, u);
sincospi(u, &s, &c);
D_RE(ans) = (c == zero)? c: c * r;
D_IM(ans) = (s == zero)? s: s * r;
} else
D_RE(ans) = pow(x, u);
} else if (((ix | lx) == 0) || ix >= hiinf || iy >= hiinf) {
if (isnan(x) || isnan(y) || isnan(u))
D_RE(ans) = D_IM(ans) = x + y + u;
else {
if ((ix | lx) == 0)
r = fabs(y);
else
r = fabs(x) + fabs(y);
t = atan2pi(y, x);
sincospi(t * u, &s, &c);
D_RE(ans) = (c == zero)? c: c * r;
D_IM(ans) = (s == zero)? s: s * r;
}
} else if (((ix - iy) | (lx - ly)) == 0) { /* |x| = |y| */
if (hx >= 0) {
t = (hy >= 0)? 0.25 : -0.25;
sincospi(t * u, &s, &c);
} else if ((lu & 3) == 0) {
t = (hy >= 0)? 0.75 : -0.75;
sincospi(t * u, &s, &c);
} else {
r = (hy >= 0)? u : -u;
t = -0.25 * r;
w1 = r + t;
w2 = t - (w1 - r);
sincospi(w1, &t1, &t2);
sincospi(w2, &t3, &t4);
s = t1 * t4 + t3 * t2;
c = t2 * t4 - t1 * t3;
}
if (ix < 0x3fe00000) /* |x| < 1/2 */
r = pow(fabs(x + x), u) * exp2(-0.5 * u);
else if (ix >= 0x3ff00000 || iu < 0x408ff800)
/* |x| >= 1 or |u| < 1023 */
r = pow(fabs(x), u) * exp2(0.5 * u);
else /* special treatment */
j = 2;
if (j == 0) {
D_RE(ans) = (c == zero)? c: c * r;
D_IM(ans) = (s == zero)? s: s * r;
}
} else
j = 1;
if (j == 0)
return (ans);
}
if (iu >= hiinf || iv >= hiinf || ix >= hiinf || iy >= hiinf) {
/*
* non-zero imag part(s) with inf component(s) yields NaN
*/
t = fabs(x) + fabs(y) + fabs(u) + fabs(v);
D_RE(ans) = D_IM(ans) = t - t;
} else {
k = 0; /* no scaling */
if (iu > 0x7f000000 || iv > 0x7f000000) {
u *= .0009765625; /* scale 2**-10 to avoid overflow */
v *= .0009765625;
k = 1; /* scale by 2**-10 */
}
/*
* Use similated higher precision arithmetic to compute:
* r = u * log(hypot(x, y)) - v * atan2(y, x)
* q = u * atan2(y, x) + v * log(hypot(x, y))
*/
t1 = __k_clog_r(x, y, &t2);
t3 = __k_atan2(y, x, &t4);
x1 = t1;
y1 = t3;
u1 = u;
v1 = v;
((int *) &u1)[LOWORD] &= 0xf8000000;
((int *) &v1)[LOWORD] &= 0xf8000000;
((int *) &x1)[LOWORD] &= 0xf8000000;
((int *) &y1)[LOWORD] &= 0xf8000000;
x2 = t2 - (x1 - t1); /* log(hypot(x,y)) = x1 + x2 */
y2 = t4 - (y1 - t3); /* atan2(y,x) = y1 + y2 */
/* compute q = u * atan2(y, x) + v * log(hypot(x, y)) */
if (j != 2) {
b[0] = u1 * y1;
b[1] = (u - u1) * y1 + u * y2;
if (j == 1) { /* v = 0 */
w1 = b[0] + b[1];
w2 = b[1] - (w1 - b[0]);
} else {
b[2] = v1 * x1;
b[3] = (v - v1) * x1 + v * x2;
w1 = sum4fp(b, &w2);
}
sincos(w1, &t1, &t2);
sincos(w2, &t3, &t4);
s = t1 * t4 + t3 * t2;
c = t2 * t4 - t1 * t3;
if (k == 1)
/*
* square (cos(q) + i sin(q)) k times to get
* (cos(2^k * q + i sin(2^k * q)
*/
for (i = 0; i < 10; i++) {
t1 = s * c;
c = (c + s) * (c - s);
s = t1 + t1;
}
}
/* compute r = u * (t1, t2) - v * (t3, t4) */
b[0] = u1 * x1;
b[1] = (u - u1) * x1 + u * x2;
if (j == 1) { /* v = 0 */
w1 = b[0] + b[1];
w2 = b[1] - (w1 - b[0]);
} else {
b[2] = -v1 * y1;
b[3] = (v1 - v) * y1 - v * y2;
w1 = sum4fp(b, &w2);
}
/* check over/underflow for exp(w1 + w2) */
if (k && fabs(w1) < 1000.0) {
w1 *= 1024; w2 *= 1024; k = 0;
}
hx = ((int *) &w1)[HIWORD];
lx = ((int *) &w1)[LOWORD];
ix = hx & 0x7fffffff;
/* compute exp(w1 + w2) */
if (ix < 0x3c900000) /* exp(tiny < 2**-54) = 1 */
r = one;
else if (ix >= 0x40880000) /* overflow/underflow */
r = (hx < 0)? tiny * tiny : huge * huge;
else { /* compute exp(w1 + w2) */
k = (int) (invln2 * w1 + ((hx >= 0)? 0.5 : -0.5));
t1 = (double) k;
t2 = w1 - t1 * ln2hi;
t3 = w2 - t1 * ln2lo;
r = exp(t2 + t3);
}
if (c != zero) c *= r;
if (s != zero) s *= r;
if (k != 0) {
c = scalbn(c, k);
s = scalbn(s, k);
}
D_RE(ans) = c;
D_IM(ans) = s;
}
return (ans);
}