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Diffstat (limited to 'usr/src/lib/libm/common/complex/cpow.c')
-rw-r--r-- | usr/src/lib/libm/common/complex/cpow.c | 337 |
1 files changed, 337 insertions, 0 deletions
diff --git a/usr/src/lib/libm/common/complex/cpow.c b/usr/src/lib/libm/common/complex/cpow.c new file mode 100644 index 0000000000..9fed91435a --- /dev/null +++ b/usr/src/lib/libm/common/complex/cpow.c @@ -0,0 +1,337 @@ +/* + * 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); +} |