/* * 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 __cacos = cacos /* INDENT OFF */ /* * dcomplex cacos(dcomplex z); * * Alogrithm * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's * paper "Implementing the Complex Arcsine and Arccosine Functins Using * Exception Handling", ACM TOMS, Vol 23, pp 299-335) * * The principal value of complex inverse cosine function cacos(z), * where z = x+iy, can be defined by * * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)), * * where the log function is the natural log, and * ____________ ____________ * 1 / 2 2 1 / 2 2 * A = --- / (x+1) + y + --- / (x-1) + y * 2 \/ 2 \/ * ____________ ____________ * 1 / 2 2 1 / 2 2 * B = --- / (x+1) + y - --- / (x-1) + y . * 2 \/ 2 \/ * * The Branch cuts are on the real line from -inf to -1 and from 1 to inf. * The real and imaginary parts are based on Abramowitz and Stegun * [Handbook of Mathematic Functions, 1972]. The sign of the imaginary * part is chosen to be the generally considered the principal value of * this function. * * Notes:1. A is the average of the distances from z to the points (1,0) * and (-1,0) in the complex z-plane, and in particular A>=1. * 2. B is in [-1,1], and A*B = x * * Basic relations * cacos(conj(z)) = conj(cacos(z)) * cacos(-z) = pi - cacos(z) * cacos( z) = pi/2 - casin(z) * * Special cases (conform to ISO/IEC 9899:1999(E)): * cacos(+-0 + i y ) = pi/2 - i y for y is +-0, +-inf, NaN * cacos( x + i inf) = pi/2 - i inf for all x * cacos( x + i NaN) = NaN + i NaN with invalid for non-zero finite x * cacos(-inf + i y ) = pi - i inf for finite +y * cacos( inf + i y ) = 0 - i inf for finite +y * cacos(-inf + i inf) = 3pi/4- i inf * cacos( inf + i inf) = pi/4 - i inf * cacos(+-inf+ i NaN) = NaN - i inf (sign of imaginary is unspecified) * cacos(NaN + i y ) = NaN + i NaN with invalid for finite y * cacos(NaN + i inf) = NaN - i inf * cacos(NaN + i NaN) = NaN + i NaN * * Special Regions (better formula for accuracy and for avoiding spurious * overflow or underflow) (all x and y are assumed nonnegative): * case 1: y = 0 * case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1| * case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number * case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5) * case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number * case 6: tiny x: x < 4 sqrt(u) * -------- * case 1 & 2. y=0 or y/|x-1| is tiny. We have * ____________ _____________ * / 2 2 / y 2 * / (x+-1) + y = |x+-1| / 1 + (------) * \/ \/ |x+-1| * * 1 y 2 * ~ |x+-1| ( 1 + --- (------) ) * 2 |x+-1| * * 2 * y * = |x+-1| + --------. * 2|x+-1| * * Consequently, it is not difficult to see that * 2 * y * [ 1 + ------------ , if x < 1, * [ 2(1+x)(1-x) * [ * [ * [ x, if x = 1 (y = 0), * [ * A ~= [ 2 * [ x * y * [ x + ------------ ~ x, if x > 1 * [ 2(x+1)(x-1) * * and hence * ______ 2 * / 2 y y * A + \/ A - 1 ~ 1 + ---------------- + -----------, if x < 1, * sqrt((x+1)(1-x)) 2(x+1)(1-x) * * * ~ x + sqrt((x-1)*(x+1)), if x >= 1. * * 2 * y * [ x(1 - -----------) ~ x, if x < 1, * [ 2(1+x)(1-x) * B = x/A ~ [ * [ 1, if x = 1, * [ * [ 2 * [ y * [ 1 - ------------ , if x > 1, * [ 2(x+1)(x-1) * Thus * [ acos(x) - i y/sqrt((x-1)*(x+1)), if x < 1, * [ * cacos(x+i*y)~ [ 0 - i 0, if x = 1, * [ * [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1. * * Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26. * case 3. y < 4 sqrt(u), where u = minimum normal x. * After case 1 and 2, this will only occurs when x=1. When x=1, we have * A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ... * and * B = 1/A = 1 - y/2 + y^2/8 + ... * Since * cos(sqrt(y)) ~ 1 - y/2 + ... * we have, for the real part, * acos(B) ~ acos(1 - y/2) ~ sqrt(y) * For the imaginary part, * log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2)) * = log(1+y/2+sqrt(y)) * = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ... * ~ sqrt(y) - y*(sqrt(y)+y/2)/2 * ~ sqrt(y) * * case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus * real part = acos(B) ~ pi/2 * and * imag part = log(y+sqrt(y*y-one)) * * case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x * In this case, * A ~ sqrt(x*x+y*y) * B ~ x/sqrt(x*x+y*y). * Thus * real part = acos(B) = atan(y/x), * imag part = log(A+sqrt(A*A-1)) ~ log(2A) * = log(2) + 0.5*log(x*x+y*y) * = log(2) + log(y) + 0.5*log(1+(x/y)^2) * * case 6. x < 4 sqrt(u). In this case, we have * A ~ sqrt(1+y*y), B = x/sqrt(1+y*y). * Since B is tiny, we have * real part = acos(B) ~ pi/2 * imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y)) * = log(y+sqrt(1+y*y)) * = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2) * = 0.5*log(1+2y(y+sqrt(1+y^2))); * = 0.5*log1p(2y(y+A)); * * cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)), */ /* INDENT ON */ #include "libm.h" #include "complex_wrapper.h" /* INDENT OFF */ static const double zero = 0.0, one = 1.0, E = 1.11022302462515654042e-16, /* 2**-53 */ ln2 = 6.93147180559945286227e-01, pi = 3.1415926535897931159979634685, pi_l = 1.224646799147353177e-16, pi_2 = 1.570796326794896558e+00, pi_2_l = 6.123233995736765886e-17, pi_4 = 0.78539816339744827899949, pi_4_l = 3.061616997868382943e-17, pi3_4 = 2.356194490192344836998, pi3_4_l = 9.184850993605148829195e-17, Foursqrtu = 5.96667258496016539463e-154, /* 2**(-509) */ Acrossover = 1.5, Bcrossover = 0.6417, half = 0.5; /* INDENT ON */ dcomplex cacos(dcomplex z) { double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx; int ix, iy, hx, hy; unsigned lx, ly; dcomplex ans; x = D_RE(z); y = D_IM(z); hx = HI_WORD(x); lx = LO_WORD(x); hy = HI_WORD(y); ly = LO_WORD(y); ix = hx & 0x7fffffff; iy = hy & 0x7fffffff; /* x is 0 */ if ((ix | lx) == 0) { if (((iy | ly) == 0) || (iy >= 0x7ff00000)) { D_RE(ans) = pi_2; D_IM(ans) = -y; return (ans); } } /* |y| is inf or NaN */ if (iy >= 0x7ff00000) { if (ISINF(iy, ly)) { /* cacos(x + i inf) = pi/2 - i inf */ D_IM(ans) = -y; if (ix < 0x7ff00000) { D_RE(ans) = pi_2 + pi_2_l; } else if (ISINF(ix, lx)) { if (hx >= 0) D_RE(ans) = pi_4 + pi_4_l; else D_RE(ans) = pi3_4 + pi3_4_l; } else { D_RE(ans) = x; } } else { /* cacos(x + i NaN) = NaN + i NaN */ D_RE(ans) = y + x; if (ISINF(ix, lx)) D_IM(ans) = -fabs(x); else D_IM(ans) = y; } return (ans); } x = fabs(x); y = fabs(y); /* x is inf or NaN */ if (ix >= 0x7ff00000) { /* x is inf or NaN */ if (ISINF(ix, lx)) { /* x is INF */ D_IM(ans) = -x; if (iy >= 0x7ff00000) { if (ISINF(iy, ly)) { /* INDENT OFF */ /* cacos(inf + i inf) = pi/4 - i inf */ /* cacos(-inf+ i inf) =3pi/4 - i inf */ /* INDENT ON */ if (hx >= 0) D_RE(ans) = pi_4 + pi_4_l; else D_RE(ans) = pi3_4 + pi3_4_l; } else /* INDENT OFF */ /* cacos(inf + i NaN) = NaN - i inf */ /* INDENT ON */ D_RE(ans) = y + y; } else /* INDENT OFF */ /* cacos(inf + iy ) = 0 - i inf */ /* cacos(-inf+ iy ) = pi - i inf */ /* INDENT ON */ if (hx >= 0) D_RE(ans) = zero; else D_RE(ans) = pi + pi_l; } else { /* x is NaN */ /* INDENT OFF */ /* * cacos(NaN + i inf) = NaN - i inf * cacos(NaN + i y ) = NaN + i NaN * cacos(NaN + i NaN) = NaN + i NaN */ /* INDENT ON */ D_RE(ans) = x + y; if (iy >= 0x7ff00000) { D_IM(ans) = -y; } else { D_IM(ans) = x; } } if (hy < 0) D_IM(ans) = -D_IM(ans); return (ans); } if ((iy | ly) == 0) { /* region 1: y=0 */ if (ix < 0x3ff00000) { /* |x| < 1 */ D_RE(ans) = acos(x); D_IM(ans) = zero; } else { D_RE(ans) = zero; if (ix >= 0x43500000) /* |x| >= 2**54 */ D_IM(ans) = ln2 + log(x); else if (ix >= 0x3ff80000) /* x > Acrossover */ D_IM(ans) = log(x + sqrt((x - one) * (x + one))); else { xm1 = x - one; D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one))); } } } else if (y <= E * fabs(x - one)) { /* region 2: y < tiny*|x-1| */ if (ix < 0x3ff00000) { /* x < 1 */ D_RE(ans) = acos(x); D_IM(ans) = y / sqrt((one + x) * (one - x)); } else if (ix >= 0x43500000) { /* |x| >= 2**54 */ D_RE(ans) = y / x; D_IM(ans) = ln2 + log(x); } else { t = sqrt((x - one) * (x + one)); D_RE(ans) = y / t; if (ix >= 0x3ff80000) /* x > Acrossover */ D_IM(ans) = log(x + t); else D_IM(ans) = log1p((x - one) + t); } } else if (y < Foursqrtu) { /* region 3 */ t = sqrt(y); D_RE(ans) = t; D_IM(ans) = t; } else if (E * y - one >= x) { /* region 4 */ D_RE(ans) = pi_2; D_IM(ans) = ln2 + log(y); } else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) { /* x,y>2**509 */ /* region 5: x+1 or y is very large (>= sqrt(max)/8) */ t = x / y; D_RE(ans) = atan(y / x); D_IM(ans) = ln2 + log(y) + half * log1p(t * t); } else if (x < Foursqrtu) { /* region 6: x is very small, < 4sqrt(min) */ D_RE(ans) = pi_2; A = sqrt(one + y * y); if (iy >= 0x3ff80000) /* if y > Acrossover */ D_IM(ans) = log(y + A); else D_IM(ans) = half * log1p((y + y) * (y + A)); } else { /* safe region */ y2 = y * y; xp1 = x + one; xm1 = x - one; R = sqrt(xp1 * xp1 + y2); S = sqrt(xm1 * xm1 + y2); A = half * (R + S); B = x / A; if (B <= Bcrossover) D_RE(ans) = acos(B); else { /* use atan and an accurate approx to a-x */ Apx = A + x; if (x <= one) D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R + xp1) + (S - xm1))) / x); else D_RE(ans) = atan((y * sqrt(half * (Apx / (R + xp1) + Apx / (S + xm1)))) / x); } if (A <= Acrossover) { /* use log1p and an accurate approx to A-1 */ if (x < one) Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1)); else Am1 = half * (y2 / (R + xp1) + (S + xm1)); D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one))); } else { D_IM(ans) = log(A + sqrt(A * A - one)); } } if (hx < 0) D_RE(ans) = pi - D_RE(ans); if (hy >= 0) D_IM(ans) = -D_IM(ans); return (ans); }
