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Diffstat (limited to 'usr/src/lib/libm/common/complex/cacos.c')
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diff --git a/usr/src/lib/libm/common/complex/cacos.c b/usr/src/lib/libm/common/complex/cacos.c new file mode 100644 index 0000000000..4fccae23bb --- /dev/null +++ b/usr/src/lib/libm/common/complex/cacos.c @@ -0,0 +1,404 @@ +/* + * 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); +} |