/* * 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 catanl = __catanl /* INDENT OFF */ /* * ldcomplex catanl(ldcomplex z); * * Atan(z) return A + Bi where, * 1 * A = --- * atan2(2x, 1-x*x-y*y) * 2 * * 1 [ x*x + (y+1)*(y+1) ] 1 4y * B = --- log [ ----------------- ] = - log (1+ -----------------) * 4 [ x*x + (y-1)*(y-1) ] 4 x*x + (y-1)*(y-1) * * 2 16 3 y * = t - 2t + -- t - ..., where t = ----------------- * 3 x*x + (y-1)*(y-1) * Proof: * Let w = atan(z=x+yi) = A + B i. Then tan(w) = z. * Since sin(w) = (exp(iw)-exp(-iw))/(2i), cos(w)=(exp(iw)+exp(-iw))/(2), * Let p = exp(iw), then z = tan(w) = ((p-1/p)/(p+1/p))/i, or * iz = (p*p-1)/(p*p+1), or, after simplification, * p*p = (1+iz)/(1-iz) ... (1) * LHS of (1) = exp(2iw) = exp(2i(A+Bi)) = exp(-2B)*exp(2iA) * = exp(-2B)*(cos(2A)+i*sin(2A)) ... (2) * 1-y+ix (1-y+ix)*(1+y+ix) 1-x*x-y*y + 2xi * RHS of (1) = ------ = ----------------- = --------------- ... (3) * 1+y-ix (1+y)**2 + x**2 (1+y)**2 + x**2 * * Comparing the real and imaginary parts of (2) and (3), we have: * cos(2A) : 1-x*x-y*y = sin(2A) : 2x * and hence * tan(2A) = 2x/(1-x*x-y*y), or * A = 0.5 * atan2(2x, 1-x*x-y*y) ... (4) * * For the imaginary part B, Note that |p*p| = exp(-2B), and * |1+iz| |i-z| hypot(x,(y-1)) * |----| = |---| = -------------- * |1-iz| |i+z| hypot(x,(y+1)) * Thus * x*x + (y+1)*(y+1) * exp(4B) = -----------------, or * x*x + (y-1)*(y-1) * * 1 [x^2+(y+1)^2] 1 4y * B = - log [-----------] = - log(1+ -------------) ... (5) * 4 [x^2+(y-1)^2] 4 x^2+(y-1)^2 * * QED. * * Note that: if catan( x, y) = ( u, v), then * catan(-x, y) = (-u, v) * catan( x,-y) = ( u,-v) * * Also, catan(x,y) = -i*catanh(-y,x), or * catanh(x,y) = i*catan(-y,x) * So, if catanh(y,x) = (v,u), then catan(x,y) = -i*(-v,u) = (u,v), i.e., * catan(x,y) = (u,v) * * EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)): * catan( 0 , 0 ) = (0 , 0 ) * catan( NaN, 0 ) = (NaN , 0 ) * catan( 0 , 1 ) = (0 , +inf) with divide-by-zero * catan( inf, y ) = (pi/2 , 0 ) for finite +y * catan( NaN, y ) = (NaN , NaN ) with invalid for finite y != 0 * catan( x , inf ) = (pi/2 , 0 ) for finite +x * catan( inf, inf ) = (pi/2 , 0 ) * catan( NaN, inf ) = (NaN , 0 ) * catan( x , NaN ) = (NaN , NaN ) with invalid for finite x * catan( inf, NaN ) = (pi/2 , +-0 ) */ /* INDENT ON */ #include "libm.h" /* atan2l/atanl/fabsl/isinfl/iszerol/log1pl/logl */ #include "complex_wrapper.h" #include "longdouble.h" /* INDENT OFF */ static const long double zero = 0.0L, one = 1.0L, two = 2.0L, half = 0.5L, ln2 = 6.931471805599453094172321214581765680755e-0001L, pi_2 = 1.570796326794896619231321691639751442098584699687552910487472L, #if defined(__x86) E = 2.910383045673370361328125000000000000000e-11L, /* 2**-35 */ Einv = 3.435973836800000000000000000000000000000e+10L; /* 2**+35 */ #else E = 8.673617379884035472059622406959533691406e-19L, /* 2**-60 */ Einv = 1.152921504606846976000000000000000000000e18L; /* 2**+60 */ #endif /* INDENT ON */ ldcomplex catanl(ldcomplex z) { ldcomplex ans; long double x, y, t1, ax, ay, t; int hx, hy, ix, iy; x = LD_RE(z); y = LD_IM(z); ax = fabsl(x); ay = fabsl(y); hx = HI_XWORD(x); hy = HI_XWORD(y); ix = hx & 0x7fffffff; iy = hy & 0x7fffffff; /* x is inf or NaN */ if (ix >= 0x7fff0000) { if (isinfl(x)) { LD_RE(ans) = pi_2; LD_IM(ans) = zero; } else { LD_RE(ans) = x + x; if (iszerol(y) || (isinfl(y))) LD_IM(ans) = zero; else LD_IM(ans) = (fabsl(y) - ay) / (fabsl(y) - ay); } } else if (iy >= 0x7fff0000) { /* y is inf or NaN */ if (isinfl(y)) { LD_RE(ans) = pi_2; LD_IM(ans) = zero; } else { LD_RE(ans) = (fabsl(x) - ax) / (fabsl(x) - ax); LD_IM(ans) = y; } } else if (iszerol(x)) { /* INDENT OFF */ /* * x = 0 * 1 1 * A = --- * atan2(2x, 1-x*x-y*y) = --- atan2(0,1-|y|) * 2 2 * * 1 [ (y+1)*(y+1) ] 1 2 1 2y * B = - log [ ----------- ] = - log (1+ ---) or - log(1+ ----) * 4 [ (y-1)*(y-1) ] 2 y-1 2 1-y */ /* INDENT ON */ t = one - ay; if (ay == one) { /* y=1: catan(0,1)=(0,+inf) with 1/0 signal */ LD_IM(ans) = ay / ax; LD_RE(ans) = zero; } else if (ay > one) { /* y>1 */ LD_IM(ans) = half * log1pl(two / (-t)); LD_RE(ans) = pi_2; } else { /* y<1 */ LD_IM(ans) = half * log1pl((ay + ay) / t); LD_RE(ans) = zero; } } else if (ay < E * (one + ax)) { /* INDENT OFF */ /* * Tiny y (relative to 1+|x|) * |y| < E*(1+|x|) * where E=2**-29, -35, -60 for double, extended, quad precision * * 1 [x<=1: atan(x) * A = - * atan2(2x,1-x*x-y*y) ~ [ 1 1+x * 2 [x>=1: - atan2(2,(1-x)*(-----)) * 2 x * * y/x * B ~ t*(1-2t), where t = ----------------- is tiny * x + (y-1)*(y-1)/x * * y * (when x < 2**-60, t = ----------- ) * (y-1)*(y-1) */ /* INDENT ON */ if (ay == zero) LD_IM(ans) = ay; else { t1 = ay - one; if (ix < 0x3fc30000) t = ay / (t1 * t1); else if (ix > 0x403b0000) t = (ay / ax) / ax; else t = ay / (ax * ax + t1 * t1); LD_IM(ans) = t * (one - two * t); } if (ix < 0x3fff0000) LD_RE(ans) = atanl(ax); else LD_RE(ans) = half * atan2l(two, (one - ax) * (one + one / ax)); } else if (ay > Einv * (one + ax)) { /* INDENT OFF */ /* * Huge y relative to 1+|x| * |y| > Einv*(1+|x|), where Einv~2**(prec/2+3), * 1 * A ~ --- * atan2(2x, -y*y) ~ pi/2 * 2 * y * B ~ t*(1-2t), where t = --------------- is tiny * (y-1)*(y-1) */ /* INDENT ON */ LD_RE(ans) = pi_2; t = (ay / (ay - one)) / (ay - one); LD_IM(ans) = t * (one - (t + t)); } else if (ay == one) { /* INDENT OFF */ /* * y=1 * 1 1 * A = - * atan2(2x, -x*x) = --- atan2(2,-x) * 2 2 * * 1 [ x*x+4] 1 4 [ 0.5(log2-logx) if * B = - log [ -----] = - log (1+ ---) = [ |x| Einv * Einv) { /* INDENT OFF */ /* * Huge x: * when |x| > 1/E^2, * 1 pi * A ~ --- * atan2(2x, -x*x-y*y) ~ --- * 2 2 * y y/x * B ~ t*(1-2t), where t = --------------- = (-------------- )/x * x*x+(y-1)*(y-1) 1+((y-1)/x)^2 */ /* INDENT ON */ LD_RE(ans) = pi_2; t = ((ay / ax) / (one + ((ay - one) / ax) * ((ay - one) / ax))) / ax; LD_IM(ans) = t * (one - (t + t)); } else if (ax < E * E * E * E) { /* INDENT OFF */ /* * Tiny x: * when |x| < E^4, (note that y != 1) * 1 1 * A = --- * atan2(2x, 1-x*x-y*y) ~ --- * atan2(2x,1-y*y) * 2 2 * * 1 [ (y+1)*(y+1) ] 1 2 1 2y * B = - log [ ----------- ] = - log (1+ ---) or - log(1+ ----) * 4 [ (y-1)*(y-1) ] 2 y-1 2 1-y */ /* INDENT ON */ LD_RE(ans) = half * atan2l(ax + ax, (one - ay) * (one + ay)); if (ay > one) /* y>1 */ LD_IM(ans) = half * log1pl(two / (ay - one)); else /* y<1 */ LD_IM(ans) = half * log1pl((ay + ay) / (one - ay)); } else { /* INDENT OFF */ /* * normal x,y * 1 * A = --- * atan2(2x, 1-x*x-y*y) * 2 * * 1 [ x*x+(y+1)*(y+1) ] 1 4y * B = - log [ --------------- ] = - log (1+ -----------------) * 4 [ x*x+(y-1)*(y-1) ] 4 x*x + (y-1)*(y-1) */ /* INDENT ON */ t = one - ay; if (iy >= 0x3ffe0000 && iy < 0x40000000) { /* y close to 1 */ LD_RE(ans) = half * (atan2l((ax + ax), (t * (one + ay) - ax * ax))); } else if (ix >= 0x3ffe0000 && ix < 0x40000000) { /* x close to 1 */ LD_RE(ans) = half * atan2l((ax + ax), ((one - ax) * (one + ax) - ay * ay)); } else LD_RE(ans) = half * atan2l((ax + ax), ((one - ax * ax) - ay * ay)); LD_IM(ans) = 0.25L * log1pl((4.0L * ay) / (ax * ax + t * t)); } if (hx < 0) LD_RE(ans) = -LD_RE(ans); if (hy < 0) LD_IM(ans) = -LD_IM(ans); return (ans); }