/*- * Copyright (c) 2013 Bruce D. Evans * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice unmodified, this list of conditions, and the following * disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ #include <complex.h> #include <float.h> #ifdef __i386__ #include <ieeefp.h> #endif #include "fpmath.h" #include "math.h" #include "math_private.h" #define MANT_DIG LDBL_MANT_DIG #define MAX_EXP LDBL_MAX_EXP #define MIN_EXP LDBL_MIN_EXP static const double ln2_hi = 6.9314718055829871e-1; /* 0x162e42fefa0000.0p-53 */ #if LDBL_MANT_DIG == 64 #define MULT_REDUX 0x1p32 /* exponent MANT_DIG / 2 rounded up */ static const double ln2l_lo = 1.6465949582897082e-12; /* 0x1cf79abc9e3b3a.0p-92 */ #elif LDBL_MANT_DIG == 113 #define MULT_REDUX 0x1p57 static const long double ln2l_lo = 1.64659495828970812809844307550013433e-12L; /* 0x1cf79abc9e3b39803f2f6af40f343.0p-152L */ #else #error "Unsupported long double format" #endif long double complex clogl(long double complex z) { long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl; long double sh, sl, t; long double x, y, v; uint16_t hax, hay; int kx, ky; ENTERIT(long double complex); x = creall(z); y = cimagl(z); v = atan2l(y, x); ax = fabsl(x); ay = fabsl(y); if (ax < ay) { t = ax; ax = ay; ay = t; } GET_LDBL_EXPSIGN(hax, ax); kx = hax - 16383; GET_LDBL_EXPSIGN(hay, ay); ky = hay - 16383; /* Handle NaNs and Infs using the general formula. */ if (kx == MAX_EXP || ky == MAX_EXP) RETURNI(CMPLXL(logl(hypotl(x, y)), v)); /* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */ if (ax == 1) { if (ky < (MIN_EXP - 1) / 2) RETURNI(CMPLXL((ay / 2) * ay, v)); RETURNI(CMPLXL(log1pl(ay * ay) / 2, v)); } /* Avoid underflow when ax is not small. Also handle zero args. */ if (kx - ky > MANT_DIG || ay == 0) RETURNI(CMPLXL(logl(ax), v)); /* Avoid overflow. */ if (kx >= MAX_EXP - 1) RETURNI(CMPLXL(logl(hypotl(x * 0x1p-16382L, y * 0x1p-16382L)) + (MAX_EXP - 2) * ln2l_lo + (MAX_EXP - 2) * ln2_hi, v)); if (kx >= (MAX_EXP - 1) / 2) RETURNI(CMPLXL(logl(hypotl(x, y)), v)); /* Reduce inaccuracies and avoid underflow when ax is denormal. */ if (kx <= MIN_EXP - 2) RETURNI(CMPLXL(logl(hypotl(x * 0x1p16383L, y * 0x1p16383L)) + (MIN_EXP - 2) * ln2l_lo + (MIN_EXP - 2) * ln2_hi, v)); /* Avoid remaining underflows (when ax is small but not denormal). */ if (ky < (MIN_EXP - 1) / 2 + MANT_DIG) RETURNI(CMPLXL(logl(hypotl(x, y)), v)); /* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */ t = (long double)(ax * (MULT_REDUX + 1)); axh = (long double)(ax - t) + t; axl = ax - axh; ax2h = ax * ax; ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl; t = (long double)(ay * (MULT_REDUX + 1)); ayh = (long double)(ay - t) + t; ayl = ay - ayh; ay2h = ay * ay; ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl; /* * When log(|z|) is far from 1, accuracy in calculating the sum * of the squares is not very important since log() reduces * inaccuracies. We depended on this to use the general * formula when log(|z|) is very far from 1. When log(|z|) is * moderately far from 1, we go through the extra-precision * calculations to reduce branches and gain a little accuracy. * * When |z| is near 1, we subtract 1 and use log1p() and don't * leave it to log() to subtract 1, since we gain at least 1 bit * of accuracy in this way. * * When |z| is very near 1, subtracting 1 can cancel almost * 3*MANT_DIG bits. We arrange that subtracting 1 is exact in * doubled precision, and then do the rest of the calculation * in sloppy doubled precision. Although large cancellations * often lose lots of accuracy, here the final result is exact * in doubled precision if the large calculation occurs (because * then it is exact in tripled precision and the cancellation * removes enough bits to fit in doubled precision). Thus the * result is accurate in sloppy doubled precision, and the only * significant loss of accuracy is when it is summed and passed * to log1p(). */ sh = ax2h; sl = ay2h; _2sumF(sh, sl); if (sh < 0.5 || sh >= 3) RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v)); sh -= 1; _2sum(sh, sl); _2sum(ax2l, ay2l); /* Briggs-Kahan algorithm (except we discard the final low term): */ _2sum(sh, ax2l); _2sum(sl, ay2l); t = ax2l + sl; _2sumF(sh, t); RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v)); }
