1333577Skib/*- 2333577Skib * Copyright (c) 2013 Bruce D. Evans 3333577Skib * All rights reserved. 4333577Skib * 5333577Skib * Redistribution and use in source and binary forms, with or without 6333577Skib * modification, are permitted provided that the following conditions 7333577Skib * are met: 8333577Skib * 1. Redistributions of source code must retain the above copyright 9333577Skib * notice unmodified, this list of conditions, and the following 10333577Skib * disclaimer. 11333577Skib * 2. Redistributions in binary form must reproduce the above copyright 12333577Skib * notice, this list of conditions and the following disclaimer in the 13333577Skib * documentation and/or other materials provided with the distribution. 14333577Skib * 15333577Skib * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16333577Skib * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17333577Skib * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18333577Skib * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19333577Skib * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20333577Skib * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21333577Skib * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22333577Skib * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23333577Skib * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24333577Skib * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25333577Skib */ 26333577Skib 27333577Skib#include <sys/cdefs.h> 28333577Skib__FBSDID("$FreeBSD: stable/11/lib/msun/src/s_clogl.c 334654 2018-06-05 13:46:18Z kib $"); 29333577Skib 30333577Skib#include <complex.h> 31333577Skib#include <float.h> 32333577Skib#ifdef __i386__ 33333577Skib#include <ieeefp.h> 34333577Skib#endif 35333577Skib 36333577Skib#include "fpmath.h" 37333577Skib#include "math.h" 38333577Skib#include "math_private.h" 39333577Skib 40333577Skib#define MANT_DIG LDBL_MANT_DIG 41333577Skib#define MAX_EXP LDBL_MAX_EXP 42333577Skib#define MIN_EXP LDBL_MIN_EXP 43333577Skib 44333577Skibstatic const double 45333577Skibln2_hi = 6.9314718055829871e-1; /* 0x162e42fefa0000.0p-53 */ 46333577Skib 47333577Skib#if LDBL_MANT_DIG == 64 48333577Skib#define MULT_REDUX 0x1p32 /* exponent MANT_DIG / 2 rounded up */ 49333577Skibstatic const double 50333577Skibln2l_lo = 1.6465949582897082e-12; /* 0x1cf79abc9e3b3a.0p-92 */ 51333577Skib#elif LDBL_MANT_DIG == 113 52333577Skib#define MULT_REDUX 0x1p57 53333577Skibstatic const long double 54333577Skibln2l_lo = 1.64659495828970812809844307550013433e-12L; /* 0x1cf79abc9e3b39803f2f6af40f343.0p-152L */ 55333577Skib#else 56333577Skib#error "Unsupported long double format" 57333577Skib#endif 58333577Skib 59333577Skiblong double complex 60333577Skibclogl(long double complex z) 61333577Skib{ 62333577Skib long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl; 63333577Skib long double sh, sl, t; 64333577Skib long double x, y, v; 65333577Skib uint16_t hax, hay; 66333577Skib int kx, ky; 67333577Skib 68333577Skib ENTERIT(long double complex); 69333577Skib 70333577Skib x = creall(z); 71333577Skib y = cimagl(z); 72333577Skib v = atan2l(y, x); 73333577Skib 74333577Skib ax = fabsl(x); 75333577Skib ay = fabsl(y); 76333577Skib if (ax < ay) { 77333577Skib t = ax; 78333577Skib ax = ay; 79333577Skib ay = t; 80333577Skib } 81333577Skib 82333577Skib GET_LDBL_EXPSIGN(hax, ax); 83333577Skib kx = hax - 16383; 84333577Skib GET_LDBL_EXPSIGN(hay, ay); 85333577Skib ky = hay - 16383; 86333577Skib 87333577Skib /* Handle NaNs and Infs using the general formula. */ 88333577Skib if (kx == MAX_EXP || ky == MAX_EXP) 89333577Skib RETURNI(CMPLXL(logl(hypotl(x, y)), v)); 90333577Skib 91333577Skib /* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */ 92333577Skib if (ax == 1) { 93333577Skib if (ky < (MIN_EXP - 1) / 2) 94333577Skib RETURNI(CMPLXL((ay / 2) * ay, v)); 95333577Skib RETURNI(CMPLXL(log1pl(ay * ay) / 2, v)); 96333577Skib } 97333577Skib 98333577Skib /* Avoid underflow when ax is not small. Also handle zero args. */ 99333577Skib if (kx - ky > MANT_DIG || ay == 0) 100333577Skib RETURNI(CMPLXL(logl(ax), v)); 101333577Skib 102333577Skib /* Avoid overflow. */ 103333577Skib if (kx >= MAX_EXP - 1) 104333577Skib RETURNI(CMPLXL(logl(hypotl(x * 0x1p-16382L, y * 0x1p-16382L)) + 105333577Skib (MAX_EXP - 2) * ln2l_lo + (MAX_EXP - 2) * ln2_hi, v)); 106333577Skib if (kx >= (MAX_EXP - 1) / 2) 107333577Skib RETURNI(CMPLXL(logl(hypotl(x, y)), v)); 108333577Skib 109333577Skib /* Reduce inaccuracies and avoid underflow when ax is denormal. */ 110333577Skib if (kx <= MIN_EXP - 2) 111333577Skib RETURNI(CMPLXL(logl(hypotl(x * 0x1p16383L, y * 0x1p16383L)) + 112333577Skib (MIN_EXP - 2) * ln2l_lo + (MIN_EXP - 2) * ln2_hi, v)); 113333577Skib 114333577Skib /* Avoid remaining underflows (when ax is small but not denormal). */ 115333577Skib if (ky < (MIN_EXP - 1) / 2 + MANT_DIG) 116333577Skib RETURNI(CMPLXL(logl(hypotl(x, y)), v)); 117333577Skib 118333577Skib /* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */ 119333577Skib t = (long double)(ax * (MULT_REDUX + 1)); 120333577Skib axh = (long double)(ax - t) + t; 121333577Skib axl = ax - axh; 122333577Skib ax2h = ax * ax; 123333577Skib ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl; 124333577Skib t = (long double)(ay * (MULT_REDUX + 1)); 125333577Skib ayh = (long double)(ay - t) + t; 126333577Skib ayl = ay - ayh; 127333577Skib ay2h = ay * ay; 128333577Skib ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl; 129333577Skib 130333577Skib /* 131333577Skib * When log(|z|) is far from 1, accuracy in calculating the sum 132333577Skib * of the squares is not very important since log() reduces 133333577Skib * inaccuracies. We depended on this to use the general 134333577Skib * formula when log(|z|) is very far from 1. When log(|z|) is 135333577Skib * moderately far from 1, we go through the extra-precision 136333577Skib * calculations to reduce branches and gain a little accuracy. 137333577Skib * 138333577Skib * When |z| is near 1, we subtract 1 and use log1p() and don't 139333577Skib * leave it to log() to subtract 1, since we gain at least 1 bit 140333577Skib * of accuracy in this way. 141333577Skib * 142333577Skib * When |z| is very near 1, subtracting 1 can cancel almost 143333577Skib * 3*MANT_DIG bits. We arrange that subtracting 1 is exact in 144333577Skib * doubled precision, and then do the rest of the calculation 145333577Skib * in sloppy doubled precision. Although large cancellations 146333577Skib * often lose lots of accuracy, here the final result is exact 147333577Skib * in doubled precision if the large calculation occurs (because 148333577Skib * then it is exact in tripled precision and the cancellation 149333577Skib * removes enough bits to fit in doubled precision). Thus the 150333577Skib * result is accurate in sloppy doubled precision, and the only 151333577Skib * significant loss of accuracy is when it is summed and passed 152333577Skib * to log1p(). 153333577Skib */ 154333577Skib sh = ax2h; 155333577Skib sl = ay2h; 156333577Skib _2sumF(sh, sl); 157333577Skib if (sh < 0.5 || sh >= 3) 158333577Skib RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v)); 159333577Skib sh -= 1; 160333577Skib _2sum(sh, sl); 161333577Skib _2sum(ax2l, ay2l); 162333577Skib /* Briggs-Kahan algorithm (except we discard the final low term): */ 163333577Skib _2sum(sh, ax2l); 164333577Skib _2sum(sl, ay2l); 165333577Skib t = ax2l + sl; 166333577Skib _2sumF(sh, t); 167333577Skib RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v)); 168333577Skib} 169