1 /*- 2 * Copyright (c) 2013 Bruce D. Evans 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice unmodified, this list of conditions, and the following 10 * disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25 */ 26 27 #include <sys/cdefs.h> 28 __FBSDID("$FreeBSD$"); 29 30 #include <complex.h> 31 #include <float.h> 32 #ifdef __i386__ 33 #include <ieeefp.h> 34 #endif 35 36 #include "fpmath.h" 37 #include "math.h" 38 #include "math_private.h" 39 40 #define MANT_DIG LDBL_MANT_DIG 41 #define MAX_EXP LDBL_MAX_EXP 42 #define MIN_EXP LDBL_MIN_EXP 43 44 static const double 45 ln2_hi = 6.9314718055829871e-1; /* 0x162e42fefa0000.0p-53 */ 46 47 #if LDBL_MANT_DIG == 64 48 #define MULT_REDUX 0x1p32 /* exponent MANT_DIG / 2 rounded up */ 49 static const double 50 ln2l_lo = 1.6465949582897082e-12; /* 0x1cf79abc9e3b3a.0p-92 */ 51 #elif LDBL_MANT_DIG == 113 52 #define MULT_REDUX 0x1p57 53 static const long double 54 ln2l_lo = 1.64659495828970812809844307550013433e-12L; /* 0x1cf79abc9e3b39803f2f6af40f343.0p-152L */ 55 #else 56 #error "Unsupported long double format" 57 #endif 58 59 long double complex 60 clogl(long double complex z) 61 { 62 long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl; 63 long double sh, sl, t; 64 long double x, y, v; 65 uint16_t hax, hay; 66 int kx, ky; 67 68 ENTERIT(long double complex); 69 70 x = creall(z); 71 y = cimagl(z); 72 v = atan2l(y, x); 73 74 ax = fabsl(x); 75 ay = fabsl(y); 76 if (ax < ay) { 77 t = ax; 78 ax = ay; 79 ay = t; 80 } 81 82 GET_LDBL_EXPSIGN(hax, ax); 83 kx = hax - 16383; 84 GET_LDBL_EXPSIGN(hay, ay); 85 ky = hay - 16383; 86 87 /* Handle NaNs and Infs using the general formula. */ 88 if (kx == MAX_EXP || ky == MAX_EXP) 89 RETURNI(CMPLXL(logl(hypotl(x, y)), v)); 90 91 /* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */ 92 if (ax == 1) { 93 if (ky < (MIN_EXP - 1) / 2) 94 RETURNI(CMPLXL((ay / 2) * ay, v)); 95 RETURNI(CMPLXL(log1pl(ay * ay) / 2, v)); 96 } 97 98 /* Avoid underflow when ax is not small. Also handle zero args. */ 99 if (kx - ky > MANT_DIG || ay == 0) 100 RETURNI(CMPLXL(logl(ax), v)); 101 102 /* Avoid overflow. */ 103 if (kx >= MAX_EXP - 1) 104 RETURNI(CMPLXL(logl(hypotl(x * 0x1p-16382L, y * 0x1p-16382L)) + 105 (MAX_EXP - 2) * ln2l_lo + (MAX_EXP - 2) * ln2_hi, v)); 106 if (kx >= (MAX_EXP - 1) / 2) 107 RETURNI(CMPLXL(logl(hypotl(x, y)), v)); 108 109 /* Reduce inaccuracies and avoid underflow when ax is denormal. */ 110 if (kx <= MIN_EXP - 2) 111 RETURNI(CMPLXL(logl(hypotl(x * 0x1p16383L, y * 0x1p16383L)) + 112 (MIN_EXP - 2) * ln2l_lo + (MIN_EXP - 2) * ln2_hi, v)); 113 114 /* Avoid remaining underflows (when ax is small but not denormal). */ 115 if (ky < (MIN_EXP - 1) / 2 + MANT_DIG) 116 RETURNI(CMPLXL(logl(hypotl(x, y)), v)); 117 118 /* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */ 119 t = (long double)(ax * (MULT_REDUX + 1)); 120 axh = (long double)(ax - t) + t; 121 axl = ax - axh; 122 ax2h = ax * ax; 123 ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl; 124 t = (long double)(ay * (MULT_REDUX + 1)); 125 ayh = (long double)(ay - t) + t; 126 ayl = ay - ayh; 127 ay2h = ay * ay; 128 ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl; 129 130 /* 131 * When log(|z|) is far from 1, accuracy in calculating the sum 132 * of the squares is not very important since log() reduces 133 * inaccuracies. We depended on this to use the general 134 * formula when log(|z|) is very far from 1. When log(|z|) is 135 * moderately far from 1, we go through the extra-precision 136 * calculations to reduce branches and gain a little accuracy. 137 * 138 * When |z| is near 1, we subtract 1 and use log1p() and don't 139 * leave it to log() to subtract 1, since we gain at least 1 bit 140 * of accuracy in this way. 141 * 142 * When |z| is very near 1, subtracting 1 can cancel almost 143 * 3*MANT_DIG bits. We arrange that subtracting 1 is exact in 144 * doubled precision, and then do the rest of the calculation 145 * in sloppy doubled precision. Although large cancellations 146 * often lose lots of accuracy, here the final result is exact 147 * in doubled precision if the large calculation occurs (because 148 * then it is exact in tripled precision and the cancellation 149 * removes enough bits to fit in doubled precision). Thus the 150 * result is accurate in sloppy doubled precision, and the only 151 * significant loss of accuracy is when it is summed and passed 152 * to log1p(). 153 */ 154 sh = ax2h; 155 sl = ay2h; 156 _2sumF(sh, sl); 157 if (sh < 0.5 || sh >= 3) 158 RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v)); 159 sh -= 1; 160 _2sum(sh, sl); 161 _2sum(ax2l, ay2l); 162 /* Briggs-Kahan algorithm (except we discard the final low term): */ 163 _2sum(sh, ax2l); 164 _2sum(sl, ay2l); 165 t = ax2l + sl; 166 _2sumF(sh, t); 167 RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v)); 168 } 169