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 <complex.h> 28 #include <float.h> 29 30 #include "fpmath.h" 31 #include "math.h" 32 #include "math_private.h" 33 34 #define MANT_DIG DBL_MANT_DIG 35 #define MAX_EXP DBL_MAX_EXP 36 #define MIN_EXP DBL_MIN_EXP 37 38 static const double 39 ln2_hi = 6.9314718055829871e-1, /* 0x162e42fefa0000.0p-53 */ 40 ln2_lo = 1.6465949582897082e-12; /* 0x1cf79abc9e3b3a.0p-92 */ 41 42 double complex 43 clog(double complex z) 44 { 45 double_t ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl, sh, sl, t; 46 double x, y, v; 47 uint32_t hax, hay; 48 int kx, ky; 49 50 x = creal(z); 51 y = cimag(z); 52 v = atan2(y, x); 53 54 ax = fabs(x); 55 ay = fabs(y); 56 if (ax < ay) { 57 t = ax; 58 ax = ay; 59 ay = t; 60 } 61 62 GET_HIGH_WORD(hax, ax); 63 kx = (hax >> 20) - 1023; 64 GET_HIGH_WORD(hay, ay); 65 ky = (hay >> 20) - 1023; 66 67 /* Handle NaNs and Infs using the general formula. */ 68 if (kx == MAX_EXP || ky == MAX_EXP) 69 return (CMPLX(log(hypot(x, y)), v)); 70 71 /* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */ 72 if (ax == 1) { 73 if (ky < (MIN_EXP - 1) / 2) 74 return (CMPLX((ay / 2) * ay, v)); 75 return (CMPLX(log1p(ay * ay) / 2, v)); 76 } 77 78 /* Avoid underflow when ax is not small. Also handle zero args. */ 79 if (kx - ky > MANT_DIG || ay == 0) 80 return (CMPLX(log(ax), v)); 81 82 /* Avoid overflow. */ 83 if (kx >= MAX_EXP - 1) 84 return (CMPLX(log(hypot(x * 0x1p-1022, y * 0x1p-1022)) + 85 (MAX_EXP - 2) * ln2_lo + (MAX_EXP - 2) * ln2_hi, v)); 86 if (kx >= (MAX_EXP - 1) / 2) 87 return (CMPLX(log(hypot(x, y)), v)); 88 89 /* Reduce inaccuracies and avoid underflow when ax is denormal. */ 90 if (kx <= MIN_EXP - 2) 91 return (CMPLX(log(hypot(x * 0x1p1023, y * 0x1p1023)) + 92 (MIN_EXP - 2) * ln2_lo + (MIN_EXP - 2) * ln2_hi, v)); 93 94 /* Avoid remaining underflows (when ax is small but not denormal). */ 95 if (ky < (MIN_EXP - 1) / 2 + MANT_DIG) 96 return (CMPLX(log(hypot(x, y)), v)); 97 98 /* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */ 99 t = (double)(ax * (0x1p27 + 1)); 100 axh = (double)(ax - t) + t; 101 axl = ax - axh; 102 ax2h = ax * ax; 103 ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl; 104 t = (double)(ay * (0x1p27 + 1)); 105 ayh = (double)(ay - t) + t; 106 ayl = ay - ayh; 107 ay2h = ay * ay; 108 ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl; 109 110 /* 111 * When log(|z|) is far from 1, accuracy in calculating the sum 112 * of the squares is not very important since log() reduces 113 * inaccuracies. We depended on this to use the general 114 * formula when log(|z|) is very far from 1. When log(|z|) is 115 * moderately far from 1, we go through the extra-precision 116 * calculations to reduce branches and gain a little accuracy. 117 * 118 * When |z| is near 1, we subtract 1 and use log1p() and don't 119 * leave it to log() to subtract 1, since we gain at least 1 bit 120 * of accuracy in this way. 121 * 122 * When |z| is very near 1, subtracting 1 can cancel almost 123 * 3*MANT_DIG bits. We arrange that subtracting 1 is exact in 124 * doubled precision, and then do the rest of the calculation 125 * in sloppy doubled precision. Although large cancellations 126 * often lose lots of accuracy, here the final result is exact 127 * in doubled precision if the large calculation occurs (because 128 * then it is exact in tripled precision and the cancellation 129 * removes enough bits to fit in doubled precision). Thus the 130 * result is accurate in sloppy doubled precision, and the only 131 * significant loss of accuracy is when it is summed and passed 132 * to log1p(). 133 */ 134 sh = ax2h; 135 sl = ay2h; 136 _2sumF(sh, sl); 137 if (sh < 0.5 || sh >= 3) 138 return (CMPLX(log(ay2l + ax2l + sl + sh) / 2, v)); 139 sh -= 1; 140 _2sum(sh, sl); 141 _2sum(ax2l, ay2l); 142 /* Briggs-Kahan algorithm (except we discard the final low term): */ 143 _2sum(sh, ax2l); 144 _2sum(sl, ay2l); 145 t = ax2l + sl; 146 _2sumF(sh, t); 147 return (CMPLX(log1p(ay2l + t + sh) / 2, v)); 148 } 149 150 #if (LDBL_MANT_DIG == 53) 151 __weak_reference(clog, clogl); 152 #endif 153