2 * Copyright (c) 2013 Bruce D. Evans
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice unmodified, this list of conditions, and the following
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.
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
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24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
35 #include "math_private.h"
37 #define MANT_DIG LDBL_MANT_DIG
38 #define MAX_EXP LDBL_MAX_EXP
39 #define MIN_EXP LDBL_MIN_EXP
42 ln2_hi = 6.9314718055829871e-1; /* 0x162e42fefa0000.0p-53 */
44 #if LDBL_MANT_DIG == 64
45 #define MULT_REDUX 0x1p32 /* exponent MANT_DIG / 2 rounded up */
47 ln2l_lo = 1.6465949582897082e-12; /* 0x1cf79abc9e3b3a.0p-92 */
48 #elif LDBL_MANT_DIG == 113
49 #define MULT_REDUX 0x1p57
50 static const long double
51 ln2l_lo = 1.64659495828970812809844307550013433e-12L; /* 0x1cf79abc9e3b39803f2f6af40f343.0p-152L */
53 #error "Unsupported long double format"
57 clogl(long double complex z)
59 long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl;
60 long double sh, sl, t;
65 ENTERIT(long double complex);
79 GET_LDBL_EXPSIGN(hax, ax);
81 GET_LDBL_EXPSIGN(hay, ay);
84 /* Handle NaNs and Infs using the general formula. */
85 if (kx == MAX_EXP || ky == MAX_EXP)
86 RETURNI(CMPLXL(logl(hypotl(x, y)), v));
88 /* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */
90 if (ky < (MIN_EXP - 1) / 2)
91 RETURNI(CMPLXL((ay / 2) * ay, v));
92 RETURNI(CMPLXL(log1pl(ay * ay) / 2, v));
95 /* Avoid underflow when ax is not small. Also handle zero args. */
96 if (kx - ky > MANT_DIG || ay == 0)
97 RETURNI(CMPLXL(logl(ax), v));
100 if (kx >= MAX_EXP - 1)
101 RETURNI(CMPLXL(logl(hypotl(x * 0x1p-16382L, y * 0x1p-16382L)) +
102 (MAX_EXP - 2) * ln2l_lo + (MAX_EXP - 2) * ln2_hi, v));
103 if (kx >= (MAX_EXP - 1) / 2)
104 RETURNI(CMPLXL(logl(hypotl(x, y)), v));
106 /* Reduce inaccuracies and avoid underflow when ax is denormal. */
107 if (kx <= MIN_EXP - 2)
108 RETURNI(CMPLXL(logl(hypotl(x * 0x1p16383L, y * 0x1p16383L)) +
109 (MIN_EXP - 2) * ln2l_lo + (MIN_EXP - 2) * ln2_hi, v));
111 /* Avoid remaining underflows (when ax is small but not denormal). */
112 if (ky < (MIN_EXP - 1) / 2 + MANT_DIG)
113 RETURNI(CMPLXL(logl(hypotl(x, y)), v));
115 /* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */
116 t = (long double)(ax * (MULT_REDUX + 1));
117 axh = (long double)(ax - t) + t;
120 ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl;
121 t = (long double)(ay * (MULT_REDUX + 1));
122 ayh = (long double)(ay - t) + t;
125 ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl;
128 * When log(|z|) is far from 1, accuracy in calculating the sum
129 * of the squares is not very important since log() reduces
130 * inaccuracies. We depended on this to use the general
131 * formula when log(|z|) is very far from 1. When log(|z|) is
132 * moderately far from 1, we go through the extra-precision
133 * calculations to reduce branches and gain a little accuracy.
135 * When |z| is near 1, we subtract 1 and use log1p() and don't
136 * leave it to log() to subtract 1, since we gain at least 1 bit
137 * of accuracy in this way.
139 * When |z| is very near 1, subtracting 1 can cancel almost
140 * 3*MANT_DIG bits. We arrange that subtracting 1 is exact in
141 * doubled precision, and then do the rest of the calculation
142 * in sloppy doubled precision. Although large cancellations
143 * often lose lots of accuracy, here the final result is exact
144 * in doubled precision if the large calculation occurs (because
145 * then it is exact in tripled precision and the cancellation
146 * removes enough bits to fit in doubled precision). Thus the
147 * result is accurate in sloppy doubled precision, and the only
148 * significant loss of accuracy is when it is summed and passed
154 if (sh < 0.5 || sh >= 3)
155 RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v));
159 /* Briggs-Kahan algorithm (except we discard the final low term): */
164 RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v));