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 #include <complex.h>
29 #include <float.h>
30
31 #include "fpmath.h"
32 #include "math.h"
33 #include "math_private.h"
34
35 #define MANT_DIG LDBL_MANT_DIG
36 #define MAX_EXP LDBL_MAX_EXP
37 #define MIN_EXP LDBL_MIN_EXP
38
39 static const double
40 ln2_hi = 6.9314718055829871e-1; /* 0x162e42fefa0000.0p-53 */
41
42 #if LDBL_MANT_DIG == 64
43 #define MULT_REDUX 0x1p32 /* exponent MANT_DIG / 2 rounded up */
44 static const double
45 ln2l_lo = 1.6465949582897082e-12; /* 0x1cf79abc9e3b3a.0p-92 */
46 #elif LDBL_MANT_DIG == 113
47 #define MULT_REDUX 0x1p57
48 static const long double
49 ln2l_lo = 1.64659495828970812809844307550013433e-12L; /* 0x1cf79abc9e3b39803f2f6af40f343.0p-152L */
50 #else
51 #error "Unsupported long double format"
52 #endif
53
54 long double complex
clogl(long double complex z)55 clogl(long double complex z)
56 {
57 long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl;
58 long double sh, sl, t;
59 long double x, y, v;
60 uint16_t hax, hay;
61 int kx, ky;
62
63 ENTERIT(long double complex);
64
65 x = creall(z);
66 y = cimagl(z);
67 v = atan2l(y, x);
68
69 ax = fabsl(x);
70 ay = fabsl(y);
71 if (ax < ay) {
72 t = ax;
73 ax = ay;
74 ay = t;
75 }
76
77 GET_LDBL_EXPSIGN(hax, ax);
78 kx = hax - 16383;
79 GET_LDBL_EXPSIGN(hay, ay);
80 ky = hay - 16383;
81
82 /* Handle NaNs and Infs using the general formula. */
83 if (kx == MAX_EXP || ky == MAX_EXP)
84 RETURNI(CMPLXL(logl(hypotl(x, y)), v));
85
86 /* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */
87 if (ax == 1) {
88 if (ky < (MIN_EXP - 1) / 2)
89 RETURNI(CMPLXL((ay / 2) * ay, v));
90 RETURNI(CMPLXL(log1pl(ay * ay) / 2, v));
91 }
92
93 /* Avoid underflow when ax is not small. Also handle zero args. */
94 if (kx - ky > MANT_DIG || ay == 0)
95 RETURNI(CMPLXL(logl(ax), v));
96
97 /* Avoid overflow. */
98 if (kx >= MAX_EXP - 1)
99 RETURNI(CMPLXL(logl(hypotl(x * 0x1p-16382L, y * 0x1p-16382L)) +
100 (MAX_EXP - 2) * ln2l_lo + (MAX_EXP - 2) * ln2_hi, v));
101 if (kx >= (MAX_EXP - 1) / 2)
102 RETURNI(CMPLXL(logl(hypotl(x, y)), v));
103
104 /* Reduce inaccuracies and avoid underflow when ax is denormal. */
105 if (kx <= MIN_EXP - 2)
106 RETURNI(CMPLXL(logl(hypotl(x * 0x1p16383L, y * 0x1p16383L)) +
107 (MIN_EXP - 2) * ln2l_lo + (MIN_EXP - 2) * ln2_hi, v));
108
109 /* Avoid remaining underflows (when ax is small but not denormal). */
110 if (ky < (MIN_EXP - 1) / 2 + MANT_DIG)
111 RETURNI(CMPLXL(logl(hypotl(x, y)), v));
112
113 /* Calculate ax*ax and ay*ay exactly using Dekker's algorithm. */
114 t = (long double)(ax * (MULT_REDUX + 1));
115 axh = (long double)(ax - t) + t;
116 axl = ax - axh;
117 ax2h = ax * ax;
118 ax2l = axh * axh - ax2h + 2 * axh * axl + axl * axl;
119 t = (long double)(ay * (MULT_REDUX + 1));
120 ayh = (long double)(ay - t) + t;
121 ayl = ay - ayh;
122 ay2h = ay * ay;
123 ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl;
124
125 /*
126 * When log(|z|) is far from 1, accuracy in calculating the sum
127 * of the squares is not very important since log() reduces
128 * inaccuracies. We depended on this to use the general
129 * formula when log(|z|) is very far from 1. When log(|z|) is
130 * moderately far from 1, we go through the extra-precision
131 * calculations to reduce branches and gain a little accuracy.
132 *
133 * When |z| is near 1, we subtract 1 and use log1p() and don't
134 * leave it to log() to subtract 1, since we gain at least 1 bit
135 * of accuracy in this way.
136 *
137 * When |z| is very near 1, subtracting 1 can cancel almost
138 * 3*MANT_DIG bits. We arrange that subtracting 1 is exact in
139 * doubled precision, and then do the rest of the calculation
140 * in sloppy doubled precision. Although large cancellations
141 * often lose lots of accuracy, here the final result is exact
142 * in doubled precision if the large calculation occurs (because
143 * then it is exact in tripled precision and the cancellation
144 * removes enough bits to fit in doubled precision). Thus the
145 * result is accurate in sloppy doubled precision, and the only
146 * significant loss of accuracy is when it is summed and passed
147 * to log1p().
148 */
149 sh = ax2h;
150 sl = ay2h;
151 _2sumF(sh, sl);
152 if (sh < 0.5 || sh >= 3)
153 RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v));
154 sh -= 1;
155 _2sum(sh, sl);
156 _2sum(ax2l, ay2l);
157 /* Briggs-Kahan algorithm (except we discard the final low term): */
158 _2sum(sh, ax2l);
159 _2sum(sl, ay2l);
160 t = ax2l + sl;
161 _2sumF(sh, t);
162 RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v));
163 }
164