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#ifdef __i386__
30#include <ieeefp.h>
31#endif
32
33#include "fpmath.h"
34#include "math.h"
35#include "math_private.h"
36
37#define	MANT_DIG	LDBL_MANT_DIG
38#define	MAX_EXP		LDBL_MAX_EXP
39#define	MIN_EXP		LDBL_MIN_EXP
40
41static const double
42ln2_hi = 6.9314718055829871e-1;		/*  0x162e42fefa0000.0p-53 */
43
44#if LDBL_MANT_DIG == 64
45#define	MULT_REDUX	0x1p32		/* exponent MANT_DIG / 2 rounded up */
46static const double
47ln2l_lo = 1.6465949582897082e-12;	/*  0x1cf79abc9e3b3a.0p-92 */
48#elif LDBL_MANT_DIG == 113
49#define	MULT_REDUX	0x1p57
50static const long double
51ln2l_lo = 1.64659495828970812809844307550013433e-12L;	/*  0x1cf79abc9e3b39803f2f6af40f343.0p-152L */
52#else
53#error "Unsupported long double format"
54#endif
55
56long double complex
57clogl(long double complex z)
58{
59	long double ax, ax2h, ax2l, axh, axl, ay, ay2h, ay2l, ayh, ayl;
60	long double sh, sl, t;
61	long double x, y, v;
62	uint16_t hax, hay;
63	int kx, ky;
64
65	ENTERIT(long double complex);
66
67	x = creall(z);
68	y = cimagl(z);
69	v = atan2l(y, x);
70
71	ax = fabsl(x);
72	ay = fabsl(y);
73	if (ax < ay) {
74		t = ax;
75		ax = ay;
76		ay = t;
77	}
78
79	GET_LDBL_EXPSIGN(hax, ax);
80	kx = hax - 16383;
81	GET_LDBL_EXPSIGN(hay, ay);
82	ky = hay - 16383;
83
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));
87
88	/* Avoid spurious underflow, and reduce inaccuracies when ax is 1. */
89	if (ax == 1) {
90		if (ky < (MIN_EXP - 1) / 2)
91			RETURNI(CMPLXL((ay / 2) * ay, v));
92		RETURNI(CMPLXL(log1pl(ay * ay) / 2, v));
93	}
94
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));
98
99	/* Avoid overflow. */
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));
105
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));
110
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));
114
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;
118	axl = ax - axh;
119	ax2h = ax * ax;
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;
123	ayl = ay - ayh;
124	ay2h = ay * ay;
125	ay2l = ayh * ayh - ay2h + 2 * ayh * ayl + ayl * ayl;
126
127	/*
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.
134	 *
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.
138	 *
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
149	 * to log1p().
150	 */
151	sh = ax2h;
152	sl = ay2h;
153	_2sumF(sh, sl);
154	if (sh < 0.5 || sh >= 3)
155		RETURNI(CMPLXL(logl(ay2l + ax2l + sl + sh) / 2, v));
156	sh -= 1;
157	_2sum(sh, sl);
158	_2sum(ax2l, ay2l);
159	/* Briggs-Kahan algorithm (except we discard the final low term): */
160	_2sum(sh, ax2l);
161	_2sum(sl, ay2l);
162	t = ax2l + sl;
163	_2sumF(sh, t);
164	RETURNI(CMPLXL(log1pl(ay2l + t + sh) / 2, v));
165}
166