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: stable/11/lib/msun/src/s_clogl.c 334654 2018-06-05 13:46:18Z kib $");
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
44static const double
45ln2_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 */
49static const double
50ln2l_lo = 1.6465949582897082e-12;	/*  0x1cf79abc9e3b3a.0p-92 */
51#elif LDBL_MANT_DIG == 113
52#define	MULT_REDUX	0x1p57
53static const long double
54ln2l_lo = 1.64659495828970812809844307550013433e-12L;	/*  0x1cf79abc9e3b39803f2f6af40f343.0p-152L */
55#else
56#error "Unsupported long double format"
57#endif
58
59long double complex
60clogl(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