1/*-
2 * Copyright (c) 2017, 2023 Steven G. Kargl
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/**
28 * tanpi(x) computes tan(pi*x) without multiplication by pi (almost).  First,
29 * note that tanpi(-x) = -tanpi(x), so the algorithm considers only |x| and
30 * includes reflection symmetry by considering the sign of x on output.  The
31 * method used depends on the magnitude of x.
32 *
33 * 1. For small |x|, tanpi(x) = pi * x where a sloppy threshold is used.  The
34 *    threshold is |x| < 0x1pN with N = -(P/2+M).  P is the precision of the
35 *    floating-point type and M = 2 to 4.  To achieve high accuracy, pi is
36 *    decomposed into high and low parts with the high part containing a
37 *    number of trailing zero bits.  x is also split into high and low parts.
38 *
39 * 2. For |x| < 1, argument reduction is not required and tanpi(x) is
40 *    computed by a direct call to a kernel, which uses the kernel for
41 *    tan(x).  See below.
42 *
43 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
44 *    |x| = j0 + r with j0 an integer and the remainder r satisfies
45 *    0 <= r < 1.  With the given domain, a simplified inline floor(x)
46 *    is used.  Also, note the following identity
47 *
48 *                                   tan(pi*j0) + tan(pi*r)
49 *    tanpi(x) = tan(pi*(j0+r)) = ---------------------------- = tanpi(r)
50 *                                 1 - tan(pi*j0) * tan(pi*r)
51 *
52 *    So, after argument reduction, the kernel is again invoked.
53 *
54 * 4. For |x| >= 0x1p(P-1), |x| is integral and tanpi(x) = copysign(0,x).
55 *
56 * 5. Special cases:
57 *
58 *    tanpi(+-0) = +-0
59 *    tanpi(n) = +0 for positive even and negative odd integer n.
60 *    tanpi(n) = -0 for positive odd and negative even integer n.
61 *    tanpi(+-n+1/4) = +-1, for positive integers n.
62 *    tanpi(n+1/2) = +inf and raises the FE_DIVBYZERO exception for
63 *    even integers n.
64 *    tanpi(n+1/2) = -inf and raises the FE_DIVBYZERO exception for
65 *    odd integers n.
66 *    tanpi(+-inf) = NaN and raises the FE_INVALID exception.
67 *    tanpi(nan) = NaN and raises the FE_INVALID exception.
68 */
69
70#include <float.h>
71#include "math.h"
72#include "math_private.h"
73
74static const double
75pi_hi =  3.1415926814079285e+00,	/* 0x400921fb 0x58000000 */
76pi_lo = -2.7818135228334233e-08;	/* 0xbe5dde97 0x3dcb3b3a */
77
78/*
79 * The kernel for tanpi(x) multiplies x by an 80-bit approximation of
80 * pi, where the hi and lo parts are used with with kernel for tan(x).
81 */
82static inline double
83__kernel_tanpi(double x)
84{
85	double_t hi, lo, t;
86
87	if (x < 0.25) {
88		hi = (float)x;
89		lo = x - hi;
90		lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
91		hi *= pi_hi;
92		_2sumF(hi, lo);
93		t = __kernel_tan(hi, lo, 1);
94	} else if (x > 0.25) {
95		x = 0.5 - x;
96		hi = (float)x;
97		lo = x - hi;
98		lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
99		hi *= pi_hi;
100		_2sumF(hi, lo);
101		t = - __kernel_tan(hi, lo, -1);
102	} else
103		t = 1;
104
105	return (t);
106}
107
108volatile static const double vzero = 0;
109
110double
111tanpi(double x)
112{
113	double ax, hi, lo, odd, t;
114	uint32_t hx, ix, j0, lx;
115
116	EXTRACT_WORDS(hx, lx, x);
117	ix = hx & 0x7fffffff;
118	INSERT_WORDS(ax, ix, lx);
119
120	if (ix < 0x3ff00000) {			/* |x| < 1 */
121		if (ix < 0x3fe00000) {		/* |x| < 0.5 */
122			if (ix < 0x3e200000) {	/* |x| < 0x1p-29 */
123				if (x == 0)
124					return (x);
125				/*
126				 * To avoid issues with subnormal values,
127				 * scale the computation and rescale on
128				 * return.
129				 */
130				INSERT_WORDS(hi, hx, 0);
131				hi *= 0x1p53;
132				lo = x * 0x1p53 - hi;
133				t = (pi_lo + pi_hi) * lo + pi_lo * hi +
134				    pi_hi * hi;
135				return (t * 0x1p-53);
136			}
137			t = __kernel_tanpi(ax);
138		} else if (ax == 0.5)
139			t = 1 / vzero;
140		else
141			t = - __kernel_tanpi(1 - ax);
142		return ((hx & 0x80000000) ? -t : t);
143	}
144
145	if (ix < 0x43300000) {		/* 1 <= |x| < 0x1p52 */
146		FFLOOR(x, j0, ix, lx);	/* Integer part of ax. */
147		odd = (uint64_t)x & 1 ? -1 : 1;
148		ax -= x;
149		EXTRACT_WORDS(ix, lx, ax);
150
151		if (ix < 0x3fe00000)		/* |x| < 0.5 */
152			t = ix == 0 ? copysign(0, odd) : __kernel_tanpi(ax);
153		else if (ax == 0.5)
154			t = odd / vzero;
155		else
156			t = - __kernel_tanpi(1 - ax);
157
158		return ((hx & 0x80000000) ? -t : t);
159	}
160
161	/* x = +-inf or nan. */
162	if (ix >= 0x7ff00000)
163		return (vzero / vzero);
164
165	/*
166	 * For 0x1p52 <= |x| < 0x1p53 need to determine if x is an even
167	 * or odd integer to set t = +0 or -0.
168	 * For |x| >= 0x1p54, it is always an even integer, so t = 0.
169	 */
170	t = ix >= 0x43400000 ? 0 : (copysign(0, (lx & 1) ? -1 : 1));
171	return ((hx & 0x80000000) ? -t : t);
172}
173
174#if LDBL_MANT_DIG == 53
175__weak_reference(tanpi, tanpil);
176#endif
177