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
74 static const double
75 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
76 pi_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 */
82 static inline double
__kernel_tanpi(double x)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
108 volatile static const double vzero = 0;
109
110 double
tanpi(double x)111 tanpi(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