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 <sys/cdefs.h>
71
72 #include "namespace.h"
73 __weak_alias(tanpi, _tanpi)
74 #include <float.h>
75 #include "math.h"
76 #include "math_private.h"
77
78 static const double
79 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
80 pi_lo = -2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
81
82 /*
83 * The kernel for tanpi(x) multiplies x by an 80-bit approximation of
84 * pi, where the hi and lo parts are used with with kernel for tan(x).
85 */
86 static inline double
__kernel_tanpi(double x)87 __kernel_tanpi(double x)
88 {
89 double_t hi, lo, t;
90
91 if (x < 0.25) {
92 hi = (float)x;
93 lo = x - hi;
94 lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
95 hi *= pi_hi;
96 _2sumF(hi, lo);
97 t = __kernel_tan(hi, lo, 1);
98 } else if (x > 0.25) {
99 x = 0.5 - x;
100 hi = (float)x;
101 lo = x - hi;
102 lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
103 hi *= pi_hi;
104 _2sumF(hi, lo);
105 t = - __kernel_tan(hi, lo, -1);
106 } else
107 t = 1;
108
109 return (t);
110 }
111
112 static volatile const double vzero = 0;
113
114 double
tanpi(double x)115 tanpi(double x)
116 {
117 double ax, hi, lo, odd, t;
118 uint32_t hx, ix, j0, lx;
119
120 EXTRACT_WORDS(hx, lx, x);
121 ix = hx & 0x7fffffff;
122 INSERT_WORDS(ax, ix, lx);
123
124 if (ix < 0x3ff00000) { /* |x| < 1 */
125 if (ix < 0x3fe00000) { /* |x| < 0.5 */
126 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
127 if (x == 0)
128 return (x);
129 /*
130 * To avoid issues with subnormal values,
131 * scale the computation and rescale on
132 * return.
133 */
134 INSERT_WORDS(hi, hx, 0);
135 hi *= 0x1p53;
136 lo = x * 0x1p53 - hi;
137 t = (pi_lo + pi_hi) * lo + pi_lo * hi +
138 pi_hi * hi;
139 return (t * 0x1p-53);
140 }
141 t = __kernel_tanpi(ax);
142 } else if (ax == 0.5)
143 t = 1 / vzero;
144 else
145 t = - __kernel_tanpi(1 - ax);
146 return ((hx & 0x80000000) ? -t : t);
147 }
148
149 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
150 FFLOOR(x, j0, ix, lx); /* Integer part of ax. */
151 odd = (uint64_t)x & 1 ? -1 : 1;
152 ax -= x;
153 EXTRACT_WORDS(ix, lx, ax);
154
155 if (ix < 0x3fe00000) /* |x| < 0.5 */
156 t = ix == 0 ? copysign(0, odd) : __kernel_tanpi(ax);
157 else if (ax == 0.5)
158 t = odd / vzero;
159 else
160 t = - __kernel_tanpi(1 - ax);
161
162 return ((hx & 0x80000000) ? -t : t);
163 }
164
165 /* x = +-inf or nan. */
166 if (ix >= 0x7ff00000)
167 return (vzero / vzero);
168
169 /*
170 * For 0x1p52 <= |x| < 0x1p53 need to determine if x is an even
171 * or odd integer to set t = +0 or -0.
172 * For |x| >= 0x1p54, it is always an even integer, so t = 0.
173 */
174 t = ix >= 0x43400000 ? 0 : (copysign(0, (lx & 1) ? -1 : 1));
175 return ((hx & 0x80000000) ? -t : t);
176 }
177