2 * Copyright (c) 2017, 2023 Steven G. Kargl
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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.
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.
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
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
48 * tan(pi*j0) + tan(pi*r)
49 * tanpi(x) = tan(pi*(j0+r)) = ---------------------------- = tanpi(r)
50 * 1 - tan(pi*j0) * tan(pi*r)
52 * So, after argument reduction, the kernel is again invoked.
54 * 4. For |x| >= 0x1p(P-1), |x| is integral and tanpi(x) = copysign(0,x).
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
64 * tanpi(n+1/2) = -inf and raises the FE_DIVBYZERO exception for
66 * tanpi(+-inf) = NaN and raises the FE_INVALID exception.
67 * tanpi(nan) = NaN and raises the FE_INVALID exception.
72 #include "math_private.h"
75 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
76 pi_lo = -2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
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).
83 __kernel_tanpi(double x)
90 lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
93 t = __kernel_tan(hi, lo, 1);
94 } else if (x > 0.25) {
98 lo = lo * (pi_lo + pi_hi) + hi * pi_lo;
101 t = - __kernel_tan(hi, lo, -1);
108 volatile static const double vzero = 0;
113 double ax, hi, lo, odd, t;
114 uint32_t hx, ix, j0, lx;
116 EXTRACT_WORDS(hx, lx, x);
117 ix = hx & 0x7fffffff;
118 INSERT_WORDS(ax, ix, lx);
120 if (ix < 0x3ff00000) { /* |x| < 1 */
121 if (ix < 0x3fe00000) { /* |x| < 0.5 */
122 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
126 * To avoid issues with subnormal values,
127 * scale the computation and rescale on
130 INSERT_WORDS(hi, hx, 0);
132 lo = x * 0x1p53 - hi;
133 t = (pi_lo + pi_hi) * lo + pi_lo * hi +
135 return (t * 0x1p-53);
137 t = __kernel_tanpi(ax);
138 } else if (ax == 0.5)
141 t = - __kernel_tanpi(1 - ax);
142 return ((hx & 0x80000000) ? -t : t);
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;
149 EXTRACT_WORDS(ix, lx, ax);
151 if (ix < 0x3fe00000) /* |x| < 0.5 */
152 t = ix == 0 ? copysign(0, odd) : __kernel_tanpi(ax);
156 t = - __kernel_tanpi(1 - ax);
158 return ((hx & 0x80000000) ? -t : t);
161 /* x = +-inf or nan. */
162 if (ix >= 0x7ff00000)
163 return (vzero / vzero);
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.
170 t = ix >= 0x43400000 ? 0 : (copysign(0, (lx & 1) ? -1 : 1));
171 return ((hx & 0x80000000) ? -t : t);
174 #if LDBL_MANT_DIG == 53
175 __weak_reference(tanpi, tanpil);