1 /* s_tanl.c -- long double version of s_tan.c.
2 * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
5 /* @(#)s_tan.c 5.1 93/09/24 */
7 * ====================================================
8 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
10 * Developed at SunPro, a Sun Microsystems, Inc. business.
11 * Permission to use, copy, modify, and distribute this
12 * software is freely granted, provided that this notice
14 * ====================================================
18 * Return tangent function of x.
21 * __kernel_tanl ... tangent function on [-pi/4,pi/4]
22 * __ieee754_rem_pio2l ... argument reduction routine
25 * Let S,C and T denote the sin, cos and tan respectively on
26 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
27 * in [-pi/4 , +pi/4], and let n = k mod 4.
30 * n sin(x) cos(x) tan(x)
31 * ----------------------------------------------------------
36 * ----------------------------------------------------------
39 * Let trig be any of sin, cos, or tan.
40 * trig(+-INF) is NaN, with signals;
41 * trig(NaN) is that NaN;
44 * TRIG(x) returns trig(x) nearly rounded
59 * ====================================================
60 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
62 * Developed at SunPro, a Sun Microsystems, Inc. business.
63 * Permission to use, copy, modify, and distribute this
64 * software is freely granted, provided that this notice
66 * ====================================================
70 Long double expansions contributed by
71 Stephen L. Moshier <moshier@na-net.ornl.gov>
74 /* __kernel_tanl( x, y, k )
75 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
76 * Input x is assumed to be bounded by ~pi/4 in magnitude.
77 * Input y is the tail of x.
78 * Input k indicates whether tan (if k=1) or
79 * -1/tan (if k= -1) is returned.
82 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
83 * 2. if x < 2^-57, return x with inexact if x!=0.
84 * 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
87 * Note: tan(x+y) = tan(x) + tan'(x)*y
88 * ~ tan(x) + (1+x*x)*y
89 * Therefore, for better accuracy in computing tan(x+y), let
92 * tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
94 * 4. For x in [0.67433,pi/4], let y = pi/4 - x, then
95 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
96 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
100 static const long double
101 pio4hi = 7.8539816339744830961566084581987569936977E-1L,
102 pio4lo = 2.1679525325309452561992610065108379921906E-35L,
104 /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
105 0 <= x <= 0.6743316650390625
106 Peak relative error 8.0e-36 */
107 TH = 3.333333333333333333333333333333333333333E-1L,
108 T0 = -1.813014711743583437742363284336855889393E7L,
109 T1 = 1.320767960008972224312740075083259247618E6L,
110 T2 = -2.626775478255838182468651821863299023956E4L,
111 T3 = 1.764573356488504935415411383687150199315E2L,
112 T4 = -3.333267763822178690794678978979803526092E-1L,
114 U0 = -1.359761033807687578306772463253710042010E8L,
115 U1 = 6.494370630656893175666729313065113194784E7L,
116 U2 = -4.180787672237927475505536849168729386782E6L,
117 U3 = 8.031643765106170040139966622980914621521E4L,
118 U4 = -5.323131271912475695157127875560667378597E2L;
119 /* 1.000000000000000000000000000000000000000E0 */
123 kernel_tanl (long double x, long double y, int iy)
125 long double z, r, v, w, s, u, u1;
136 if (x < 0.000000000000000006938893903907228377647697925567626953125L) /* x < 2**-57 */
139 { /* generate inexact */
140 if (iy == -1 && x == 0.0)
141 return 1.0L / fabs (x);
143 return (iy == 1) ? x : -1.0L / x;
146 if (x >= 0.6743316650390625) /* |x| >= 0.6743316650390625 */
156 r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
157 v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
161 r = y + z * (s * r + y);
166 v = (long double) iy;
167 w = (v - 2.0 * (x - (w * w / (w + v) - r)));
175 { /* if allow error up to 2 ulp,
176 simply return -1.0/(x+r) here */
177 /* compute -1.0/(x+r) accurately */
183 return u + z * (s + u * v);
190 long double y[2], z = 0.0L;
194 if (x >= -0.7853981633974483096156608458198757210492 &&
195 x <= 0.7853981633974483096156608458198757210492)
196 return kernel_tanl (x, z, 1);
198 /* tanl(Inf or NaN) is NaN, tanl(0) is 0 */
199 else if (x + x == x || x != x)
200 return x - x; /* NaN */
202 /* argument reduction needed */
205 n = ieee754_rem_pio2l (x, y);
206 /* 1 -- n even, -1 -- n odd */
207 return kernel_tanl (y[0], y[1], 1 - ((n & 1) << 1));
215 printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492));
216 printf ("%.16Lg\n", tanl(-0.7853981633974483096156608458198757210492));
217 printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 *3));
218 printf ("%.16Lg\n", tanl(-0.7853981633974483096156608458198757210492 *31));
219 printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 / 2));
220 printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 * 3/2));
221 printf ("%.16Lg\n", tanl(0.7853981633974483096156608458198757210492 * 5/2));