1/* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12/* 13 Long double expansions are 14 Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> 15 and are incorporated herein by permission of the author. The author 16 reserves the right to distribute this material elsewhere under different 17 copying permissions. These modifications are distributed here under 18 the following terms: 19 20 This library is free software; you can redistribute it and/or 21 modify it under the terms of the GNU Lesser General Public 22 License as published by the Free Software Foundation; either 23 version 2.1 of the License, or (at your option) any later version. 24 25 This library is distributed in the hope that it will be useful, 26 but WITHOUT ANY WARRANTY; without even the implied warranty of 27 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 28 Lesser General Public License for more details. 29 30 You should have received a copy of the GNU Lesser General Public 31 License along with this library; if not, see 32 <http://www.gnu.org/licenses/>. */ 33 34/* __quadmath_kernel_tanq( x, y, k ) 35 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 36 * Input x is assumed to be bounded by ~pi/4 in magnitude. 37 * Input y is the tail of x. 38 * Input k indicates whether tan (if k=1) or 39 * -1/tan (if k= -1) is returned. 40 * 41 * Algorithm 42 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 43 * 2. if x < 2^-57, return x with inexact if x!=0. 44 * 3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2) 45 * on [0,0.67433]. 46 * 47 * Note: tan(x+y) = tan(x) + tan'(x)*y 48 * ~ tan(x) + (1+x*x)*y 49 * Therefore, for better accuracy in computing tan(x+y), let 50 * r = x^3 * R(x^2) 51 * then 52 * tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y)) 53 * 54 * 4. For x in [0.67433,pi/4], let y = pi/4 - x, then 55 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 56 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 57 */ 58 59#include "quadmath-imp.h" 60 61static const __float128 62 one = 1, 63 pio4hi = 7.8539816339744830961566084581987569936977E-1Q, 64 pio4lo = 2.1679525325309452561992610065108379921906E-35Q, 65 66 /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2) 67 0 <= x <= 0.6743316650390625 68 Peak relative error 8.0e-36 */ 69 TH = 3.333333333333333333333333333333333333333E-1Q, 70 T0 = -1.813014711743583437742363284336855889393E7Q, 71 T1 = 1.320767960008972224312740075083259247618E6Q, 72 T2 = -2.626775478255838182468651821863299023956E4Q, 73 T3 = 1.764573356488504935415411383687150199315E2Q, 74 T4 = -3.333267763822178690794678978979803526092E-1Q, 75 76 U0 = -1.359761033807687578306772463253710042010E8Q, 77 U1 = 6.494370630656893175666729313065113194784E7Q, 78 U2 = -4.180787672237927475505536849168729386782E6Q, 79 U3 = 8.031643765106170040139966622980914621521E4Q, 80 U4 = -5.323131271912475695157127875560667378597E2Q; 81 /* 1.000000000000000000000000000000000000000E0 */ 82 83 84__float128 85__quadmath_kernel_tanq (__float128 x, __float128 y, int iy) 86{ 87 __float128 z, r, v, w, s; 88 int32_t ix, sign; 89 ieee854_float128 u, u1; 90 91 u.value = x; 92 ix = u.words32.w0 & 0x7fffffff; 93 if (ix < 0x3fc60000) /* x < 2**-57 */ 94 { 95 if ((int) x == 0) 96 { /* generate inexact */ 97 if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3 98 | (iy + 1)) == 0) 99 return one / fabsq (x); 100 else if (iy == 1) 101 { 102 math_check_force_underflow (x); 103 return x; 104 } 105 else 106 return -one / x; 107 } 108 } 109 if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */ 110 { 111 if ((u.words32.w0 & 0x80000000) != 0) 112 { 113 x = -x; 114 y = -y; 115 sign = -1; 116 } 117 else 118 sign = 1; 119 z = pio4hi - x; 120 w = pio4lo - y; 121 x = z + w; 122 y = 0.0; 123 } 124 z = x * x; 125 r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4))); 126 v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z)))); 127 r = r / v; 128 129 s = z * x; 130 r = y + z * (s * r + y); 131 r += TH * s; 132 w = x + r; 133 if (ix >= 0x3ffe5942) 134 { 135 v = (__float128) iy; 136 w = (v - 2.0 * (x - (w * w / (w + v) - r))); 137 /* SIGN is set for arguments that reach this code, but not 138 otherwise, resulting in warnings that it may be used 139 uninitialized although in the cases where it is used it has 140 always been set. */ 141 142 143 if (sign < 0) 144 w = -w; 145 146 return w; 147 } 148 if (iy == 1) 149 return w; 150 else 151 { /* if allow error up to 2 ulp, 152 simply return -1.0/(x+r) here */ 153 /* compute -1.0/(x+r) accurately */ 154 u1.value = w; 155 u1.words32.w2 = 0; 156 u1.words32.w3 = 0; 157 v = r - (u1.value - x); /* u1+v = r+x */ 158 z = -1.0 / w; 159 u.value = z; 160 u.words32.w2 = 0; 161 u.words32.w3 = 0; 162 s = 1.0 + u.value * u1.value; 163 return u.value + z * (s + u.value * v); 164 } 165} 166