1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 22 /* 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 24 */ 25 /* 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 /* INDENT OFF */ 31 /* 32 * __k_tan( double x; double y; int k ) 33 * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164 34 * Input x is assumed to be bounded by ~pi/4 in magnitude. 35 * Input y is the tail of x. 36 * Input k indicate -- tan if k=0; else -1/tan 37 * 38 * Table look up algorithm 39 * 1. by tan(-x) = -tan(x), need only to consider positive x 40 * 2. if x < 5/32 = [0x3fc40000, 0] = 0.15625 , then 41 * if x < 2^-27 (hx < 0x3e400000 0), set w=x with inexact if x != 0 42 * else 43 * z = x*x; 44 * w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6)))))) 45 * return (k == 0)? w: 1/w; 46 * 3. else 47 * ht = (hx + 0x4000)&0x7fff8000 (round x to a break point t) 48 * lt = 0 49 * i = (hy-0x3fc40000)>>15; (i<=64) 50 * x' = (x - t)+y (|x'| ~<= 2^-7) 51 * By 52 * tan(t+x') 53 * = (tan(t)+tan(x'))/(1-tan(x')tan(t)) 54 * We have 55 * sin(x')+tan(t)*(tan(t)*sin(x')) 56 * = tan(t) + ------------------------------- for k=0 57 * cos(x') - tan(t)*sin(x') 58 * 59 * cos(x') - tan(t)*sin(x') 60 * = - -------------------------------------- for k=1 61 * tan(t) + tan(t)*(cos(x')-1) + sin(x') 62 * 63 * 64 * where tan(t) is from the table, 65 * sin(x') = x + pp1*x^3 + pp2*x^5 66 * cos(x') = 1 + qq1*x^2 + qq2*x^4 67 */ 68 69 #include "libm.h" 70 71 extern const double _TBL_tan_hi[], _TBL_tan_lo[]; 72 static const double q[] = { 73 /* one = */ 1.0, 74 /* 75 * 2 2 -59.56 76 * |sin(x) - pp1*x*(pp2+x *(pp3+x )| <= 2 for |x|<1/64 77 */ 78 /* pp1 = */ 8.33326120969096230395312119298978359438478946686e-0003, 79 /* pp2 = */ 1.20001038589438965215025680596868692381425944526e+0002, 80 /* pp3 = */ -2.00001730975089451192161504877731204032897949219e+0001, 81 82 /* 83 * 2 2 -56.19 84 * |cos(x) - (1+qq1*x (qq2+x ))| <= 2 for |x|<=1/128 85 */ 86 /* qq1 = */ 4.16665486385721928197511942926212213933467864990e-0002, 87 /* qq2 = */ -1.20000339921340035687080671777948737144470214844e+0001, 88 89 /* 90 * |tan(x) - PF(x)| 91 * |--------------| <= 2^-58.57 for |x|<0.15625 92 * | x | 93 * 94 * where (let z = x*x) 95 * PF(x) = x + (t1*x*z)(t2 + z(t3 + z))(t4 + z)(t5 + z(t6 + z)) 96 */ 97 /* t1 = */ 3.71923358986516816929168705030406272271648049355e-0003, 98 /* t2 = */ 6.02645120354857866118436504621058702468872070312e+0000, 99 /* t3 = */ 2.42627327587398156083509093150496482849121093750e+0000, 100 /* t4 = */ 2.44968983934252770851003333518747240304946899414e+0000, 101 /* t5 = */ 6.07089252571767978849948121933266520500183105469e+0000, 102 /* t6 = */ -2.49403756995593761658369658107403665781021118164e+0000, 103 }; 104 105 106 #define one q[0] 107 #define pp1 q[1] 108 #define pp2 q[2] 109 #define pp3 q[3] 110 #define qq1 q[4] 111 #define qq2 q[5] 112 #define t1 q[6] 113 #define t2 q[7] 114 #define t3 q[8] 115 #define t4 q[9] 116 #define t5 q[10] 117 #define t6 q[11] 118 119 /* INDENT ON */ 120 121 122 double 123 __k_tan(double x, double y, int k) { 124 double a, t, z, w = 0.0L, s, c, r, rh, xh, xl; 125 int i, j, hx, ix; 126 127 t = one; 128 hx = ((int *) &x)[HIWORD]; 129 ix = hx & 0x7fffffff; 130 if (ix < 0x3fc40000) { /* 0.15625 */ 131 if (ix < 0x3e400000) { /* 2^-27 */ 132 if ((i = (int) x) == 0) /* generate inexact */ 133 w = x; 134 t = y; 135 } else { 136 z = x * x; 137 t = y + (((t1 * x) * z) * (t2 + z * (t3 + z))) * 138 ((t4 + z) * (t5 + z * (t6 + z))); 139 w = x + t; 140 } 141 if (k == 0) 142 return (w); 143 /* 144 * Compute -1/(x+T) with great care 145 * Let r = -1/(x+T), rh = r chopped to 20 bits. 146 * Also let xh = x+T chopped to 20 bits, xl = (x-xh)+T. Then 147 * -1/(x+T) = rh + (-1/(x+T)-rh) = rh + r*(1+rh*(x+T)) 148 * = rh + r*((1+rh*xh)+rh*xl). 149 */ 150 rh = r = -one / w; 151 ((int *) &rh)[LOWORD] = 0; 152 xh = w; 153 ((int *) &xh)[LOWORD] = 0; 154 xl = (x - xh) + t; 155 return (rh + r * ((one + rh * xh) + rh * xl)); 156 } 157 j = (ix + 0x4000) & 0x7fff8000; 158 i = (j - 0x3fc40000) >> 15; 159 ((int *) &t)[HIWORD] = j; 160 if (hx > 0) 161 x = y - (t - x); 162 else 163 x = -y - (t + x); 164 a = _TBL_tan_hi[i]; 165 z = x * x; 166 s = (pp1 * x) * (pp2 + z * (pp3 + z)); /* sin(x) */ 167 t = (qq1 * z) * (qq2 + z); /* cos(x) - 1 */ 168 if (k == 0) { 169 w = a * s; 170 t = _TBL_tan_lo[i] + (s + a * w) / (one - (w - t)); 171 return (hx < 0 ? -a - t : a + t); 172 } else { 173 w = s + a * t; 174 c = w + _TBL_tan_lo[i]; 175 t = a * s - t; 176 /* 177 * Now try to compute [(1-T)/(a+c)] accurately 178 * 179 * Let r = 1/(a+c), rh = (1-T)*r chopped to 20 bits. 180 * Also let xh = a+c chopped to 20 bits, xl = (a-xh)+c. Then 181 * (1-T)/(a+c) = rh + ((1-T)/(a+c)-rh) 182 * = rh + r*(1-T-rh*(a+c)) 183 * = rh + r*((1-T-rh*xh)-rh*xl) 184 * = rh + r*(((1-rh*xh)-T)-rh*xl) 185 */ 186 r = one / (a + c); 187 rh = (one - t) * r; 188 ((int *) &rh)[LOWORD] = 0; 189 xh = a + c; 190 ((int *) &xh)[LOWORD] = 0; 191 xl = (a - xh) + c; 192 z = rh + r * (((one - rh * xh) - t) - rh * xl); 193 return (hx >= 0 ? -z : z); 194 } 195 } 196