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 #pragma weak __atan = atan 31 32 /* INDENT OFF */ 33 /* 34 * atan(x) 35 * Accurate Table look-up algorithm with polynomial approximation in 36 * partially product form. 37 * 38 * -- K.C. Ng, October 17, 2004 39 * 40 * Algorithm 41 * 42 * (1). Purge off Inf and NaN and 0 43 * (2). Reduce x to positive by atan(x) = -atan(-x). 44 * (3). For x <= 1/8 and let z = x*x, return 45 * (2.1) if x < 2^(-prec/2), atan(x) = x with inexact flag raised 46 * (2.2) if x < 2^(-prec/4-1), atan(x) = x+(x/3)(x*x) 47 * (2.3) if x < 2^(-prec/6-2), atan(x) = x+(z-5/3)(z*x/5) 48 * (2.4) Otherwise 49 * atan(x) = poly1(x) = x + A * B, 50 * where 51 * A = (p1*x*z) * (p2+z(p3+z)) 52 * B = (p4+z)+z*z) * (p5+z(p6+z)) 53 * Note: (i) domain of poly1 is [0, 1/8], (ii) remez relative 54 * approximation error of poly1 is bounded by 55 * |(atan(x)-poly1(x))/x| <= 2^-57.61 56 * (4). For x >= 8 then 57 * (3.1) if x >= 2^prec, atan(x) = atan(inf) - pio2lo 58 * (3.2) if x >= 2^(prec/3), atan(x) = atan(inf) - 1/x 59 * (3.3) if x <= 65, atan(x) = atan(inf) - poly1(1/x) 60 * (3.4) otherwise atan(x) = atan(inf) - poly2(1/x) 61 * where 62 * poly2(r) = (q1*r) * (q2+z(q3+z)) * (q4+z), 63 * its domain is [0, 0.0154]; and its remez absolute 64 * approximation error is bounded by 65 * |atan(x)-poly2(x)|<= 2^-59.45 66 * 67 * (5). Now x is in (0.125, 8). 68 * Recall identity 69 * atan(x) = atan(y) + atan((x-y)/(1+x*y)). 70 * Let j = (ix - 0x3fc00000) >> 16, 0 <= j < 96, where ix is the high 71 * part of x in IEEE double format. Then 72 * atan(x) = atan(y[j]) + poly2((x-y[j])/(1+x*y[j])) 73 * where y[j] are carefully chosen so that it matches x to around 4.5 74 * bits and at the same time atan(y[j]) is very close to an IEEE double 75 * floating point number. Calculation indicates that 76 * max|(x-y[j])/(1+x*y[j])| < 0.0154 77 * j,x 78 * 79 * Accuracy: Maximum error observed is bounded by 0.6 ulp after testing 80 * more than 10 million random arguments 81 */ 82 /* INDENT ON */ 83 84 #include "libm.h" 85 #include "libm_protos.h" 86 87 extern const double _TBL_atan[]; 88 static const double g[] = { 89 /* one = */ 1.0, 90 /* p1 = */ 8.02176624254765935351230154992663301527500152588e-0002, 91 /* p2 = */ 1.27223421700559402580665846471674740314483642578e+0000, 92 /* p3 = */ -1.20606901800503640842521235754247754812240600586e+0000, 93 /* p4 = */ -2.36088967922325565496066701598465442657470703125e+0000, 94 /* p5 = */ 1.38345799501389166152875986881554126739501953125e+0000, 95 /* p6 = */ 1.06742368078953453469637224770849570631980895996e+0000, 96 /* q1 = */ -1.42796626333911796935538518482644576579332351685e-0001, 97 /* q2 = */ 3.51427110447873227059810477159863497078605962912e+0000, 98 /* q3 = */ 5.92129112708164262457444237952586263418197631836e-0001, 99 /* q4 = */ -1.99272234785683144409063061175402253866195678711e+0000, 100 /* pio2hi */ 1.570796326794896558e+00, 101 /* pio2lo */ 6.123233995736765886e-17, 102 /* t1 = */ -0.333333333333333333333333333333333, 103 /* t2 = */ 0.2, 104 /* t3 = */ -1.666666666666666666666666666666666, 105 }; 106 107 #define one g[0] 108 #define p1 g[1] 109 #define p2 g[2] 110 #define p3 g[3] 111 #define p4 g[4] 112 #define p5 g[5] 113 #define p6 g[6] 114 #define q1 g[7] 115 #define q2 g[8] 116 #define q3 g[9] 117 #define q4 g[10] 118 #define pio2hi g[11] 119 #define pio2lo g[12] 120 #define t1 g[13] 121 #define t2 g[14] 122 #define t3 g[15] 123 124 125 double 126 atan(double x) { 127 double y, z, r, p, s; 128 int ix, lx, hx, j; 129 130 hx = ((int *) &x)[HIWORD]; 131 lx = ((int *) &x)[LOWORD]; 132 ix = hx & ~0x80000000; 133 j = ix >> 20; 134 135 /* for |x| < 1/8 */ 136 if (j < 0x3fc) { 137 if (j < 0x3f5) { /* when |x| < 2**(-prec/6-2) */ 138 if (j < 0x3e3) { /* if |x| < 2**(-prec/2-2) */ 139 return ((int) x == 0 ? x : one); 140 } 141 if (j < 0x3f1) { /* if |x| < 2**(-prec/4-1) */ 142 return (x + (x * t1) * (x * x)); 143 } else { /* if |x| < 2**(-prec/6-2) */ 144 z = x * x; 145 s = t2 * x; 146 return (x + (t3 + z) * (s * z)); 147 } 148 } 149 z = x * x; s = p1 * x; 150 return (x + ((s * z) * (p2 + z * (p3 + z))) * 151 (((p4 + z) + z * z) * (p5 + z * (p6 + z)))); 152 } 153 154 /* for |x| >= 8.0 */ 155 if (j >= 0x402) { 156 if (j < 0x436) { 157 r = one / x; 158 if (hx >= 0) { 159 y = pio2hi; p = pio2lo; 160 } else { 161 y = -pio2hi; p = -pio2lo; 162 } 163 if (ix < 0x40504000) { /* x < 65 */ 164 z = r * r; 165 s = p1 * r; 166 return (y + ((p - r) - ((s * z) * 167 (p2 + z * (p3 + z))) * 168 (((p4 + z) + z * z) * 169 (p5 + z * (p6 + z))))); 170 } else if (j < 0x412) { 171 z = r * r; 172 return (y + (p - ((q1 * r) * (q4 + z)) * 173 (q2 + z * (q3 + z)))); 174 } else 175 return (y + (p - r)); 176 } else { 177 if (j >= 0x7ff) /* x is inf or NaN */ 178 if (((ix - 0x7ff00000) | lx) != 0) 179 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN) 180 return (ix >= 0x7ff80000 ? x : x - x); 181 /* assumes sparc-like QNaN */ 182 #else 183 return (x - x); 184 #endif 185 y = -pio2lo; 186 return (hx >= 0 ? pio2hi - y : y - pio2hi); 187 } 188 } else { /* now x is between 1/8 and 8 */ 189 double *w, w0, w1, s, z; 190 w = (double *) _TBL_atan + (((ix - 0x3fc00000) >> 16) << 1); 191 w0 = (hx >= 0)? w[0] : -w[0]; 192 s = (x - w0) / (one + x * w0); 193 w1 = (hx >= 0)? w[1] : -w[1]; 194 z = s * s; 195 return (((q1 * s) * (q4 + z)) * (q2 + z * (q3 + z)) + w1); 196 } 197 } 198