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 asin = __asin 31 32 /* INDENT OFF */ 33 /* 34 * asin(x) 35 * Method : 36 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 37 * we approximate asin(x) on [0,0.5] by 38 * asin(x) = x + x*x^2*R(x^2) 39 * where 40 * R(x^2) is a rational approximation of (asin(x)-x)/x^3 41 * and its remez error is bounded by 42 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 43 * 44 * For x in [0.5,1] 45 * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 46 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 47 * then for x>0.98 48 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 49 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 50 * For x<=0.98, let pio4_hi = pio2_hi/2, then 51 * f = hi part of s; 52 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 53 * and 54 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 55 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 56 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 57 * 58 * Special cases: 59 * if x is NaN, return x itself; 60 * if |x|>1, return NaN with invalid signal. 61 * 62 */ 63 /* INDENT ON */ 64 65 #include "libm_synonyms.h" /* __asin, __sqrt, __isnan */ 66 #include "libm_protos.h" /* _SVID_libm_error */ 67 #include "libm_macros.h" 68 #include <math.h> 69 70 /* INDENT OFF */ 71 static const double xxx[] = { 72 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 73 /* huge */ 1.000e+300, 74 /* pio2_hi */ 1.57079632679489655800e+00, /* 3FF921FB, 54442D18 */ 75 /* pio2_lo */ 6.12323399573676603587e-17, /* 3C91A626, 33145C07 */ 76 /* pio4_hi */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 77 /* coefficient for R(x^2) */ 78 /* pS0 */ 1.66666666666666657415e-01, /* 3FC55555, 55555555 */ 79 /* pS1 */ -3.25565818622400915405e-01, /* BFD4D612, 03EB6F7D */ 80 /* pS2 */ 2.01212532134862925881e-01, /* 3FC9C155, 0E884455 */ 81 /* pS3 */ -4.00555345006794114027e-02, /* BFA48228, B5688F3B */ 82 /* pS4 */ 7.91534994289814532176e-04, /* 3F49EFE0, 7501B288 */ 83 /* pS5 */ 3.47933107596021167570e-05, /* 3F023DE1, 0DFDF709 */ 84 /* qS1 */ -2.40339491173441421878e+00, /* C0033A27, 1C8A2D4B */ 85 /* qS2 */ 2.02094576023350569471e+00, /* 40002AE5, 9C598AC8 */ 86 /* qS3 */ -6.88283971605453293030e-01, /* BFE6066C, 1B8D0159 */ 87 /* qS4 */ 7.70381505559019352791e-02 /* 3FB3B8C5, B12E9282 */ 88 }; 89 #define one xxx[0] 90 #define huge xxx[1] 91 #define pio2_hi xxx[2] 92 #define pio2_lo xxx[3] 93 #define pio4_hi xxx[4] 94 #define pS0 xxx[5] 95 #define pS1 xxx[6] 96 #define pS2 xxx[7] 97 #define pS3 xxx[8] 98 #define pS4 xxx[9] 99 #define pS5 xxx[10] 100 #define qS1 xxx[11] 101 #define qS2 xxx[12] 102 #define qS3 xxx[13] 103 #define qS4 xxx[14] 104 /* INDENT ON */ 105 106 double 107 asin(double x) { 108 double t, w, p, q, c, r, s; 109 int hx, ix, i; 110 111 hx = ((int *) &x)[HIWORD]; 112 ix = hx & 0x7fffffff; 113 if (ix >= 0x3ff00000) { /* |x| >= 1 */ 114 if (((ix - 0x3ff00000) | ((int *) &x)[LOWORD]) == 0) 115 /* asin(1)=+-pi/2 with inexact */ 116 return (x * pio2_hi + x * pio2_lo); 117 else if (isnan(x)) 118 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN) 119 return (ix >= 0x7ff80000 ? x : (x - x) / (x - x)); 120 /* assumes sparc-like QNaN */ 121 #else 122 return (x - x) / (x - x); /* asin(|x|>1) is NaN */ 123 #endif 124 else 125 return (_SVID_libm_err(x, x, 2)); 126 } else if (ix < 0x3fe00000) { /* |x| < 0.5 */ 127 if (ix < 0x3e400000) { /* if |x| < 2**-27 */ 128 if ((i = (int) x) == 0) 129 /* return x with inexact if x != 0 */ 130 return (x); 131 } 132 t = x * x; 133 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + 134 t * (pS4 + t * pS5))))); 135 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); 136 w = p / q; 137 return (x + x * w); 138 } 139 /* 1 > |x| >= 0.5 */ 140 w = one - fabs(x); 141 t = w * 0.5; 142 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); 143 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); 144 s = sqrt(t); 145 if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */ 146 w = p / q; 147 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); 148 } else { 149 w = s; 150 ((int *) &w)[LOWORD] = 0; 151 c = (t - w * w) / (s + w); 152 r = p / q; 153 p = 2.0 * s * r - (pio2_lo - 2.0 * c); 154 q = pio4_hi - 2.0 * w; 155 t = pio4_hi - (p - q); 156 } 157 return (hx > 0 ? t : -t); 158 } 159