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 /*
  27  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  28  * Use is subject to license terms.
  29  */
  30 
  31 #pragma weak __asin = asin
  32 
  33 
  34 /*
  35  * asin(x)
  36  * Method :
  37  *      Since  asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
  38  *      we approximate asin(x) on [0,0.5] by
  39  *              asin(x) = x + x*x^2*R(x^2)
  40  *      where
  41  *              R(x^2) is a rational approximation of (asin(x)-x)/x^3
  42  *      and its remez error is bounded by
  43  *              |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
  44  *
  45  *      For x in [0.5,1]
  46  *              asin(x) = pi/2-2*asin(sqrt((1-x)/2))
  47  *      Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
  48  *      then for x>0.98
  49  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  50  *                      = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
  51  *      For x<=0.98, let pio4_hi = pio2_hi/2, then
  52  *              f = hi part of s;
  53  *              c = sqrt(z) - f = (z-f*f)/(s+f)         ...f+c=sqrt(z)
  54  *      and
  55  *              asin(x) = pi/2 - 2*(s+s*z*R(z))
  56  *                      = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
  57  *                      = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
  58  *
  59  * Special cases:
  60  *      if x is NaN, return x itself;
  61  *      if |x|>1, return NaN with invalid signal.
  62  *
  63  */
  64 
  65 #include "libm_protos.h"                /* _SVID_libm_error */
  66 #include "libm_macros.h"
  67 #include <math.h>
  68 
  69 static const double xxx[] = {
  70 /* one */
  71         1.00000000000000000000e+00,             /* 3FF00000, 00000000 */
  72 /* huge */ 1.000e+300,
  73 /* pio2_hi */ 1.57079632679489655800e+00,       /* 3FF921FB, 54442D18 */
  74 /* pio2_lo */ 6.12323399573676603587e-17,       /* 3C91A626, 33145C07 */
  75 /* pio4_hi */ 7.85398163397448278999e-01,       /* 3FE921FB, 54442D18 */
  76 
  77 /*
  78  * coefficient for R(x^2)
  79  * pS0
  80  */
  81         1.66666666666666657415e-01,             /* 3FC55555, 55555555 */
  82 /* pS1 */ -3.25565818622400915405e-01,          /* BFD4D612, 03EB6F7D */
  83 /* pS2 */ 2.01212532134862925881e-01,           /* 3FC9C155, 0E884455 */
  84 /* pS3 */ -4.00555345006794114027e-02,          /* BFA48228, B5688F3B */
  85 /* pS4 */ 7.91534994289814532176e-04,           /* 3F49EFE0, 7501B288 */
  86 /* pS5 */ 3.47933107596021167570e-05,           /* 3F023DE1, 0DFDF709 */
  87 /* qS1 */ -2.40339491173441421878e+00,          /* C0033A27, 1C8A2D4B */
  88 /* qS2 */ 2.02094576023350569471e+00,           /* 40002AE5, 9C598AC8 */
  89 /* qS3 */ -6.88283971605453293030e-01,          /* BFE6066C, 1B8D0159 */
  90 /* qS4 */ 7.70381505559019352791e-02            /* 3FB3B8C5, B12E9282 */
  91 };
  92 
  93 #define one             xxx[0]
  94 #define huge            xxx[1]
  95 #define pio2_hi         xxx[2]
  96 #define pio2_lo         xxx[3]
  97 #define pio4_hi         xxx[4]
  98 #define pS0             xxx[5]
  99 #define pS1             xxx[6]
 100 #define pS2             xxx[7]
 101 #define pS3             xxx[8]
 102 #define pS4             xxx[9]
 103 #define pS5             xxx[10]
 104 #define qS1             xxx[11]
 105 #define qS2             xxx[12]
 106 #define qS3             xxx[13]
 107 #define qS4             xxx[14]
 108 
 109 double
 110 asin(double x)
 111 {
 112         double t, w, p, q, c, r, s;
 113         int hx, ix, i;
 114 
 115         hx = ((int *)&x)[HIWORD];
 116         ix = hx & 0x7fffffff;
 117 
 118         if (ix >= 0x3ff00000) {              /* |x| >= 1 */
 119                 if (((ix - 0x3ff00000) | ((int *)&x)[LOWORD]) == 0)
 120                         /* asin(1)=+-pi/2 with inexact */
 121                         return (x * pio2_hi + x * pio2_lo);
 122                 else if (isnan(x))
 123 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
 124                         return (ix >= 0x7ff80000 ? x : (x - x) / (x - x));
 125 
 126                 /* assumes sparc-like QNaN */
 127 #else
 128                         return ((x - x) / (x - x));     /* asin(|x|>1) is NaN */
 129 #endif
 130                 else
 131                         return (_SVID_libm_err(x, x, 2));
 132         } else if (ix < 0x3fe00000) {        /* |x| < 0.5 */
 133                 if (ix < 0x3e400000) {       /* if |x| < 2**-27 */
 134                         if ((i = (int)x) == 0)
 135                                 /* return x with inexact if x != 0 */
 136                                 return (x);
 137                 }
 138 
 139                 t = x * x;
 140                 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t *
 141                     pS5)))));
 142                 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
 143                 w = p / q;
 144                 return (x + x * w);
 145         }
 146 
 147         /* 1 > |x| >= 0.5 */
 148         w = one - fabs(x);
 149         t = w * 0.5;
 150         p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
 151         q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
 152         s = sqrt(t);
 153 
 154         if (ix >= 0x3FEF3333) {              /* if |x| > 0.975 */
 155                 w = p / q;
 156                 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
 157         } else {
 158                 w = s;
 159                 ((int *)&w)[LOWORD] = 0;
 160                 c = (t - w * w) / (s + w);
 161                 r = p / q;
 162                 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
 163                 q = pio4_hi - 2.0 * w;
 164                 t = pio4_hi - (p - q);
 165         }
 166 
 167         return (hx > 0 ? t : -t);
 168 }