usr/src/lib/libm/common/C/asin.c

   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_protos.h"        /* _SVID_libm_error */
  66 #include "libm_macros.h"
  67 #include <math.h>
  68
  69 /* INDENT OFF */
  70 static const double xxx[] = {
  71 /* one */        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 /* coefficient for R(x^2) */
  77 /* pS0 */        1.66666666666666657415e-01,    /* 3FC55555, 55555555 */
  78 /* pS1 */       -3.25565818622400915405e-01,    /* BFD4D612, 03EB6F7D */
  79 /* pS2 */        2.01212532134862925881e-01,    /* 3FC9C155, 0E884455 */
  80 /* pS3 */       -4.00555345006794114027e-02,    /* BFA48228, B5688F3B */
  81 /* pS4 */        7.91534994289814532176e-04,    /* 3F49EFE0, 7501B288 */
  82 /* pS5 */        3.47933107596021167570e-05,    /* 3F023DE1, 0DFDF709 */
  83 /* qS1 */       -2.40339491173441421878e+00,    /* C0033A27, 1C8A2D4B */
  84 /* qS2 */        2.02094576023350569471e+00,    /* 40002AE5, 9C598AC8 */
  85 /* qS3 */       -6.88283971605453293030e-01,    /* BFE6066C, 1B8D0159 */
  86 /* qS4 */        7.70381505559019352791e-02     /* 3FB3B8C5, B12E9282 */
  87 };
  88 #define one     xxx[0]
  89 #define huge    xxx[1]
  90 #define pio2_hi xxx[2]
  91 #define pio2_lo xxx[3]
  92 #define pio4_hi xxx[4]
  93 #define pS0     xxx[5]
  94 #define pS1     xxx[6]
  95 #define pS2     xxx[7]
  96 #define pS3     xxx[8]
  97 #define pS4     xxx[9]
  98 #define pS5     xxx[10]
  99 #define qS1     xxx[11]
 100 #define qS2     xxx[12]
 101 #define qS3     xxx[13]
 102 #define qS4     xxx[14]
 103 /* INDENT ON */
 104
 105 double
 106 asin(double x) {
 107         double t, w, p, q, c, r, s;
 108         int hx, ix, i;
 109
 110         hx = ((int *) &x)[HIWORD];
 111         ix = hx & 0x7fffffff;
 112         if (ix >= 0x3ff00000) { /* |x| >= 1 */
 113                 if (((ix - 0x3ff00000) | ((int *) &x)[LOWORD]) == 0)
 114                         /* asin(1)=+-pi/2 with inexact */
 115                         return (x * pio2_hi + x * pio2_lo);
 116                 else if (isnan(x))
 117 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
 118                         return (ix >= 0x7ff80000 ? x : (x - x) / (x - x));
 119                         /* assumes sparc-like QNaN */
 120 #else
 121                         return (x - x) / (x - x);       /* asin(|x|>1) is NaN */
 122 #endif
 123                 else
 124                         return (_SVID_libm_err(x, x, 2));
 125         } else if (ix < 0x3fe00000) {   /* |x| < 0.5 */
 126                 if (ix < 0x3e400000) {  /* if |x| < 2**-27 */
 127                         if ((i = (int) x) == 0)
 128                                 /* return x with inexact if x != 0 */
 129                                 return (x);
 130                 }
 131                 t = x * x;
 132                 p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 +
 133                         t * (pS4 + t * pS5)))));
 134                 q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
 135                 w = p / q;
 136                 return (x + x * w);
 137         }
 138         /* 1 > |x| >= 0.5 */
 139         w = one - fabs(x);
 140         t = w * 0.5;
 141         p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
 142         q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
 143         s = sqrt(t);
 144         if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
 145                 w = p / q;
 146                 t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
 147         } else {
 148                 w = s;
 149                 ((int *) &w)[LOWORD] = 0;
 150                 c = (t - w * w) / (s + w);
 151                 r = p / q;
 152                 p = 2.0 * s * r - (pio2_lo - 2.0 * c);
 153                 q = pio4_hi - 2.0 * w;
 154                 t = pio4_hi - (p - q);
 155         }
 156         return (hx > 0 ? t : -t);
 157 }