| 1 |
148 |
jeremybenn |
|
| 2 |
|
|
/* @(#)e_asin.c 5.1 93/09/24 */
|
| 3 |
|
|
/*
|
| 4 |
|
|
* ====================================================
|
| 5 |
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
| 6 |
|
|
*
|
| 7 |
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
| 8 |
|
|
* Permission to use, copy, modify, and distribute this
|
| 9 |
|
|
* software is freely granted, provided that this notice
|
| 10 |
|
|
* is preserved.
|
| 11 |
|
|
* ====================================================
|
| 12 |
|
|
*/
|
| 13 |
|
|
|
| 14 |
|
|
/* __ieee754_asin(x)
|
| 15 |
|
|
* Method :
|
| 16 |
|
|
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
|
| 17 |
|
|
* we approximate asin(x) on [0,0.5] by
|
| 18 |
|
|
* asin(x) = x + x*x^2*R(x^2)
|
| 19 |
|
|
* where
|
| 20 |
|
|
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
|
| 21 |
|
|
* and its remez error is bounded by
|
| 22 |
|
|
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
|
| 23 |
|
|
*
|
| 24 |
|
|
* For x in [0.5,1]
|
| 25 |
|
|
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
|
| 26 |
|
|
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
|
| 27 |
|
|
* then for x>0.98
|
| 28 |
|
|
* asin(x) = pi/2 - 2*(s+s*z*R(z))
|
| 29 |
|
|
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
|
| 30 |
|
|
* For x<=0.98, let pio4_hi = pio2_hi/2, then
|
| 31 |
|
|
* f = hi part of s;
|
| 32 |
|
|
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
|
| 33 |
|
|
* and
|
| 34 |
|
|
* asin(x) = pi/2 - 2*(s+s*z*R(z))
|
| 35 |
|
|
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
|
| 36 |
|
|
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
|
| 37 |
|
|
*
|
| 38 |
|
|
* Special cases:
|
| 39 |
|
|
* if x is NaN, return x itself;
|
| 40 |
|
|
* if |x|>1, return NaN with invalid signal.
|
| 41 |
|
|
*
|
| 42 |
|
|
*/
|
| 43 |
|
|
|
| 44 |
|
|
|
| 45 |
|
|
#include "fdlibm.h"
|
| 46 |
|
|
|
| 47 |
|
|
#ifndef _DOUBLE_IS_32BITS
|
| 48 |
|
|
|
| 49 |
|
|
#ifdef __STDC__
|
| 50 |
|
|
static const double
|
| 51 |
|
|
#else
|
| 52 |
|
|
static double
|
| 53 |
|
|
#endif
|
| 54 |
|
|
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
| 55 |
|
|
huge = 1.000e+300,
|
| 56 |
|
|
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
|
| 57 |
|
|
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
|
| 58 |
|
|
pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
|
| 59 |
|
|
/* coefficient for R(x^2) */
|
| 60 |
|
|
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
|
| 61 |
|
|
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
|
| 62 |
|
|
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
|
| 63 |
|
|
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
|
| 64 |
|
|
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
|
| 65 |
|
|
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
|
| 66 |
|
|
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
|
| 67 |
|
|
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
|
| 68 |
|
|
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
|
| 69 |
|
|
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
|
| 70 |
|
|
|
| 71 |
|
|
#ifdef __STDC__
|
| 72 |
|
|
double __ieee754_asin(double x)
|
| 73 |
|
|
#else
|
| 74 |
|
|
double __ieee754_asin(x)
|
| 75 |
|
|
double x;
|
| 76 |
|
|
#endif
|
| 77 |
|
|
{
|
| 78 |
|
|
double t,w,p,q,c,r,s;
|
| 79 |
|
|
__int32_t hx,ix;
|
| 80 |
|
|
GET_HIGH_WORD(hx,x);
|
| 81 |
|
|
ix = hx&0x7fffffff;
|
| 82 |
|
|
if(ix>= 0x3ff00000) { /* |x|>= 1 */
|
| 83 |
|
|
__uint32_t lx;
|
| 84 |
|
|
GET_LOW_WORD(lx,x);
|
| 85 |
|
|
if(((ix-0x3ff00000)|lx)==0)
|
| 86 |
|
|
/* asin(1)=+-pi/2 with inexact */
|
| 87 |
|
|
return x*pio2_hi+x*pio2_lo;
|
| 88 |
|
|
return (x-x)/(x-x); /* asin(|x|>1) is NaN */
|
| 89 |
|
|
} else if (ix<0x3fe00000) { /* |x|<0.5 */
|
| 90 |
|
|
if(ix<0x3e400000) { /* if |x| < 2**-27 */
|
| 91 |
|
|
if(huge+x>one) return x;/* return x with inexact if x!=0*/
|
| 92 |
|
|
} else {
|
| 93 |
|
|
t = x*x;
|
| 94 |
|
|
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
| 95 |
|
|
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
| 96 |
|
|
w = p/q;
|
| 97 |
|
|
return x+x*w;
|
| 98 |
|
|
}
|
| 99 |
|
|
}
|
| 100 |
|
|
/* 1> |x|>= 0.5 */
|
| 101 |
|
|
w = one-fabs(x);
|
| 102 |
|
|
t = w*0.5;
|
| 103 |
|
|
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
| 104 |
|
|
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
| 105 |
|
|
s = __ieee754_sqrt(t);
|
| 106 |
|
|
if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
|
| 107 |
|
|
w = p/q;
|
| 108 |
|
|
t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
|
| 109 |
|
|
} else {
|
| 110 |
|
|
w = s;
|
| 111 |
|
|
SET_LOW_WORD(w,0);
|
| 112 |
|
|
c = (t-w*w)/(s+w);
|
| 113 |
|
|
r = p/q;
|
| 114 |
|
|
p = 2.0*s*r-(pio2_lo-2.0*c);
|
| 115 |
|
|
q = pio4_hi-2.0*w;
|
| 116 |
|
|
t = pio4_hi-(p-q);
|
| 117 |
|
|
}
|
| 118 |
|
|
if(hx>0) return t; else return -t;
|
| 119 |
|
|
}
|
| 120 |
|
|
|
| 121 |
|
|
#endif /* defined(_DOUBLE_IS_32BITS) */
|
