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e_asin.c
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1 
2 /* @(#)e_asin.c 1.4 96/03/07 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunSoft, 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 Remes 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 #ifdef __STDC__
48 static const double
49 #else
50 static double
51 #endif
52 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
53 huge = 1.000e+300,
54 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
55 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
56 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
57  /* coefficient for R(x^2) */
58 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
59 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
60 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
61 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
62 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
63 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
64 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
65 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
66 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
67 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
68 
69 #ifdef __STDC__
70  double __ieee754_asin(double x)
71 #else
72  double __ieee754_asin(x)
73  double x;
74 #endif
75 {
76  double t=0,w,p,q,c,r,s;
77  int hx,ix;
78  hx = __HI(x);
79  ix = hx&0x7fffffff;
80  if(ix>= 0x3ff00000) { /* |x|>= 1 */
81  if(((ix-0x3ff00000)|__LO(x))==0)
82  /* asin(1)=+-pi/2 with inexact */
83  return x*pio2_hi+x*pio2_lo;
84  return (x-x)/(x-x); /* asin(|x|>1) is NaN */
85  } else if (ix<0x3fe00000) { /* |x|<0.5 */
86  if(ix<0x3e400000) { /* if |x| < 2**-27 */
87  if(huge+x>one) return x;/* return x with inexact if x!=0*/
88  } else
89  t = x*x;
90  p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
91  q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
92  w = p/q;
93  return x+x*w;
94  }
95  /* 1> |x|>= 0.5 */
96  w = one-fabs(x);
97  t = w*0.5;
98  p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
99  q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
100  s = sqrt(t);
101  if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
102  w = p/q;
103  t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
104  } else {
105  w = s;
106  __LO(w) = 0;
107  c = (t-w*w)/(s+w);
108  r = p/q;
109  p = 2.0*s*r-(pio2_lo-2.0*c);
110  q = pio4_hi-2.0*w;
111  t = pio4_hi-(p-q);
112  }
113  if(hx>0) return t; else return -t;
114 }