1 /* @(#)e_asin.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_asin.c,v 1.13 2017/05/09 02:04:38 maya Exp $");
16 #endif
17
18 /* __ieee754_asin(x)
19 * Method :
20 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
21 * we approximate asin(x) on [0,0.5] by
22 * asin(x) = x + x*x^2*R(x^2)
23 * where
24 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
25 * and its remez error is bounded by
26 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
27 *
28 * For x in [0.5,1]
29 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
30 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
31 * then for x>0.98
32 * asin(x) = pi/2 - 2*(s+s*z*R(z))
33 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
34 * For x<=0.98, let pio4_hi = pio2_hi/2, then
35 * f = hi part of s;
36 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
37 * and
38 * asin(x) = pi/2 - 2*(s+s*z*R(z))
39 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
40 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
41 *
42 * Special cases:
43 * if x is NaN, return x itself;
44 * if |x|>1, return NaN with invalid signal.
45 *
46 */
47
48
49 #include "math.h"
50 #include "math_private.h"
51
52 static const double
53 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
54 huge = 1.000e+300,
55 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
56 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
57 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
58 /* coefficient for R(x^2) */
59 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
60 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
61 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
62 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
63 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
64 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
65 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
66 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
67 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
68 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
69
70 double
__ieee754_asin(double x)71 __ieee754_asin(double x)
72 {
73 double t,w,p,q,c,r,s;
74 int32_t hx,ix;
75
76 t = 0;
77 GET_HIGH_WORD(hx,x);
78 ix = hx&0x7fffffff;
79 if(ix>= 0x3ff00000) { /* |x|>= 1 */
80 u_int32_t lx;
81 GET_LOW_WORD(lx,x);
82 if(((ix-0x3ff00000)|lx)==0)
83 /* asin(1)=+-pi/2 with inexact */
84 return x*pio2_hi+x*pio2_lo;
85 return (x-x)/(x-x); /* asin(|x|>1) is NaN */
86 } else if (ix<0x3fe00000) { /* |x|<0.5 */
87 if(ix<0x3e400000) { /* if |x| < 2**-27 */
88 if(huge+x>one) return x;/* return x with inexact if x!=0*/
89 } else {
90 t = x*x;
91 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
92 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
93 w = p/q;
94 return x+x*w;
95 }
96 }
97 /* 1> |x|>= 0.5 */
98 w = one-fabs(x);
99 t = w*0.5;
100 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
101 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
102 s = __ieee754_sqrt(t);
103 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
104 w = p/q;
105 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
106 } else {
107 w = s;
108 SET_LOW_WORD(w,0);
109 c = (t-w*w)/(s+w);
110 r = p/q;
111 p = 2.0*s*r-(pio2_lo-2.0*c);
112 q = pio4_hi-2.0*w;
113 t = pio4_hi-(p-q);
114 }
115 if(hx>0) return t; else return -t;
116 }
117