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 #include <LibConfig.h>
13 #include <sys/EfiCdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_asin.c,v 1.12 2002/05/26 22:01:48 wiz Exp $");
16 #endif
17
18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
19 // C4723: potential divide by zero.
20 #pragma warning ( disable : 4723 )
21 #endif
22
23 /* __ieee754_asin(x)
24 * Method :
25 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
26 * we approximate asin(x) on [0,0.5] by
27 * asin(x) = x + x*x^2*R(x^2)
28 * where
29 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
30 * and its remez error is bounded by
31 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
32 *
33 * For x in [0.5,1]
34 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
35 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
36 * then for x>0.98
37 * asin(x) = pi/2 - 2*(s+s*z*R(z))
38 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
39 * For x<=0.98, let pio4_hi = pio2_hi/2, then
40 * f = hi part of s;
41 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
42 * and
43 * asin(x) = pi/2 - 2*(s+s*z*R(z))
44 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
45 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
46 *
47 * Special cases:
48 * if x is NaN, return x itself;
49 * if |x|>1, return NaN with invalid signal.
50 *
51 */
52
53
54 #include "math.h"
55 #include "math_private.h"
56
57 static const double
58 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
59 huge = 1.000e+300,
60 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
61 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
62 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
63 /* coefficient for R(x^2) */
64 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
65 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
66 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
67 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
68 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
69 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
70 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
71 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
72 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
73 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
74
75 double
__ieee754_asin(double x)76 __ieee754_asin(double x)
77 {
78 double t,w,p,q,c,r,s;
79 int32_t hx,ix;
80
81 t = 0;
82 GET_HIGH_WORD(hx,x);
83 ix = hx&0x7fffffff;
84 if(ix>= 0x3ff00000) { /* |x|>= 1 */
85 u_int32_t lx;
86 GET_LOW_WORD(lx,x);
87 if(((ix-0x3ff00000)|lx)==0)
88 /* asin(1)=+-pi/2 with inexact */
89 return x*pio2_hi+x*pio2_lo;
90 return (x-x)/(x-x); /* asin(|x|>1) is NaN */
91 } else if (ix<0x3fe00000) { /* |x|<0.5 */
92 if(ix<0x3e400000) { /* if |x| < 2**-27 */
93 if(huge+x>one) return x;/* return x with inexact if x!=0*/
94 } else
95 t = x*x;
96 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
97 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
98 w = p/q;
99 return x+x*w;
100 }
101 /* 1> |x|>= 0.5 */
102 w = one-fabs(x);
103 t = w*0.5;
104 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
105 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
106 s = __ieee754_sqrt(t);
107 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
108 w = p/q;
109 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
110 } else {
111 w = s;
112 SET_LOW_WORD(w,0);
113 c = (t-w*w)/(s+w);
114 r = p/q;
115 p = 2.0*s*r-(pio2_lo-2.0*c);
116 q = pio4_hi-2.0*w;
117 t = pio4_hi-(p-q);
118 }
119 if(hx>0) return t; else return -t;
120 }
121