1
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, 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 /* asin(x)
14 * Method :
15 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
16 * we approximate asin(x) on [0,0.5] by
17 * asin(x) = x + x*x^2*R(x^2)
18 * where
19 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
20 * and its remez error is bounded by
21 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
22 *
23 * For x in [0.5,1]
24 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
25 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
26 * then for x>0.98
27 * asin(x) = pi/2 - 2*(s+s*z*R(z))
28 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
29 * For x<=0.98, let pio4_hi = pio2_hi/2, then
30 * f = hi part of s;
31 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
32 * and
33 * asin(x) = pi/2 - 2*(s+s*z*R(z))
34 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
35 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
36 *
37 * Special cases:
38 * if x is NaN, return x itself;
39 * if |x|>1, return NaN with invalid signal.
40 *
41 */
42
43 #include <float.h>
44
45 #include "math.h"
46 #include "math_private.h"
47
48 static const double
49 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
50 huge = 1.000e+300,
51 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
52 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
53 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
54 /* coefficient for R(x^2) */
55 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
56 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
57 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
58 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
59 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
60 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
61 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
62 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
63 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
64 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
65
66 double
asin(double x)67 asin(double x)
68 {
69 double t=0.0,w,p,q,c,r,s;
70 int32_t hx,ix;
71 GET_HIGH_WORD(hx,x);
72 ix = hx&0x7fffffff;
73 if(ix>= 0x3ff00000) { /* |x|>= 1 */
74 u_int32_t lx;
75 GET_LOW_WORD(lx,x);
76 if(((ix-0x3ff00000)|lx)==0)
77 /* asin(1)=+-pi/2 with inexact */
78 return x*pio2_hi+x*pio2_lo;
79 return (x-x)/(x-x); /* asin(|x|>1) is NaN */
80 } else if (ix<0x3fe00000) { /* |x|<0.5 */
81 if(ix<0x3e500000) { /* if |x| < 2**-26 */
82 if(huge+x>one) return x;/* return x with inexact if x!=0*/
83 }
84 t = x*x;
85 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
86 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
87 w = p/q;
88 return x+x*w;
89 }
90 /* 1> |x|>= 0.5 */
91 w = one-fabs(x);
92 t = w*0.5;
93 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
94 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
95 s = sqrt(t);
96 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
97 w = p/q;
98 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
99 } else {
100 w = s;
101 SET_LOW_WORD(w,0);
102 c = (t-w*w)/(s+w);
103 r = p/q;
104 p = 2.0*s*r-(pio2_lo-2.0*c);
105 q = pio4_hi-2.0*w;
106 t = pio4_hi-(p-q);
107 }
108 if(hx>0) return t; else return -t;
109 }
110
111 #if LDBL_MANT_DIG == 53
112 __weak_reference(asin, asinl);
113 #endif
114