1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /* __ieee754_asin(x)
13 * Method :
14 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
15 * we approximate asin(x) on [0,0.5] by
16 * asin(x) = x + x*x^2*R(x^2)
17 * where
18 * R(x^2) is a rational approximation of (asin(x)-x)/x^3
19 * and its remez error is bounded by
20 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
21 *
22 * For x in [0.5,1]
23 * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
24 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
25 * then for x>0.98
26 * asin(x) = pi/2 - 2*(s+s*z*R(z))
27 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
28 * For x<=0.98, let pio4_hi = pio2_hi/2, then
29 * f = hi part of s;
30 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
31 * and
32 * asin(x) = pi/2 - 2*(s+s*z*R(z))
33 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
34 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
35 *
36 * Special cases:
37 * if x is NaN, return x itself;
38 * if |x|>1, return NaN with invalid signal.
39 *
40 */
41
42 #include "math.h"
43 #include "math_private.h"
44
45 static const double
46 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
47 huge = 1.000e+300,
48 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
49 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
50 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
51 /* coefficient for R(x^2) */
52 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
53 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
54 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
55 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
56 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
57 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
58 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
59 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
60 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
61 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
62
__ieee754_asin(double x)63 double __ieee754_asin(double x)
64 {
65 double t=0.0,w,p,q,c,r,s;
66 int32_t hx,ix;
67 GET_HIGH_WORD(hx,x);
68 ix = hx&0x7fffffff;
69 if(ix>= 0x3ff00000) { /* |x|>= 1 */
70 u_int32_t lx;
71 GET_LOW_WORD(lx,x);
72 if(((ix-0x3ff00000)|lx)==0)
73 /* asin(1)=+-pi/2 with inexact */
74 return x*pio2_hi+x*pio2_lo;
75 return (x-x)/(x-x); /* asin(|x|>1) is NaN */
76 } else if (ix<0x3fe00000) { /* |x|<0.5 */
77 if(ix<0x3e400000) { /* if |x| < 2**-27 */
78 if(huge+x>one) return x;/* return x with inexact if x!=0*/
79 } else {
80 t = x*x;
81 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
82 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
83 w = p/q;
84 return x+x*w;
85 }
86 }
87 /* 1> |x|>= 0.5 */
88 w = one-fabs(x);
89 t = w*0.5;
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 s = __ieee754_sqrt(t);
93 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
94 w = p/q;
95 t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
96 } else {
97 w = s;
98 SET_LOW_WORD(w,0);
99 c = (t-w*w)/(s+w);
100 r = p/q;
101 p = 2.0*s*r-(pio2_lo-2.0*c);
102 q = pio4_hi-2.0*w;
103 t = pio4_hi-(p-q);
104 }
105 if(hx>0) return t; else return -t;
106 }
107
108 /*
109 * wrapper asin(x)
110 */
111 #ifndef _IEEE_LIBM
asin(double x)112 double asin(double x)
113 {
114 double z = __ieee754_asin(x);
115 if (_LIB_VERSION == _IEEE_ || isnan(x))
116 return z;
117 if (fabs(x) > 1.0)
118 return __kernel_standard(x, x, 2); /* asin(|x|>1) */
119 return z;
120 }
121 #else
122 strong_alias(__ieee754_asin, asin)
123 #endif
124 libm_hidden_def(asin)
125