1*05a0b428SJohn Marino /* @(#)e_asin.c 5.1 93/09/24 */
2*05a0b428SJohn Marino /*
3*05a0b428SJohn Marino * ====================================================
4*05a0b428SJohn Marino * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*05a0b428SJohn Marino *
6*05a0b428SJohn Marino * Developed at SunPro, a Sun Microsystems, Inc. business.
7*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this
8*05a0b428SJohn Marino * software is freely granted, provided that this notice
9*05a0b428SJohn Marino * is preserved.
10*05a0b428SJohn Marino * ====================================================
11*05a0b428SJohn Marino */
12*05a0b428SJohn Marino
13*05a0b428SJohn Marino /* asin(x)
14*05a0b428SJohn Marino * Method :
15*05a0b428SJohn Marino * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
16*05a0b428SJohn Marino * we approximate asin(x) on [0,0.5] by
17*05a0b428SJohn Marino * asin(x) = x + x*x^2*R(x^2)
18*05a0b428SJohn Marino * where
19*05a0b428SJohn Marino * R(x^2) is a rational approximation of (asin(x)-x)/x^3
20*05a0b428SJohn Marino * and its Remes error is bounded by
21*05a0b428SJohn Marino * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
22*05a0b428SJohn Marino *
23*05a0b428SJohn Marino * For x in [0.5,1]
24*05a0b428SJohn Marino * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
25*05a0b428SJohn Marino * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
26*05a0b428SJohn Marino * then for x>0.98
27*05a0b428SJohn Marino * asin(x) = pi/2 - 2*(s+s*z*R(z))
28*05a0b428SJohn Marino * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
29*05a0b428SJohn Marino * For x<=0.98, let pio4_hi = pio2_hi/2, then
30*05a0b428SJohn Marino * f = hi part of s;
31*05a0b428SJohn Marino * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
32*05a0b428SJohn Marino * and
33*05a0b428SJohn Marino * asin(x) = pi/2 - 2*(s+s*z*R(z))
34*05a0b428SJohn Marino * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
35*05a0b428SJohn Marino * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
36*05a0b428SJohn Marino *
37*05a0b428SJohn Marino * Special cases:
38*05a0b428SJohn Marino * if x is NaN, return x itself;
39*05a0b428SJohn Marino * if |x|>1, return NaN with invalid signal.
40*05a0b428SJohn Marino *
41*05a0b428SJohn Marino */
42*05a0b428SJohn Marino
43*05a0b428SJohn Marino #include <float.h>
44*05a0b428SJohn Marino #include <math.h>
45*05a0b428SJohn Marino
46*05a0b428SJohn Marino #include "math_private.h"
47*05a0b428SJohn Marino
48*05a0b428SJohn Marino static const double
49*05a0b428SJohn Marino one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
50*05a0b428SJohn Marino huge = 1.000e+300,
51*05a0b428SJohn Marino pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
52*05a0b428SJohn Marino pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
53*05a0b428SJohn Marino pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
54*05a0b428SJohn Marino /* coefficient for R(x^2) */
55*05a0b428SJohn Marino pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
56*05a0b428SJohn Marino pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
57*05a0b428SJohn Marino pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
58*05a0b428SJohn Marino pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
59*05a0b428SJohn Marino pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
60*05a0b428SJohn Marino pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
61*05a0b428SJohn Marino qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
62*05a0b428SJohn Marino qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
63*05a0b428SJohn Marino qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
64*05a0b428SJohn Marino qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
65*05a0b428SJohn Marino
66*05a0b428SJohn Marino double
asin(double x)67*05a0b428SJohn Marino asin(double x)
68*05a0b428SJohn Marino {
69*05a0b428SJohn Marino double t,w,p,q,c,r,s;
70*05a0b428SJohn Marino int32_t hx,ix;
71*05a0b428SJohn Marino GET_HIGH_WORD(hx,x);
72*05a0b428SJohn Marino ix = hx&0x7fffffff;
73*05a0b428SJohn Marino if(ix>= 0x3ff00000) { /* |x|>= 1 */
74*05a0b428SJohn Marino u_int32_t lx;
75*05a0b428SJohn Marino GET_LOW_WORD(lx,x);
76*05a0b428SJohn Marino if(((ix-0x3ff00000)|lx)==0)
77*05a0b428SJohn Marino /* asin(1)=+-pi/2 with inexact */
78*05a0b428SJohn Marino return x*pio2_hi+x*pio2_lo;
79*05a0b428SJohn Marino return (x-x)/(x-x); /* asin(|x|>1) is NaN */
80*05a0b428SJohn Marino } else if (ix<0x3fe00000) { /* |x|<0.5 */
81*05a0b428SJohn Marino if(ix<0x3e400000) { /* if |x| < 2**-27 */
82*05a0b428SJohn Marino if(huge+x>one) return x;/* return x with inexact if x!=0*/
83*05a0b428SJohn Marino }
84*05a0b428SJohn Marino t = x*x;
85*05a0b428SJohn Marino p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
86*05a0b428SJohn Marino q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
87*05a0b428SJohn Marino w = p/q;
88*05a0b428SJohn Marino return x+x*w;
89*05a0b428SJohn Marino }
90*05a0b428SJohn Marino /* 1> |x|>= 0.5 */
91*05a0b428SJohn Marino w = one-fabs(x);
92*05a0b428SJohn Marino t = w*0.5;
93*05a0b428SJohn Marino p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
94*05a0b428SJohn Marino q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
95*05a0b428SJohn Marino s = sqrt(t);
96*05a0b428SJohn Marino if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
97*05a0b428SJohn Marino w = p/q;
98*05a0b428SJohn Marino t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
99*05a0b428SJohn Marino } else {
100*05a0b428SJohn Marino w = s;
101*05a0b428SJohn Marino SET_LOW_WORD(w,0);
102*05a0b428SJohn Marino c = (t-w*w)/(s+w);
103*05a0b428SJohn Marino r = p/q;
104*05a0b428SJohn Marino p = 2.0*s*r-(pio2_lo-2.0*c);
105*05a0b428SJohn Marino q = pio4_hi-2.0*w;
106*05a0b428SJohn Marino t = pio4_hi-(p-q);
107*05a0b428SJohn Marino }
108*05a0b428SJohn Marino if(hx>0) return t; else return -t;
109*05a0b428SJohn Marino }
110*05a0b428SJohn Marino
111*05a0b428SJohn Marino #if LDBL_MANT_DIG == DBL_MANT_DIG
112*05a0b428SJohn Marino __strong_alias(asinl, asin);
113*05a0b428SJohn Marino #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */
114