/src/moddable/xs/tools/fdlibm/e_asin.c
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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 "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 | | |
63 | | double |
64 | | __ieee754_asin(double x) |
65 | 2.47k | { |
66 | 2.47k | double t=0.0,w,p,q,c,r,s; |
67 | 2.47k | int32_t hx,ix; |
68 | 2.47k | GET_HIGH_WORD(hx,x); |
69 | 2.47k | ix = hx&0x7fffffff; |
70 | 2.47k | if(ix>= 0x3ff00000) { /* |x|>= 1 */ |
71 | 1.26k | u_int32_t lx; |
72 | 1.26k | GET_LOW_WORD(lx,x); |
73 | 1.26k | if(((ix-0x3ff00000)|lx)==0) |
74 | | /* asin(1)=+-pi/2 with inexact */ |
75 | 1.22k | return x*pio2_hi+x*pio2_lo; |
76 | 38 | return (x-x)/(x-x); /* asin(|x|>1) is NaN */ |
77 | 1.26k | } else if (ix<0x3fe00000) { /* |x|<0.5 */ |
78 | 329 | if(ix<0x3e500000) { /* if |x| < 2**-26 */ |
79 | 254 | if(huge+x>one) return x;/* return x with inexact if x!=0*/ |
80 | 254 | } |
81 | 75 | t = x*x; |
82 | 75 | p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); |
83 | 75 | q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); |
84 | 75 | w = p/q; |
85 | 75 | return x+x*w; |
86 | 329 | } |
87 | | /* 1> |x|>= 0.5 */ |
88 | 883 | w = one-fabs(x); |
89 | 883 | t = w*0.5; |
90 | 883 | p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); |
91 | 883 | q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); |
92 | 883 | s = sqrt(t); |
93 | 883 | if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ |
94 | 0 | w = p/q; |
95 | 0 | t = pio2_hi-(2.0*(s+s*w)-pio2_lo); |
96 | 883 | } else { |
97 | 883 | w = s; |
98 | 883 | SET_LOW_WORD(w,0); |
99 | 883 | c = (t-w*w)/(s+w); |
100 | 883 | r = p/q; |
101 | 883 | p = 2.0*s*r-(pio2_lo-2.0*c); |
102 | 883 | q = pio4_hi-2.0*w; |
103 | 883 | t = pio4_hi-(p-q); |
104 | 883 | } |
105 | 883 | if(hx>0) return t; else return -t; |
106 | 883 | } |