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| 2 | /* @(#)e_asin.c 5.1 93/09/24 */ |
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| 3 | /* |
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| 4 | * ==================================================== |
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| 5 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 6 | * |
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| 7 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 8 | * Permission to use, copy, modify, and distribute this |
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| 9 | * software is freely granted, provided that this notice |
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| 10 | * is preserved. |
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| 11 | * ==================================================== |
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| 12 | */ |
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| 13 | |
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| 14 | /* __ieee754_asin(x) |
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| 15 | * Method : |
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| 16 | * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... |
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| 17 | * we approximate asin(x) on [0,0.5] by |
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| 18 | * asin(x) = x + x*x^2*R(x^2) |
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| 19 | * where |
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| 20 | * R(x^2) is a rational approximation of (asin(x)-x)/x^3 |
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| 21 | * and its remez error is bounded by |
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| 22 | * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) |
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| 23 | * |
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| 24 | * For x in [0.5,1] |
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| 25 | * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) |
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| 26 | * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; |
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| 27 | * then for x>0.98 |
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| 28 | * asin(x) = pi/2 - 2*(s+s*z*R(z)) |
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| 29 | * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) |
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| 30 | * For x<=0.98, let pio4_hi = pio2_hi/2, then |
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| 31 | * f = hi part of s; |
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| 32 | * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) |
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| 33 | * and |
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| 34 | * asin(x) = pi/2 - 2*(s+s*z*R(z)) |
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| 35 | * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) |
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| 36 | * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) |
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| 37 | * |
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| 38 | * Special cases: |
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| 39 | * if x is NaN, return x itself; |
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| 40 | * if |x|>1, return NaN with invalid signal. |
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| 41 | * |
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| 42 | */ |
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| 43 | |
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| 44 | |
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| 45 | #include "fdlibm.h" |
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| 46 | |
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| 47 | #ifndef _DOUBLE_IS_32BITS |
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| 48 | |
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| 49 | #ifdef __STDC__ |
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| 50 | static const double |
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| 51 | #else |
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| 52 | static double |
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| 53 | #endif |
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| 54 | one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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| 55 | huge = 1.000e+300, |
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| 56 | pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ |
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| 57 | pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ |
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| 58 | pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ |
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| 59 | /* coefficient for R(x^2) */ |
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| 60 | pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ |
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| 61 | pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ |
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| 62 | pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ |
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| 63 | pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ |
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| 64 | pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ |
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| 65 | pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ |
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| 66 | qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ |
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| 67 | qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ |
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| 68 | qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ |
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| 69 | qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ |
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| 70 | |
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| 71 | #ifdef __STDC__ |
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| 72 | double __ieee754_asin(double x) |
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| 73 | #else |
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| 74 | double __ieee754_asin(x) |
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| 75 | double x; |
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| 76 | #endif |
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| 77 | { |
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| 78 | double t,w,p,q,c,r,s; |
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| 79 | __int32_t hx,ix; |
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| 80 | GET_HIGH_WORD(hx,x); |
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| 81 | ix = hx&0x7fffffff; |
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| 82 | if(ix>= 0x3ff00000) { /* |x|>= 1 */ |
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| 83 | __uint32_t lx; |
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| 84 | GET_LOW_WORD(lx,x); |
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| 85 | if(((ix-0x3ff00000)|lx)==0) |
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| 86 | /* asin(1)=+-pi/2 with inexact */ |
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| 87 | return x*pio2_hi+x*pio2_lo; |
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| 88 | return (x-x)/(x-x); /* asin(|x|>1) is NaN */ |
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| 89 | } else if (ix<0x3fe00000) { /* |x|<0.5 */ |
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| 90 | if(ix<0x3e400000) { /* if |x| < 2**-27 */ |
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| 91 | if(huge+x>one) return x;/* return x with inexact if x!=0*/ |
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| 92 | } else { |
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| 93 | t = x*x; |
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| 94 | p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); |
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| 95 | q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); |
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| 96 | w = p/q; |
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| 97 | return x+x*w; |
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| 98 | } |
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| 99 | } |
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| 100 | /* 1> |x|>= 0.5 */ |
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| 101 | w = one-fabs(x); |
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| 102 | t = w*0.5; |
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| 103 | p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); |
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| 104 | q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); |
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| 105 | s = __ieee754_sqrt(t); |
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| 106 | if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ |
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| 107 | w = p/q; |
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| 108 | t = pio2_hi-(2.0*(s+s*w)-pio2_lo); |
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| 109 | } else { |
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| 110 | w = s; |
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| 111 | SET_LOW_WORD(w,0); |
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| 112 | c = (t-w*w)/(s+w); |
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| 113 | r = p/q; |
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| 114 | p = 2.0*s*r-(pio2_lo-2.0*c); |
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| 115 | q = pio4_hi-2.0*w; |
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| 116 | t = pio4_hi-(p-q); |
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| 117 | } |
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| 118 | if(hx>0) return t; else return -t; |
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| 119 | } |
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| 120 | |
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| 121 | #endif /* defined(_DOUBLE_IS_32BITS) */ |
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