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path: root/src/math/asin.c
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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 /* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* asin(x) * Method : * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... * we approximate asin(x) on [0,0.5] by * asin(x) = x + x*x^2*R(x^2) * where * R(x^2) is a rational approximation of (asin(x)-x)/x^3 * and its remez error is bounded by * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) * * For x in [0.5,1] * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; * then for x>0.98 * asin(x) = pi/2 - 2*(s+s*z*R(z)) * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) * For x<=0.98, let pio4_hi = pio2_hi/2, then * f = hi part of s; * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) * and * asin(x) = pi/2 - 2*(s+s*z*R(z)) * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) * * Special cases: * if x is NaN, return x itself; * if |x|>1, return NaN with invalid signal. * */ #include "libm.h" static const double pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ /* coefficients for R(x^2) */ pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ static double R(double z) { double_t p, q; p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4))); return p/q; } double asin(double x) { double z,r,s; uint32_t hx,ix; GET_HIGH_WORD(hx, x); ix = hx & 0x7fffffff; /* |x| >= 1 or nan */ if (ix >= 0x3ff00000) { uint32_t lx; GET_LOW_WORD(lx, x); if ((ix-0x3ff00000 | lx) == 0) /* asin(1) = +-pi/2 with inexact */ return x*pio2_hi + 0x1p-120f; return 0/(x-x); } /* |x| < 0.5 */ if (ix < 0x3fe00000) { /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */ if (ix < 0x3e500000 && ix >= 0x00100000) return x; return x + x*R(x*x); } /* 1 > |x| >= 0.5 */ z = (1 - fabs(x))*0.5; s = sqrt(z); r = R(z); if (ix >= 0x3fef3333) { /* if |x| > 0.975 */ x = pio2_hi-(2*(s+s*r)-pio2_lo); } else { double f,c; /* f+c = sqrt(z) */ f = s; SET_LOW_WORD(f,0); c = (z-f*f)/(s+f); x = 0.5*pio2_hi - (2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f)); } if (hx >> 31) return -x; return x; }