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 ```/* 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; ``` ```} ```