1 /* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */ 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 /* asin(x) 13 * Method : 14 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 15 * we approximate asin(x) on [0,0.5] by 16 * asin(x) = x + x*x^2*R(x^2) 17 * where 18 * R(x^2) is a rational approximation of (asin(x)-x)/x^3 19 * and its remez error is bounded by 20 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 21 * 22 * For x in [0.5,1] 23 * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 24 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 25 * then for x>0.98 26 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 27 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 28 * For x<=0.98, let pio4_hi = pio2_hi/2, then 29 * f = hi part of s; 30 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 31 * and 32 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 33 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 34 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 35 * 36 * Special cases: 37 * if x is NaN, return x itself; 38 * if |x|>1, return NaN with invalid signal. 39 * 40 */ 41 42 #include "libm.h" 43 44 static const double 45 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 46 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 47 /* coefficients for R(x^2) */ 48 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 49 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 50 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 51 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 52 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 53 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 54 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 55 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 56 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 57 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 58 59 static double R(double z) 60 { 61 double_t p, q; 62 p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5))))); 63 q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4))); 64 return p/q; 65 } 66 67 double asin(double x) 68 { 69 double z,r,s; 70 uint32_t hx,ix; 71 72 GET_HIGH_WORD(hx, x); 73 ix = hx & 0x7fffffff; 74 /* |x| >= 1 or nan */ 75 if (ix >= 0x3ff00000) { 76 uint32_t lx; 77 GET_LOW_WORD(lx, x); 78 if ((ix-0x3ff00000 | lx) == 0) 79 /* asin(1) = +-pi/2 with inexact */ 80 return x*pio2_hi + 0x1p-120f; 81 return 0/(x-x); 82 } 83 /* |x| < 0.5 */ 84 if (ix < 0x3fe00000) { 85 /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */ 86 if (ix < 0x3e500000 && ix >= 0x00100000) 87 return x; 88 return x + x*R(x*x); 89 } 90 /* 1 > |x| >= 0.5 */ 91 z = (1 - fabs(x))*0.5; 92 s = sqrt(z); 93 r = R(z); 94 if (ix >= 0x3fef3333) { /* if |x| > 0.975 */ 95 x = pio2_hi-(2*(s+s*r)-pio2_lo); 96 } else { 97 double f,c; 98 /* f+c = sqrt(z) */ 99 f = s; 100 SET_LOW_WORD(f,0); 101 c = (z-f*f)/(s+f); 102 x = 0.5*pio2_hi - (2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f)); 103 } 104 if (hx >> 31) 105 return -x; 106 return x; 107 } 108