1 /* @(#)e_asin.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, 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 Remes 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 <float.h> 44 #include <math.h> 45 46 #include "math_private.h" 47 48 static const double 49 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 50 huge = 1.000e+300, 51 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 52 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 53 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 54 /* coefficient for R(x^2) */ 55 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 56 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 57 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 58 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 59 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 60 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 61 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 62 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 63 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 64 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 65 66 double 67 asin(double x) 68 { 69 double t,w,p,q,c,r,s; 70 int32_t hx,ix; 71 GET_HIGH_WORD(hx,x); 72 ix = hx&0x7fffffff; 73 if(ix>= 0x3ff00000) { /* |x|>= 1 */ 74 u_int32_t lx; 75 GET_LOW_WORD(lx,x); 76 if(((ix-0x3ff00000)|lx)==0) 77 /* asin(1)=+-pi/2 with inexact */ 78 return x*pio2_hi+x*pio2_lo; 79 return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 80 } else if (ix<0x3fe00000) { /* |x|<0.5 */ 81 if(ix<0x3e400000) { /* if |x| < 2**-27 */ 82 if(huge+x>one) return x;/* return x with inexact if x!=0*/ 83 } 84 t = x*x; 85 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 86 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 87 w = p/q; 88 return x+x*w; 89 } 90 /* 1> |x|>= 0.5 */ 91 w = one-fabs(x); 92 t = w*0.5; 93 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 94 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 95 s = sqrt(t); 96 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 97 w = p/q; 98 t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 99 } else { 100 w = s; 101 SET_LOW_WORD(w,0); 102 c = (t-w*w)/(s+w); 103 r = p/q; 104 p = 2.0*s*r-(pio2_lo-2.0*c); 105 q = pio4_hi-2.0*w; 106 t = pio4_hi-(p-q); 107 } 108 if(hx>0) return t; else return -t; 109 } 110 111 #if LDBL_MANT_DIG == DBL_MANT_DIG 112 __strong_alias(asinl, asin); 113 #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */ 114