1 2 /* @(#)e_asin.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 #include "cdefs-compat.h" 15 //__FBSDID("$FreeBSD: src/lib/msun/src/e_asin.c,v 1.15 2011/02/10 07:37:50 das Exp $"); 16 17 /* __ieee754_asin(x) 18 * Method : 19 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 20 * we approximate asin(x) on [0,0.5] by 21 * asin(x) = x + x*x^2*R(x^2) 22 * where 23 * R(x^2) is a rational approximation of (asin(x)-x)/x^3 24 * and its remez error is bounded by 25 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 26 * 27 * For x in [0.5,1] 28 * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 29 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 30 * then for x>0.98 31 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 32 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 33 * For x<=0.98, let pio4_hi = pio2_hi/2, then 34 * f = hi part of s; 35 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 36 * and 37 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 38 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 39 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 40 * 41 * Special cases: 42 * if x is NaN, return x itself; 43 * if |x|>1, return NaN with invalid signal. 44 * 45 */ 46 47 #include <float.h> 48 #include <openlibm_math.h> 49 50 #include "math_private.h" 51 52 static const double 53 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 54 huge = 1.000e+300, 55 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 56 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 57 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 58 /* coefficient for R(x^2) */ 59 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 60 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 61 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 62 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 63 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 64 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 65 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 66 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 67 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 68 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 69 70 OLM_DLLEXPORT double 71 __ieee754_asin(double x) 72 { 73 double t=0.0,w,p,q,c,r,s; 74 int32_t hx,ix; 75 GET_HIGH_WORD(hx,x); 76 ix = hx&0x7fffffff; 77 if(ix>= 0x3ff00000) { /* |x|>= 1 */ 78 u_int32_t lx; 79 GET_LOW_WORD(lx,x); 80 if(((ix-0x3ff00000)|lx)==0) 81 /* asin(1)=+-pi/2 with inexact */ 82 return x*pio2_hi+x*pio2_lo; 83 return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 84 } else if (ix<0x3fe00000) { /* |x|<0.5 */ 85 if(ix<0x3e500000) { /* if |x| < 2**-26 */ 86 if(huge+x>one) return x;/* return x with inexact if x!=0*/ 87 } 88 t = x*x; 89 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 90 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 91 w = p/q; 92 return x+x*w; 93 } 94 /* 1> |x|>= 0.5 */ 95 w = one-fabs(x); 96 t = w*0.5; 97 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 98 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 99 s = sqrt(t); 100 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 101 w = p/q; 102 t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 103 } else { 104 w = s; 105 SET_LOW_WORD(w,0); 106 c = (t-w*w)/(s+w); 107 r = p/q; 108 p = 2.0*s*r-(pio2_lo-2.0*c); 109 q = pio4_hi-2.0*w; 110 t = pio4_hi-(p-q); 111 } 112 if(hx>0) return t; else return -t; 113 } 114 115 #if LDBL_MANT_DIG == 53 116 __weak_reference(asin, asinl); 117 #endif 118