RTAI API: base/math/e_asin.c Source File

00001 /* @(#)e_asin.c 5.1 93/09/24 */ 00002 /* 00003 * ==================================================== 00004 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 00005 * 00006 * Developed at SunPro, a Sun Microsystems, Inc. business. 00007 * Permission to use, copy, modify, and distribute this 00008 * software is freely granted, provided that this notice 00009 * is preserved. 00010 * ==================================================== 00011 */ 00012 00013 #if defined(LIBM_SCCS) && !defined(lint) 00014 static char rcsid[] = "$NetBSD: e_asin.c,v 1.9 1995/05/12 04:57:22 jtc Exp $"; 00015 #endif 00016 00017 /* __ieee754_asin(x) 00018 * Method : 00019 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 00020 * we approximate asin(x) on [0,0.5] by 00021 * asin(x) = x + x*x^2*R(x^2) 00022 * where 00023 * R(x^2) is a rational approximation of (asin(x)-x)/x^3 00024 * and its remez error is bounded by 00025 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 00026 * 00027 * For x in [0.5,1] 00028 * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 00029 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 00030 * then for x>0.98 00031 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 00032 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 00033 * For x<=0.98, let pio4_hi = pio2_hi/2, then 00034 * f = hi part of s; 00035 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 00036 * and 00037 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 00038 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 00039 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 00040 * 00041 * Special cases: 00042 * if x is NaN, return x itself; 00043 * if |x|>1, return NaN with invalid signal. 00044 * 00045 */ 00046 00047 00048 #include "math.h" 00049 #include "mathP.h" 00050 00051 #ifdef __STDC__ 00052 static const double 00053 #else 00054 static double 00055 #endif 00056 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 00057 huge = 1.000e+300, 00058 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 00059 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 00060 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 00061 /* coefficient for R(x^2) */ 00062 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 00063 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 00064 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 00065 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 00066 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 00067 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 00068 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 00069 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 00070 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 00071 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 00072 00073 #ifdef __STDC__ 00074 double __ieee754_asin(double x) 00075 #else 00076 double __ieee754_asin(x) 00077 double x; 00078 #endif 00079 { 00080 double t,w,p,q,c,r,s; 00081 int32_t hx,ix; 00082 00083 /* XXX t is not properly initialized. Bug? */ 00084 t = 0; 00085 00086 GET_HIGH_WORD(hx,x); 00087 ix = hx&0x7fffffff; 00088 if(ix>= 0x3ff00000) { /* |x|>= 1 */ 00089 u_int32_t lx; 00090 GET_LOW_WORD(lx,x); 00091 if(((ix-0x3ff00000)|lx)==0) 00092 /* asin(1)=+-pi/2 with inexact */ 00093 return x*pio2_hi+x*pio2_lo; 00094 return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 00095 } else if (ix<0x3fe00000) { /* |x|<0.5 */ 00096 if(ix<0x3e400000) { /* if |x| < 2**-27 */ 00097 if(huge+x>one) return x;/* return x with inexact if x!=0*/ 00098 } else 00099 t = x*x; 00100 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 00101 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 00102 w = p/q; 00103 return x+x*w; 00104 } 00105 /* 1> |x|>= 0.5 */ 00106 w = one-fabs(x); 00107 t = w*0.5; 00108 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 00109 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 00110 s = __ieee754_sqrt(t); 00111 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 00112 w = p/q; 00113 t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 00114 } else { 00115 w = s; 00116 SET_LOW_WORD(w,0); 00117 c = (t-w*w)/(s+w); 00118 r = p/q; 00119 p = 2.0*s*r-(pio2_lo-2.0*c); 00120 q = pio4_hi-2.0*w; 00121 t = pio4_hi-(p-q); 00122 } 00123 if(hx>0) return t; else return -t; 00124 }