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 #include <LibConfig.h> 13 #include <sys/EfiCdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: e_asin.c,v 1.12 2002/05/26 22:01:48 wiz Exp $"); 16 #endif 17 18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ 19 // C4723: potential divide by zero. 20 #pragma warning ( disable : 4723 ) 21 #endif 22 23 /* __ieee754_asin(x) 24 * Method : 25 * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 26 * we approximate asin(x) on [0,0.5] by 27 * asin(x) = x + x*x^2*R(x^2) 28 * where 29 * R(x^2) is a rational approximation of (asin(x)-x)/x^3 30 * and its remez error is bounded by 31 * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 32 * 33 * For x in [0.5,1] 34 * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 35 * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 36 * then for x>0.98 37 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 38 * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 39 * For x<=0.98, let pio4_hi = pio2_hi/2, then 40 * f = hi part of s; 41 * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 42 * and 43 * asin(x) = pi/2 - 2*(s+s*z*R(z)) 44 * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 45 * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 46 * 47 * Special cases: 48 * if x is NaN, return x itself; 49 * if |x|>1, return NaN with invalid signal. 50 * 51 */ 52 53 54 #include "math.h" 55 #include "math_private.h" 56 57 static const double 58 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 59 huge = 1.000e+300, 60 pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 61 pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 62 pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 63 /* coefficient for R(x^2) */ 64 pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 65 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 66 pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 67 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 68 pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 69 pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 70 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 71 qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 72 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 73 qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 74 75 double 76 __ieee754_asin(double x) 77 { 78 double t,w,p,q,c,r,s; 79 int32_t hx,ix; 80 81 t = 0; 82 GET_HIGH_WORD(hx,x); 83 ix = hx&0x7fffffff; 84 if(ix>= 0x3ff00000) { /* |x|>= 1 */ 85 u_int32_t lx; 86 GET_LOW_WORD(lx,x); 87 if(((ix-0x3ff00000)|lx)==0) 88 /* asin(1)=+-pi/2 with inexact */ 89 return x*pio2_hi+x*pio2_lo; 90 return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 91 } else if (ix<0x3fe00000) { /* |x|<0.5 */ 92 if(ix<0x3e400000) { /* if |x| < 2**-27 */ 93 if(huge+x>one) return x;/* return x with inexact if x!=0*/ 94 } else 95 t = x*x; 96 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 97 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 98 w = p/q; 99 return x+x*w; 100 } 101 /* 1> |x|>= 0.5 */ 102 w = one-fabs(x); 103 t = w*0.5; 104 p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 105 q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 106 s = __ieee754_sqrt(t); 107 if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 108 w = p/q; 109 t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 110 } else { 111 w = s; 112 SET_LOW_WORD(w,0); 113 c = (t-w*w)/(s+w); 114 r = p/q; 115 p = 2.0*s*r-(pio2_lo-2.0*c); 116 q = pio4_hi-2.0*w; 117 t = pio4_hi-(p-q); 118 } 119 if(hx>0) return t; else return -t; 120 } 121
