1141296Sdas 2141296Sdas/* @(#)e_asin.c 1.3 95/01/18 */ 32116Sjkh/* 42116Sjkh * ==================================================== 52116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 62116Sjkh * 7141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 82116Sjkh * Permission to use, copy, modify, and distribute this 9141296Sdas * software is freely granted, provided that this notice 102116Sjkh * is preserved. 112116Sjkh * ==================================================== 122116Sjkh */ 132116Sjkh 14176451Sdas#include <sys/cdefs.h> 15176451Sdas__FBSDID("$FreeBSD$"); 162116Sjkh 172116Sjkh/* __ieee754_asin(x) 18141296Sdas * Method : 192116Sjkh * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 202116Sjkh * we approximate asin(x) on [0,0.5] by 212116Sjkh * asin(x) = x + x*x^2*R(x^2) 222116Sjkh * where 23141296Sdas * R(x^2) is a rational approximation of (asin(x)-x)/x^3 242116Sjkh * and its remez error is bounded by 252116Sjkh * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 262116Sjkh * 272116Sjkh * For x in [0.5,1] 282116Sjkh * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 292116Sjkh * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 302116Sjkh * then for x>0.98 312116Sjkh * asin(x) = pi/2 - 2*(s+s*z*R(z)) 322116Sjkh * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 332116Sjkh * For x<=0.98, let pio4_hi = pio2_hi/2, then 342116Sjkh * f = hi part of s; 352116Sjkh * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 362116Sjkh * and 372116Sjkh * asin(x) = pi/2 - 2*(s+s*z*R(z)) 382116Sjkh * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 392116Sjkh * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 402116Sjkh * 412116Sjkh * Special cases: 422116Sjkh * if x is NaN, return x itself; 432116Sjkh * if |x|>1, return NaN with invalid signal. 442116Sjkh * 452116Sjkh */ 462116Sjkh 47181074Sdas#include <float.h> 482116Sjkh 492116Sjkh#include "math.h" 502116Sjkh#include "math_private.h" 512116Sjkh 528870Srgrimesstatic const double 532116Sjkhone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 542116Sjkhhuge = 1.000e+300, 552116Sjkhpio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 562116Sjkhpio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 572116Sjkhpio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 582116Sjkh /* coefficient for R(x^2) */ 592116SjkhpS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 602116SjkhpS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 612116SjkhpS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 622116SjkhpS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 632116SjkhpS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 642116SjkhpS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 652116SjkhqS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 662116SjkhqS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 672116SjkhqS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 682116SjkhqS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 692116Sjkh 7097407Salfreddouble 71117912Speter__ieee754_asin(double x) 722116Sjkh{ 7317141Sjkh double t=0.0,w,p,q,c,r,s; 742116Sjkh int32_t hx,ix; 752116Sjkh GET_HIGH_WORD(hx,x); 762116Sjkh ix = hx&0x7fffffff; 772116Sjkh if(ix>= 0x3ff00000) { /* |x|>= 1 */ 782116Sjkh u_int32_t lx; 792116Sjkh GET_LOW_WORD(lx,x); 802116Sjkh if(((ix-0x3ff00000)|lx)==0) 812116Sjkh /* asin(1)=+-pi/2 with inexact */ 82141296Sdas return x*pio2_hi+x*pio2_lo; 83141296Sdas return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 842116Sjkh } else if (ix<0x3fe00000) { /* |x|<0.5 */ 85218509Sdas if(ix<0x3e500000) { /* if |x| < 2**-26 */ 862116Sjkh if(huge+x>one) return x;/* return x with inexact if x!=0*/ 87181258Sdas } 88181258Sdas t = x*x; 89181258Sdas p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 90181258Sdas q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 91181258Sdas w = p/q; 92181258Sdas return x+x*w; 932116Sjkh } 942116Sjkh /* 1> |x|>= 0.5 */ 952116Sjkh w = one-fabs(x); 962116Sjkh t = w*0.5; 972116Sjkh p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 982116Sjkh q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 99141296Sdas s = sqrt(t); 1002116Sjkh if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 1012116Sjkh w = p/q; 1022116Sjkh t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 1032116Sjkh } else { 1042116Sjkh w = s; 1052116Sjkh SET_LOW_WORD(w,0); 1062116Sjkh c = (t-w*w)/(s+w); 1072116Sjkh r = p/q; 1082116Sjkh p = 2.0*s*r-(pio2_lo-2.0*c); 1092116Sjkh q = pio4_hi-2.0*w; 1102116Sjkh t = pio4_hi-(p-q); 111141296Sdas } 112141296Sdas if(hx>0) return t; else return -t; 1132116Sjkh} 114181074Sdas 115181074Sdas#if LDBL_MANT_DIG == 53 116181074Sdas__weak_reference(asin, asinl); 117181074Sdas#endif 118