11dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 21dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* @(#)e_asin.c 1.3 95/01/18 */ 31dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 41dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 51dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 61dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 71dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Developed at SunSoft, a Sun Microsystems, Inc. business. 81dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Permission to use, copy, modify, and distribute this 91dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * software is freely granted, provided that this notice 101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * is preserved. 111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 14a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <sys/cdefs.h> 15a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* __ieee754_asin(x) 181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Method : 191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ... 201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * we approximate asin(x) on [0,0.5] by 211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * asin(x) = x + x*x^2*R(x^2) 221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where 231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * R(x^2) is a rational approximation of (asin(x)-x)/x^3 241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * and its remez error is bounded by 251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75) 261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * For x in [0.5,1] 281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * asin(x) = pi/2-2*asin(sqrt((1-x)/2)) 291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2; 301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * then for x>0.98 311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * asin(x) = pi/2 - 2*(s+s*z*R(z)) 321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) 331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * For x<=0.98, let pio4_hi = pio2_hi/2, then 341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * f = hi part of s; 351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) 361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * and 371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * asin(x) = pi/2 - 2*(s+s*z*R(z)) 381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) 391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) 401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Special cases: 421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * if x is NaN, return x itself; 431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * if |x|>1, return NaN with invalid signal. 441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 47a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <float.h> 481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math.h" 501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math_private.h" 511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double 531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecthuge = 1.000e+300, 551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ 561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ 571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* coefficient for R(x^2) */ 591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectpS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ 601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectpS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ 611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectpS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ 621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectpS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ 631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectpS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ 641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectpS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ 651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectqS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ 661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectqS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ 671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectqS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ 681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source ProjectqS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ 691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectdouble 711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project__ieee754_asin(double x) 721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double t=0.0,w,p,q,c,r,s; 741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project int32_t hx,ix; 751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hx,x); 761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ix = hx&0x7fffffff; 771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>= 0x3ff00000) { /* |x|>= 1 */ 781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project u_int32_t lx; 791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_LOW_WORD(lx,x); 801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(((ix-0x3ff00000)|lx)==0) 811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* asin(1)=+-pi/2 with inexact */ 821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return x*pio2_hi+x*pio2_lo; 831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (x-x)/(x-x); /* asin(|x|>1) is NaN */ 841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else if (ix<0x3fe00000) { /* |x|<0.5 */ 85a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes if(ix<0x3e500000) { /* if |x| < 2**-26 */ 861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(huge+x>one) return x;/* return x with inexact if x!=0*/ 87a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes } 88a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes t = x*x; 89a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 90a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 91a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes w = p/q; 92a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes return x+x*w; 931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* 1> |x|>= 0.5 */ 951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = one-fabs(x); 961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t = w*0.5; 971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5))))); 981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4))); 991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = sqrt(t); 1001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x3FEF3333) { /* if |x| > 0.975 */ 1011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = p/q; 1021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t = pio2_hi-(2.0*(s+s*w)-pio2_lo); 1031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 1041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project w = s; 1051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_LOW_WORD(w,0); 1061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project c = (t-w*w)/(s+w); 1071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = p/q; 1081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project p = 2.0*s*r-(pio2_lo-2.0*c); 1091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project q = pio4_hi-2.0*w; 1101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project t = pio4_hi-(p-q); 1111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 1121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>0) return t; else return -t; 1131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 114a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes 115a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#if LDBL_MANT_DIG == 53 116a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__weak_reference(asin, asinl); 117a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#endif 118