libm/e_acos.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* __ieee754_acos(x)
  13  * Method :
  14  *      acos(x)  = pi/2 - asin(x)
  15  *      acos(-x) = pi/2 + asin(x)
  16  * For |x|<=0.5
  17  *      acos(x) = pi/2 - (x + x*x^2*R(x^2))     (see asin.c)
  18  * For x>0.5
  19  *      acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
  20  *              = 2asin(sqrt((1-x)/2))
  21  *              = 2s + 2s*z*R(z)        ...z=(1-x)/2, s=sqrt(z)
  22  *              = 2f + (2c + 2s*z*R(z))
  23  *     where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
  24  *     for f so that f+c ~ sqrt(z).
  25  * For x<-0.5
  26  *      acos(x) = pi - 2asin(sqrt((1-|x|)/2))
  27  *              = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
  28  *
  29  * Special cases:
  30  *      if x is NaN, return x itself;
  31  *      if |x|>1, return NaN with invalid signal.
  32  *
  33  * Function needed: __ieee754_sqrt
  34  */
  35
  36 #include "math.h"
  37 #include "math_private.h"
  38
  39 static const double
  40 one=  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  41 pi =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  42 pio2_hi =  1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
  43 pio2_lo =  6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
  44 pS0 =  1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
  45 pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
  46 pS2 =  2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
  47 pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
  48 pS4 =  7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
  49 pS5 =  3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
  50 qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
  51 qS2 =  2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
  52 qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
  53 qS4 =  7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
  54
  55 double __ieee754_acos(double x)
  56 {
  57         double z,p,q,r,w,s,c,df;
  58         int32_t hx,ix;
  59         GET_HIGH_WORD(hx,x);
  60         ix = hx&0x7fffffff;
  61         if(ix>=0x3ff00000) {    /* |x| >= 1 */
  62             u_int32_t lx;
  63             GET_LOW_WORD(lx,x);
  64             if(((ix-0x3ff00000)|lx)==0) {       /* |x|==1 */
  65                 if(hx>0) return 0.0;            /* acos(1) = 0  */
  66                 else return pi+2.0*pio2_lo;     /* acos(-1)= pi */
  67             }
  68             return (x-x)/(x-x);         /* acos(|x|>1) is NaN */
  69         }
  70         if(ix<0x3fe00000) {     /* |x| < 0.5 */
  71             if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
  72             z = x*x;
  73             p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
  74             q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
  75             r = p/q;
  76             return pio2_hi - (x - (pio2_lo-x*r));
  77         } else  if (hx<0) {             /* x < -0.5 */
  78             z = (one+x)*0.5;
  79             p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
  80             q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
  81             s = __ieee754_sqrt(z);
  82             r = p/q;
  83             w = r*s-pio2_lo;
  84             return pi - 2.0*(s+w);
  85         } else {                        /* x > 0.5 */
  86             z = (one-x)*0.5;
  87             s = __ieee754_sqrt(z);
  88             df = s;
  89             SET_LOW_WORD(df,0);
  90             c  = (z-df*df)/(s+df);
  91             p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
  92             q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
  93             r = p/q;
  94             w = r*s+c;
  95             return 2.0*(df+w);
  96         }
  97 }
  98
  99 /*
 100  * wrap_acos(x)
 101  */
 102 #ifndef _IEEE_LIBM
 103 double acos(double x)
 104 {
 105         double z = __ieee754_acos(x);
 106         if (_LIB_VERSION == _IEEE_ || isnan(x))
 107                 return z;
 108         if (fabs(x) > 1.0)
 109                 return __kernel_standard(x, x, 1); /* acos(|x|>1) */
 110         return z;
 111 }
 112 #else
 113 strong_alias(__ieee754_acos, acos)
 114 #endif
 115 libm_hidden_def(acos)