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s_expm1.c
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00001 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
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00015  *
00016  * The Original Code is Mozilla Communicator client code, released
00017  * March 31, 1998.
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00019  * The Initial Developer of the Original Code is
00020  * Sun Microsystems, Inc.
00021  * Portions created by the Initial Developer are Copyright (C) 1998
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00026  * Alternatively, the contents of this file may be used under the terms of
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00039 
00040 /* @(#)s_expm1.c 1.3 95/01/18 */
00041 /*
00042  * ====================================================
00043  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00044  *
00045  * Developed at SunSoft, a Sun Microsystems, Inc. business.
00046  * Permission to use, copy, modify, and distribute this
00047  * software is freely granted, provided that this notice 
00048  * is preserved.
00049  * ====================================================
00050  */
00051 
00052 /* expm1(x)
00053  * Returns exp(x)-1, the exponential of x minus 1.
00054  *
00055  * Method
00056  *   1. Argument reduction:
00057  *     Given x, find r and integer k such that
00058  *
00059  *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658  
00060  *
00061  *      Here a correction term c will be computed to compensate 
00062  *     the error in r when rounded to a floating-point number.
00063  *
00064  *   2. Approximating expm1(r) by a special rational function on
00065  *     the interval [0,0.34658]:
00066  *     Since
00067  *         r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
00068  *     we define R1(r*r) by
00069  *         r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
00070  *     That is,
00071  *         R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
00072  *                 = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
00073  *                 = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
00074  *      We use a special Reme algorithm on [0,0.347] to generate 
00075  *     a polynomial of degree 5 in r*r to approximate R1. The 
00076  *     maximum error of this polynomial approximation is bounded 
00077  *     by 2**-61. In other words,
00078  *         R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
00079  *     where  Q1  =  -1.6666666666666567384E-2,
00080  *            Q2  =   3.9682539681370365873E-4,
00081  *            Q3  =  -9.9206344733435987357E-6,
00082  *            Q4  =   2.5051361420808517002E-7,
00083  *            Q5  =  -6.2843505682382617102E-9;
00084  *     (where z=r*r, and the values of Q1 to Q5 are listed below)
00085  *     with error bounded by
00086  *         |                  5           |     -61
00087  *         | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2 
00088  *         |                              |
00089  *     
00090  *     expm1(r) = exp(r)-1 is then computed by the following 
00091  *     specific way which minimize the accumulation rounding error: 
00092  *                          2     3
00093  *                         r     r    [ 3 - (R1 + R1*r/2)  ]
00094  *           expm1(r) = r + --- + --- * [--------------------]
00095  *                          2     2    [ 6 - r*(3 - R1*r/2) ]
00096  *     
00097  *     To compensate the error in the argument reduction, we use
00098  *            expm1(r+c) = expm1(r) + c + expm1(r)*c 
00099  *                      ~ expm1(r) + c + r*c 
00100  *     Thus c+r*c will be added in as the correction terms for
00101  *     expm1(r+c). Now rearrange the term to avoid optimization 
00102  *     screw up:
00103  *                    (      2                                    2 )
00104  *                    ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
00105  *      expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
00106  *                     ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
00107  *                      (                                             )
00108  *     
00109  *               = r - E
00110  *   3. Scale back to obtain expm1(x):
00111  *     From step 1, we have
00112  *        expm1(x) = either 2^k*[expm1(r)+1] - 1
00113  *                = or     2^k*[expm1(r) + (1-2^-k)]
00114  *   4. Implementation notes:
00115  *     (A). To save one multiplication, we scale the coefficient Qi
00116  *          to Qi*2^i, and replace z by (x^2)/2.
00117  *     (B). To achieve maximum accuracy, we compute expm1(x) by
00118  *       (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
00119  *       (ii)  if k=0, return r-E
00120  *       (iii) if k=-1, return 0.5*(r-E)-0.5
00121  *        (iv)       if k=1 if r < -0.25, return 2*((r+0.5)- E)
00122  *                          else        return  1.0+2.0*(r-E);
00123  *       (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
00124  *       (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
00125  *       (vii) return 2^k(1-((E+2^-k)-r)) 
00126  *
00127  * Special cases:
00128  *     expm1(INF) is INF, expm1(NaN) is NaN;
00129  *     expm1(-INF) is -1, and
00130  *     for finite argument, only expm1(0)=0 is exact.
00131  *
00132  * Accuracy:
00133  *     according to an error analysis, the error is always less than
00134  *     1 ulp (unit in the last place).
00135  *
00136  * Misc. info.
00137  *     For IEEE double 
00138  *         if x >  7.09782712893383973096e+02 then expm1(x) overflow
00139  *
00140  * Constants:
00141  * The hexadecimal values are the intended ones for the following 
00142  * constants. The decimal values may be used, provided that the 
00143  * compiler will convert from decimal to binary accurately enough
00144  * to produce the hexadecimal values shown.
00145  */
00146 
00147 #include "fdlibm.h"
00148 
00149 #ifdef __STDC__
00150 static const double
00151 #else
00152 static double
00153 #endif
00154 one           = 1.0,
00155 really_big           = 1.0e+300,
00156 tiny          = 1.0e-300,
00157 o_threshold   = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
00158 ln2_hi        = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
00159 ln2_lo        = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
00160 invln2        = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
00161        /* scaled coefficients related to expm1 */
00162 Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
00163 Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
00164 Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
00165 Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
00166 Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
00167 
00168 #ifdef __STDC__
00169        double fd_expm1(double x)
00170 #else
00171        double fd_expm1(x)
00172        double x;
00173 #endif
00174 {
00175         fd_twoints u;
00176        double y,hi,lo,c,t,e,hxs,hfx,r1;
00177        int k,xsb;
00178        unsigned hx;
00179 
00180         u.d = x;
00181        hx  = __HI(u);       /* high word of x */
00182        xsb = hx&0x80000000;        /* sign bit of x */
00183        if(xsb==0) y=x; else y= -x; /* y = |x| */
00184        hx &= 0x7fffffff;           /* high word of |x| */
00185 
00186     /* filter out huge and non-finite argument */
00187        if(hx >= 0x4043687A) {                    /* if |x|>=56*ln2 */
00188            if(hx >= 0x40862E42) {         /* if |x|>=709.78... */
00189                 if(hx>=0x7ff00000) {
00190                     u.d = x;
00191                   if(((hx&0xfffff)|__LO(u))!=0) 
00192                        return x+x;         /* NaN */
00193                   else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
00194                }
00195                if(x > o_threshold) return really_big*really_big; /* overflow */
00196            }
00197            if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
00198               if(x+tiny<0.0)              /* raise inexact */
00199               return tiny-one;     /* return -1 */
00200            }
00201        }
00202 
00203     /* argument reduction */
00204        if(hx > 0x3fd62e42) {              /* if  |x| > 0.5 ln2 */ 
00205            if(hx < 0x3FF0A2B2) {   /* and |x| < 1.5 ln2 */
00206               if(xsb==0)
00207                   {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
00208               else
00209                   {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
00210            } else {
00211               k  = (int)(invln2*x+((xsb==0)?0.5:-0.5));
00212               t  = k;
00213               hi = x - t*ln2_hi;   /* t*ln2_hi is exact here */
00214               lo = t*ln2_lo;
00215            }
00216            x  = hi - lo;
00217            c  = (hi-x)-lo;
00218        } 
00219        else if(hx < 0x3c900000) {         /* when |x|<2**-54, return x */
00220            t = really_big+x;       /* return x with inexact flags when x!=0 */
00221            return x - (t-(really_big+x)); 
00222        }
00223        else k = 0;
00224 
00225     /* x is now in primary range */
00226        hfx = 0.5*x;
00227        hxs = x*hfx;
00228        r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
00229        t  = 3.0-r1*hfx;
00230        e  = hxs*((r1-t)/(6.0 - x*t));
00231        if(k==0) return x - (x*e-hxs);            /* c is 0 */
00232        else {
00233            e  = (x*(e-c)-c);
00234            e -= hxs;
00235            if(k== -1) return 0.5*(x-e)-0.5;
00236            if(k==1) 
00237                      if(x < -0.25) return -2.0*(e-(x+0.5));
00238                      else         return  one+2.0*(x-e);
00239            if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
00240                y = one-(e-x);
00241                 u.d = y;
00242                __HI(u) += (k<<20); /* add k to y's exponent */
00243                 y = u.d;
00244                return y-one;
00245            }
00246            t = one;
00247            if(k<20) {
00248                 u.d = t;
00249                      __HI(u) = 0x3ff00000 - (0x200000>>k);  /* t=1-2^-k */
00250                 t = u.d;
00251                      y = t-(e-x);
00252                 u.d = y;
00253                      __HI(u) += (k<<20);  /* add k to y's exponent */
00254                 y = u.d;
00255           } else {
00256                u.d = t;
00257                      __HI(u)  = ((0x3ff-k)<<20); /* 2^-k */
00258                 t = u.d;
00259                      y = x-(e+t);
00260                      y += one;
00261                 u.d = y;
00262                      __HI(u) += (k<<20);  /* add k to y's exponent */
00263                 y = u.d;
00264            }
00265        }
00266        return y;
00267 }