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[/] [or1k/] [trunk/] [linux/] [uClibc/] [libm/] [s_expm1.c] - Blame information for rev 1765

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1 1325 phoenix
/* @(#)s_expm1.c 5.1 93/09/24 */
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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#if defined(LIBM_SCCS) && !defined(lint)
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static char rcsid[] = "$NetBSD: s_expm1.c,v 1.8 1995/05/10 20:47:09 jtc Exp $";
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#endif
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/* expm1(x)
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 * Returns exp(x)-1, the exponential of x minus 1.
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 *
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 * Method
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 *   1. Argument reduction:
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 *      Given x, find r and integer k such that
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 *
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 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
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 *
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 *      Here a correction term c will be computed to compensate
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 *      the error in r when rounded to a floating-point number.
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 *
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 *   2. Approximating expm1(r) by a special rational function on
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 *      the interval [0,0.34658]:
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 *      Since
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 *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
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 *      we define R1(r*r) by
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 *          r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
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 *      That is,
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 *          R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
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 *                   = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
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 *                   = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
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 *      We use a special Reme algorithm on [0,0.347] to generate
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 *      a polynomial of degree 5 in r*r to approximate R1. The
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 *      maximum error of this polynomial approximation is bounded
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 *      by 2**-61. In other words,
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 *          R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
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 *      where   Q1  =  -1.6666666666666567384E-2,
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 *              Q2  =   3.9682539681370365873E-4,
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 *              Q3  =  -9.9206344733435987357E-6,
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 *              Q4  =   2.5051361420808517002E-7,
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 *              Q5  =  -6.2843505682382617102E-9;
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 *      (where z=r*r, and the values of Q1 to Q5 are listed below)
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 *      with error bounded by
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 *          |                  5           |     -61
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 *          | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
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 *          |                              |
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 *
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 *      expm1(r) = exp(r)-1 is then computed by the following
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 *      specific way which minimize the accumulation rounding error:
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 *                             2     3
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 *                            r     r    [ 3 - (R1 + R1*r/2)  ]
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 *            expm1(r) = r + --- + --- * [--------------------]
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 *                            2     2    [ 6 - r*(3 - R1*r/2) ]
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 *
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 *      To compensate the error in the argument reduction, we use
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 *              expm1(r+c) = expm1(r) + c + expm1(r)*c
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 *                         ~ expm1(r) + c + r*c
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 *      Thus c+r*c will be added in as the correction terms for
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 *      expm1(r+c). Now rearrange the term to avoid optimization
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 *      screw up:
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 *                      (      2                                    2 )
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 *                      ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
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 *       expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
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 *                      ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
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 *                      (                                             )
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 *
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 *                 = r - E
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 *   3. Scale back to obtain expm1(x):
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 *      From step 1, we have
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 *         expm1(x) = either 2^k*[expm1(r)+1] - 1
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 *                  = or     2^k*[expm1(r) + (1-2^-k)]
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 *   4. Implementation notes:
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 *      (A). To save one multiplication, we scale the coefficient Qi
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 *           to Qi*2^i, and replace z by (x^2)/2.
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 *      (B). To achieve maximum accuracy, we compute expm1(x) by
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 *        (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
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 *        (ii)  if k=0, return r-E
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 *        (iii) if k=-1, return 0.5*(r-E)-0.5
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 *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)
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 *                     else          return  1.0+2.0*(r-E);
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 *        (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
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 *        (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
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 *        (vii) return 2^k(1-((E+2^-k)-r))
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 *
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 * Special cases:
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 *      expm1(INF) is INF, expm1(NaN) is NaN;
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 *      expm1(-INF) is -1, and
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 *      for finite argument, only expm1(0)=0 is exact.
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 *
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 * Accuracy:
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 *      according to an error analysis, the error is always less than
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 *      1 ulp (unit in the last place).
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 *
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 * Misc. info.
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 *      For IEEE double
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 *          if x >  7.09782712893383973096e+02 then expm1(x) overflow
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 *
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 * Constants:
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 * The hexadecimal values are the intended ones for the following
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 * constants. The decimal values may be used, provided that the
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 * compiler will convert from decimal to binary accurately enough
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 * to produce the hexadecimal values shown.
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 */
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#include "math.h"
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#include "math_private.h"
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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one             = 1.0,
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huge            = 1.0e+300,
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tiny            = 1.0e-300,
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o_threshold     = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
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ln2_hi          = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
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ln2_lo          = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
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invln2          = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
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        /* scaled coefficients related to expm1 */
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Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
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Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
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Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
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Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
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Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
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#ifdef __STDC__
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        double expm1(double x)
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#else
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        double expm1(x)
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        double x;
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#endif
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{
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        double y,hi,lo,c=0.0,t,e,hxs,hfx,r1;
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        int32_t k,xsb;
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        u_int32_t hx;
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        GET_HIGH_WORD(hx,x);
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        xsb = hx&0x80000000;            /* sign bit of x */
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        if(xsb==0) y=x; else y= -x;      /* y = |x| */
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        hx &= 0x7fffffff;               /* high word of |x| */
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    /* filter out huge and non-finite argument */
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        if(hx >= 0x4043687A) {                  /* if |x|>=56*ln2 */
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            if(hx >= 0x40862E42) {              /* if |x|>=709.78... */
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                if(hx>=0x7ff00000) {
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                    u_int32_t low;
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                    GET_LOW_WORD(low,x);
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                    if(((hx&0xfffff)|low)!=0)
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                         return x+x;     /* NaN */
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                    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
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                }
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                if(x > o_threshold) return huge*huge; /* overflow */
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            }
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            if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
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                if(x+tiny<0.0)          /* raise inexact */
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                return tiny-one;        /* return -1 */
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            }
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        }
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    /* argument reduction */
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        if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
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            if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
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                if(xsb==0)
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                    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
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                else
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                    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
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            } else {
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                k  = invln2*x+((xsb==0)?0.5:-0.5);
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                t  = k;
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                hi = x - t*ln2_hi;      /* t*ln2_hi is exact here */
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                lo = t*ln2_lo;
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            }
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            x  = hi - lo;
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            c  = (hi-x)-lo;
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        }
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        else if(hx < 0x3c900000) {      /* when |x|<2**-54, return x */
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            t = huge+x; /* return x with inexact flags when x!=0 */
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            return x - (t-(huge+x));
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        }
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        else k = 0;
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    /* x is now in primary range */
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        hfx = 0.5*x;
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        hxs = x*hfx;
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        r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
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        t  = 3.0-r1*hfx;
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        e  = hxs*((r1-t)/(6.0 - x*t));
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        if(k==0) return x - (x*e-hxs);           /* c is 0 */
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        else {
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            e  = (x*(e-c)-c);
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            e -= hxs;
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            if(k== -1) return 0.5*(x-e)-0.5;
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            if(k==1) {
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                if(x < -0.25) return -2.0*(e-(x+0.5));
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                else          return  one+2.0*(x-e);
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            }
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            if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
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                u_int32_t high;
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                y = one-(e-x);
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                GET_HIGH_WORD(high,y);
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                SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
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                return y-one;
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            }
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            t = one;
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            if(k<20) {
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                u_int32_t high;
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                SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
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                y = t-(e-x);
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                GET_HIGH_WORD(high,y);
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                SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
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           } else {
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                u_int32_t high;
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                SET_HIGH_WORD(t,((0x3ff-k)<<20));       /* 2^-k */
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                y = x-(e+t);
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                y += one;
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                GET_HIGH_WORD(high,y);
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                SET_HIGH_WORD(y,high+(k<<20));  /* add k to y's exponent */
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            }
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        }
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        return y;
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}

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