/*

 * ====================================================

 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.

 *

 * Developed at SunPro, a Sun Microsystems, Inc. business.

 * Permission to use, copy, modify, and distribute this

 * software is freely granted, provided that this notice

 * is preserved.

 * ====================================================

 */

/* expm1(x)

 * Returns exp(x)-1, the exponential of x minus 1.

 *

 * Method

 *   1. Argument reduction:

 *  Given x, find r and integer k such that

 *

 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658

 *

 *      Here a correction term c will be computed to compensate

 *  the error in r when rounded to a floating-point number.

 *

 *   2. Approximating expm1(r) by a special rational function on

 *  the interval [0,0.34658]:

 *  Since

 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...

 *  we define R1(r*r) by

 *      r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)

 *  That is,

 *      R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)

 *         = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))

 *         = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...

 *      We use a special Reme algorithm on [0,0.347] to generate

 *  a polynomial of degree 5 in r*r to approximate R1. The

 *  maximum error of this polynomial approximation is bounded

 *  by 2**-61. In other words,

 *      R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5

 *  where   Q1  =  -1.6666666666666567384E-2,

 *    Q2  =   3.9682539681370365873E-4,

 *    Q3  =  -9.9206344733435987357E-6,

 *    Q4  =   2.5051361420808517002E-7,

 *    Q5  =  -6.2843505682382617102E-9;

 *    z   =  r*r,

 *  with error bounded by

 *      |                  5           |     -61

 *      | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2

 *      |                              |

 *

 *  expm1(r) = exp(r)-1 is then computed by the following

 *  specific way which minimize the accumulation rounding error:

 *             2     3

 *            r     r    [ 3 - (R1 + R1*r/2)  ]

 *        expm1(r) = r + --- + --- * [--------------------]

 *                  2     2    [ 6 - r*(3 - R1*r/2) ]

 *

 *  To compensate the error in the argument reduction, we use

 *    expm1(r+c) = expm1(r) + c + expm1(r)*c

 *         ~ expm1(r) + c + r*c

 *  Thus c+r*c will be added in as the correction terms for

 *  expm1(r+c). Now rearrange the term to avoid optimization

 *  screw up:

 *            (      2                                    2 )

 *            ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )

 *   expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )

 *                  ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )

 *                      (                                             )

 *

 *       = r - E

 *   3. Scale back to obtain expm1(x):

 *  From step 1, we have

 *     expm1(x) = either 2^k*[expm1(r)+1] - 1

 *        = or     2^k*[expm1(r) + (1-2^-k)]

 *   4. Implementation notes:

 *  (A). To save one multiplication, we scale the coefficient Qi

 *       to Qi*2^i, and replace z by (x^2)/2.

 *  (B). To achieve maximum accuracy, we compute expm1(x) by

 *    (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)

 *    (ii)  if k=0, return r-E

 *    (iii) if k=-1, return 0.5*(r-E)-0.5

 *        (iv)  if k=1 if r < -0.25, return 2*((r+0.5)- E)

 *                 else      return  1.0+2.0*(r-E);

 *    (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)

 *    (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else

 *    (vii) return 2^k(1-((E+2^-k)-r))

 *

 * Special cases:

 *  expm1(INF) is INF, expm1(NaN) is NaN;

 *  expm1(-INF) is -1, and

 *  for finite argument, only expm1(0)=0 is exact.

 *

 * Accuracy:

 *  according to an error analysis, the error is always less than

 *  1 ulp (unit in the last place).

 *

 * Misc. info.

 *  For IEEE double

 *      if x >  7.09782712893383973096e+02 then expm1(x) overflow

 *

 * Constants:

 * The hexadecimal values are the intended ones for the following

 * constants. The decimal values may be used, provided that the

 * compiler will convert from decimal to binary accurately enough

 * to produce the hexadecimal values shown.

 */

#include "math_private.h"

static const double

one   = 1.0,

tiny    = 1.0e-300,

o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */

ln2_hi    = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */

ln2_lo    = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */

invln2    = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */

/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */

Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */

Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */

Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */

Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */

Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */

static volatile double huge = 1.0e+300;

double

s_expm1(double x)

{

  double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;

  int32_t k,xsb;

  u_int32_t hx;

  GET_HIGH_WORD(hx,x);

  xsb = hx&0x80000000;    /* sign bit of x */

  hx &= 0x7fffffff;   /* high word of |x| */

    /* filter out huge and non-finite argument */

  if(hx >= 0x4043687A) {     /* if |x|>=56*ln2 */

      if(hx >= 0x40862E42) {   /* if |x|>=709.78... */

                if(hx>=0x7ff00000) {

        u_int32_t low;

        GET_LOW_WORD(low,x);

        if(((hx&0xfffff)|low)!=0)

             return x+x;   /* NaN */

        else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */

          }

          if(x > o_threshold) return huge*huge; /* overflow */

      }

      if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */

    if(x+tiny<0.0)   /* raise inexact */

    return tiny-one; /* return -1 */

      }

  }

    /* argument reduction */

  if(hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */

      if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */

    if(xsb==0)

        {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}

    else

        {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}

      } else {

    k  = (int32_t)(invln2*x+((xsb==0)?0.5:-0.5));

    t  = k;

    hi = x - t*ln2_hi;  /* t*ln2_hi is exact here */

    lo = t*ln2_lo;

      }

      STRICT_ASSIGN(double, x, hi - lo);

      c  = (hi-x)-lo;

  }

  else if(hx < 0x3c900000) {   /* when |x|<2**-54, return x */

      t = huge+x; /* return x with inexact flags when x!=0 */

      return x - (t-(huge+x));

  }

  else k = 0;

    /* x is now in primary range */

  hfx = 0.5*x;

  hxs = x*hfx;

  r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));

  t  = 3.0-r1*hfx;

  e  = hxs*((r1-t)/(6.0 - x*t));

  if(k==0) return x - (x*e-hxs);    /* c is 0 */

  else {

      INSERT_WORDS(twopk,((u_int32_t)(0x3ff+k))<<20,0); /* 2^k */

      e  = (x*(e-c)-c);

      e -= hxs;

      if(k== -1) return 0.5*(x-e)-0.5;

      if(k==1) {

          if(x < -0.25) return -2.0*(e-(x+0.5));

          else        return  one+2.0*(x-e);

      }

      if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */

          y = one-(e-x);

    if (k == 1024) {

        double const_0x1p1023 = __ieee754_pow(2, 1023);

        y = y*2.0*const_0x1p1023;

    }

    else y = y*twopk;

          return y-one;

      }

      t = one;

      if(k<20) {

          SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */

          y = t-(e-x);

    y = y*twopk;

     } else {

    SET_HIGH_WORD(t,((0x3ff-k)<<20));  /* 2^-k */

          y = x-(e+t);

          y += one;

    y = y*twopk;

      }

  }

  return y;

}