/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */
/*
 * ====================================================
 * 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 Remez 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 "libm.h"

static const double 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 */

double expm1(double x) {
  double_t y, hi, lo, c, t, e, hxs, hfx, r1, twopk;
  union {
    double f;
    uint64_t i;
  } u = {x};
  uint32_t hx = u.i >> 32 & 0x7fffffff;
  int k, sign = u.i >> 63;

  /* filter out huge and non-finite argument */
  if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
    if (isnan(x))
      return x;
    if (sign)
      return -1;
    if (x > o_threshold) {
      x *= 0x1p1023;
      return x;
    }
  }

  /* argument reduction */
  if (hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
    if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
      if (!sign) {
        hi = x - ln2_hi;
        lo = ln2_lo;
        k = 1;
      } else {
        hi = x + ln2_hi;
        lo = -ln2_lo;
        k = -1;
      }
    } else {
      k = invln2 * x + (sign ? -0.5 : 0.5);
      t = k;
      hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
      lo = t * ln2_lo;
    }
    x = hi - lo;
    c = (hi - x) - lo;
  } else if (hx < 0x3c900000) { /* |x| < 2**-54, return x */
    if (hx < 0x00100000)
      FORCE_EVAL((float)x);
    return x;
  } else
    k = 0;

  /* x is now in primary range */
  hfx = 0.5 * x;
  hxs = x * hfx;
  r1 = 1.0 + 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) /* c is 0 */
    return x - (x * e - hxs);
  e = x * (e - c) - c;
  e -= hxs;
  /* exp(x) ~ 2^k (x_reduced - e + 1) */
  if (k == -1)
    return 0.5 * (x - e) - 0.5;
  if (k == 1) {
    if (x < -0.25)
      return -2.0 * (e - (x + 0.5));
    return 1.0 + 2.0 * (x - e);
  }
  u.i = (uint64_t)(0x3ff + k) << 52; /* 2^k */
  twopk = u.f;
  if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
    y = x - e + 1.0;
    if (k == 1024)
      y = y * 2.0 * 0x1p1023;
    else
      y = y * twopk;
    return y - 1.0;
  }
  u.i = (uint64_t)(0x3ff - k) << 52; /* 2^-k */
  if (k < 20)
    y = (x - e + (1 - u.f)) * twopk;
  else
    y = (x - (e + u.f) + 1) * twopk;
  return y;
}