text: /*
   *                               POK header
   *
   * The following file is a part of the POK project. Any modification should
   * be made according to the POK licence. You CANNOT use this file or a part
   * of a file for your own project.
   *
   * For more information on the POK licence, please see our LICENCE FILE
   *
   * Please follow the coding guidelines described in doc/CODING_GUIDELINES
   *
   *                                      Copyright (c) 2007-2020 POK team
   */
  
  /* @(#)s_expm1.c 5.1 93/09/24 */
  /*
   * ====================================================
   * 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;
   *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
   *	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.
   */
  
  #ifdef POK_NEEDS_LIBMATH
  
  #include "math_private.h"
  #include <libm.h>
  
  static const double one = 1.0, huge = 1.0e+300, 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 coefficients related to expm1 */
      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 y, hi, lo, c, t, e, hxs, hfx, r1;
    int32_t k, xsb;
    uint32_t hx;
  
    c = 0;
    GET_HIGH_WORD(hx, x);
    xsb = hx & 0x80000000; /* sign bit of x */
    if (xsb == 0)
      y = x;
    else
      y = -x;         /* y = |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) {
          uint32_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 = 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;
      }
      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 {
      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 */
        uint32_t high;
        y = one - (e - x);
        GET_HIGH_WORD(high, y);
        SET_HIGH_WORD(y, high + (k << 20)); /* add k to y's exponent */
        return y - one;
      }
      t = one;
      if (k < 20) {
        uint32_t high;
        SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
        y = t - (e - x);
        GET_HIGH_WORD(high, y);
        SET_HIGH_WORD(y, high + (k << 20)); /* add k to y's exponent */
      } else {
        uint32_t high;
        SET_HIGH_WORD(t, ((0x3ff - k) << 20)); /* 2^-k */
        y = x - (e + t);
        y += one;
        GET_HIGH_WORD(high, y);
        SET_HIGH_WORD(y, high + (k << 20)); /* add k to y's exponent */
      }
    }
    return y;
  }
  
  #endif
  