/* @(#)s_expm1.c 1.5 04/04/22 */
/*
 * ====================================================
 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
 *
 * 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 Remes 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.
 */

package kotlin.math.fdlibm


private const val one = 1.0
private const val huge = 1.0e+300
private const val tiny = 1.0e-300
private const val o_threshold = 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
private const val ln2_hi = 6.93147180369123816490e-01 /* 0x3fe62e42, 0xfee00000 */
private const val ln2_lo = 1.90821492927058770002e-10 /* 0x3dea39ef, 0x35793c76 */
private const val invln2 = 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */

/* scaled coefficients related to expm1 */
private const val Q1 = -3.33333333333331316428e-02 /* BFA11111 111110F4 */
private const val Q2 = 1.58730158725481460165e-03 /* 3F5A01A0 19FE5585 */
private const val Q3 = -7.93650757867487942473e-05 /* BF14CE19 9EAADBB7 */
private const val Q4 = 4.00821782732936239552e-06 /* 3ED0CFCA 86E65239 */
private const val Q5 = -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */

internal fun expm1(_x: Double): Double {
    var x: Double = _x
    var y: Double
    var hi: Double
    var lo: Double
    var c: Double = 0.0
    var t: Double
    var e: Double
    var hxs: Double
    var hfx: Double
    var r1: Double
    var k: Int
    var xsb: Int
    var hx: UInt

    hx = __HIu(x)    /* high word of x */
    xsb = (hx and Int.MIN_VALUE.toUInt()).toInt()        /* sign bit of x */
    hx = (hx and 0x7fffffffU)        /* high word of |x| */

    /* filter out huge and non-finite argument */
    if (hx >= 0x4043687AU) {            /* if |x|>=56*ln2 */
        if (hx >= 0x40862E42U) {        /* if |x|>=709.78... */
            if (hx >= 0x7ff00000U) {
                if (((hx and 0xfffffU) or __LOu(x)) != 0U)
                    return x + x     /* NaN */
                else return if (xsb == 0) x else -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 > 0x3fd62e42U) {        /* if  |x| > 0.5 ln2 */
        if (hx < 0x3FF0A2B2U) {    /* 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 + (if (xsb == 0) 0.5 else -0.5)).toInt()
            t = k.toDouble()
            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 < 0x3c900000U) {    /* 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 */
            y = one - (e - x)
            y = doubleSetWord(d = y, hi = __HI(y) + (k shl 20)) /* add k to y's exponent */
            return y - one
        }
        t = one
        if (k < 20) {
            t = doubleSetWord(d = t, hi = 0x3ff00000 - (0x200000 shr k))  /* t=1-2^-k */
            y = t - (e - x)
            y = doubleSetWord(d = y, hi = __HI(y) + (k shl 20)) /* add k to y's exponent */
        } else {
            t = doubleSetWord(d = t, hi = ((0x3ff - k) shl 20)) /* 2^-k */
            y = x - (e + t)
            y += one
            y = doubleSetWord(d = y, hi = __HI(y) + (k shl 20)) /* add k to y's exponent */
        }
    }
    return y
}
