\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

exp_eps.hpp

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exp_eps: Implementation

template <class Type>
Type exp_eps(const Type &x, const Type &epsilon)
{   // abs_x = |x|
    Type abs_x = x;
    if( Type(0) > x )
        abs_x = - x;
    // initialize
    int  k    = 0;          // initial order
    Type term = 1.;         // term = |x|^k / k !
    Type sum  = term;       // initial sum
    while(term > epsilon)
    {   k         = k + 1;          // order for next term
        Type temp = term * abs_x;   // term = |x|^k / (k-1)!
        term      = temp / Type(k); // term = |x|^k / k !
        sum       = sum + term;     // sum  = 1 + ... + |x|^k / k !
    }
    // In the case where x is negative, use exp(x) = 1 / exp(-|x|)
    if( Type(0) > x )
        sum = Type(1) / sum;
    return sum;
}