Average Error: 29.6 → 1.0
Time: 8.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.625672270910783:\\ \;\;\;\;\left(\log \left(e^{0.33333333333333337 \cdot {x}^{3}}\right) + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.625672270910783:\\
\;\;\;\;\left(\log \left(e^{0.33333333333333337 \cdot {x}^{3}}\right) + 1\right) - 0.5 \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\

\end{array}
double f(double x, double eps) {
        double r57695 = 1.0;
        double r57696 = eps;
        double r57697 = r57695 / r57696;
        double r57698 = r57695 + r57697;
        double r57699 = r57695 - r57696;
        double r57700 = x;
        double r57701 = r57699 * r57700;
        double r57702 = -r57701;
        double r57703 = exp(r57702);
        double r57704 = r57698 * r57703;
        double r57705 = r57697 - r57695;
        double r57706 = r57695 + r57696;
        double r57707 = r57706 * r57700;
        double r57708 = -r57707;
        double r57709 = exp(r57708);
        double r57710 = r57705 * r57709;
        double r57711 = r57704 - r57710;
        double r57712 = 2.0;
        double r57713 = r57711 / r57712;
        return r57713;
}
↓
double f(double x, double eps) {
        double r57714 = x;
        double r57715 = 1.6256722709107834;
        bool r57716 = r57714 <= r57715;
        double r57717 = 0.33333333333333337;
        double r57718 = 3.0;
        double r57719 = pow(r57714, r57718);
        double r57720 = r57717 * r57719;
        double r57721 = exp(r57720);
        double r57722 = log(r57721);
        double r57723 = 1.0;
        double r57724 = r57722 + r57723;
        double r57725 = 0.5;
        double r57726 = 2.0;
        double r57727 = pow(r57714, r57726);
        double r57728 = r57725 * r57727;
        double r57729 = r57724 - r57728;
        double r57730 = eps;
        double r57731 = r57723 / r57730;
        double r57732 = r57723 + r57731;
        double r57733 = r57723 - r57730;
        double r57734 = r57733 * r57714;
        double r57735 = exp(r57734);
        double r57736 = r57732 / r57735;
        double r57737 = 2.0;
        double r57738 = r57736 / r57737;
        double r57739 = r57723 + r57730;
        double r57740 = r57739 * r57714;
        double r57741 = exp(r57740);
        double r57742 = r57731 / r57741;
        double r57743 = r57742 / r57737;
        double r57744 = r57738 - r57743;
        double r57745 = r57723 / r57741;
        double r57746 = r57745 / r57737;
        double r57747 = r57744 + r57746;
        double r57748 = r57716 ? r57729 : r57747;
        return r57748;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 1.6256722709107834

    1. Initial program 39.6

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Simplified39.6

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}}\]

    3. Taylor expanded around 0 1.2

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied add-log-exp1.2

      \[\leadsto \left(\color{blue}{\log \left(e^{0.33333333333333337 \cdot {x}^{3}}\right)} + 1\right) - 0.5 \cdot {x}^{2}\]

    if 1.6256722709107834 < x

    1. Initial program 0.5

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Simplified0.5

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}}\]
    3. Using strategy rm

    4. Applied div-sub0.5

      \[\leadsto \frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\color{blue}{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}} - \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]

    5. Applied div-sub0.5

      \[\leadsto \frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \color{blue}{\left(\frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} - \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right)}\]

    6. Applied associate--r-0.4

      \[\leadsto \color{blue}{\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.625672270910783:\\ \;\;\;\;\left(\log \left(e^{0.33333333333333337 \cdot {x}^{3}}\right) + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))