Average Error: 30.1 → 1.0
Time: 13.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 95.541640315829881:\\
\;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r27999 = 1.0;
        double r28000 = eps;
        double r28001 = r27999 / r28000;
        double r28002 = r27999 + r28001;
        double r28003 = r27999 - r28000;
        double r28004 = x;
        double r28005 = r28003 * r28004;
        double r28006 = -r28005;
        double r28007 = exp(r28006);
        double r28008 = r28002 * r28007;
        double r28009 = r28001 - r27999;
        double r28010 = r27999 + r28000;
        double r28011 = r28010 * r28004;
        double r28012 = -r28011;
        double r28013 = exp(r28012);
        double r28014 = r28009 * r28013;
        double r28015 = r28008 - r28014;
        double r28016 = 2.0;
        double r28017 = r28015 / r28016;
        return r28017;
}
↓
double f(double x, double eps) {
        double r28018 = x;
        double r28019 = 95.54164031582988;
        bool r28020 = r28018 <= r28019;
        double r28021 = 0.6666666666666667;
        double r28022 = 3.0;
        double r28023 = pow(r28018, r28022);
        double r28024 = r28021 * r28023;
        double r28025 = cbrt(r28024);
        double r28026 = r28025 * r28025;
        double r28027 = r28026 * r28025;
        double r28028 = 2.0;
        double r28029 = 1.0;
        double r28030 = 2.0;
        double r28031 = pow(r28018, r28030);
        double r28032 = r28029 * r28031;
        double r28033 = r28028 - r28032;
        double r28034 = r28027 + r28033;
        double r28035 = r28034 / r28028;
        double r28036 = eps;
        double r28037 = r28029 / r28036;
        double r28038 = r28029 + r28037;
        double r28039 = 1.0;
        double r28040 = r28029 - r28036;
        double r28041 = r28040 * r28018;
        double r28042 = exp(r28041);
        double r28043 = r28039 / r28042;
        double r28044 = r28038 * r28043;
        double r28045 = r28037 - r28029;
        double r28046 = r28029 + r28036;
        double r28047 = r28046 * r28018;
        double r28048 = -r28047;
        double r28049 = exp(r28048);
        double r28050 = r28045 * r28049;
        double r28051 = r28044 - r28050;
        double r28052 = r28051 / r28028;
        double r28053 = r28020 ? r28035 : r28052;
        return r28053;
}

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 < 95.54164031582988

    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. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied fma-udef1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)} - 1 \cdot {x}^{2}}{2}\]

    6. Applied associate--l+1.3

      \[\leadsto \frac{\color{blue}{0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)}}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\]

    if 95.54164031582988 < x

    1. Initial program 0.3

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

    3. Applied exp-neg0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))