Average Error: 29.7 → 1.0
Time: 21.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 378.4237213355006019810389261692762374878:\\ \;\;\;\;\frac{\sqrt[3]{{\left(2 + {x}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, \left(-e^{-\left(1 + \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}{2}\\ \end{array}\]
\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 378.4237213355006019810389261692762374878:\\
\;\;\;\;\frac{\sqrt[3]{{\left(2 + {x}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right)}^{3}}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r62026 = 1.0;
        double r62027 = eps;
        double r62028 = r62026 / r62027;
        double r62029 = r62026 + r62028;
        double r62030 = r62026 - r62027;
        double r62031 = x;
        double r62032 = r62030 * r62031;
        double r62033 = -r62032;
        double r62034 = exp(r62033);
        double r62035 = r62029 * r62034;
        double r62036 = r62028 - r62026;
        double r62037 = r62026 + r62027;
        double r62038 = r62037 * r62031;
        double r62039 = -r62038;
        double r62040 = exp(r62039);
        double r62041 = r62036 * r62040;
        double r62042 = r62035 - r62041;
        double r62043 = 2.0;
        double r62044 = r62042 / r62043;
        return r62044;
}
↓
double f(double x, double eps) {
        double r62045 = x;
        double r62046 = 378.4237213355006;
        bool r62047 = r62045 <= r62046;
        double r62048 = 2.0;
        double r62049 = 2.0;
        double r62050 = pow(r62045, r62049);
        double r62051 = 0.6666666666666667;
        double r62052 = r62051 * r62045;
        double r62053 = 1.0;
        double r62054 = r62052 - r62053;
        double r62055 = r62050 * r62054;
        double r62056 = r62048 + r62055;
        double r62057 = 3.0;
        double r62058 = pow(r62056, r62057);
        double r62059 = cbrt(r62058);
        double r62060 = r62059 / r62048;
        double r62061 = eps;
        double r62062 = r62053 / r62061;
        double r62063 = r62053 + r62062;
        double r62064 = r62053 - r62061;
        double r62065 = r62064 * r62045;
        double r62066 = -r62065;
        double r62067 = exp(r62066);
        double r62068 = r62053 + r62061;
        double r62069 = r62068 * r62045;
        double r62070 = -r62069;
        double r62071 = exp(r62070);
        double r62072 = -r62071;
        double r62073 = r62062 - r62053;
        double r62074 = r62072 * r62073;
        double r62075 = fma(r62063, r62067, r62074);
        double r62076 = r62075 / r62048;
        double r62077 = r62047 ? r62060 : r62076;
        return r62077;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 378.4237213355006

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}\right)}}}{2}\]

    6. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(2 + {x}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right)}^{3}}}}{2}\]

    if 378.4237213355006 < x

    1. Initial program 0.0

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

    3. Applied fma-neg0.0

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

    4. Simplified0.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 378.4237213355006019810389261692762374878:\\ \;\;\;\;\frac{\sqrt[3]{{\left(2 + {x}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, \left(-e^{-\left(1 + \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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))