Average Error: 30.1 → 1.0
Time: 26.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.8329629569573715:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\right)\right)\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\mathsf{fma}\left(e^{x \cdot \left(\varepsilon + -1\right)}, \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}\right) - \frac{\frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) + \frac{1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)\\ \end{array}\]
\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 1.8329629569573715:\\
\;\;\;\;\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\right)\right)\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\\

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

\end{array}
double f(double x, double eps) {
        double r2144016 = 1.0;
        double r2144017 = eps;
        double r2144018 = r2144016 / r2144017;
        double r2144019 = r2144016 + r2144018;
        double r2144020 = r2144016 - r2144017;
        double r2144021 = x;
        double r2144022 = r2144020 * r2144021;
        double r2144023 = -r2144022;
        double r2144024 = exp(r2144023);
        double r2144025 = r2144019 * r2144024;
        double r2144026 = r2144018 - r2144016;
        double r2144027 = r2144016 + r2144017;
        double r2144028 = r2144027 * r2144021;
        double r2144029 = -r2144028;
        double r2144030 = exp(r2144029);
        double r2144031 = r2144026 * r2144030;
        double r2144032 = r2144025 - r2144031;
        double r2144033 = 2.0;
        double r2144034 = r2144032 / r2144033;
        return r2144034;
}
↓
double f(double x, double eps) {
        double r2144035 = x;
        double r2144036 = 1.8329629569573715;
        bool r2144037 = r2144035 <= r2144036;
        double r2144038 = 0.5;
        double r2144039 = r2144035 * r2144035;
        double r2144040 = 0.6666666666666666;
        double r2144041 = r2144039 * r2144040;
        double r2144042 = 2.0;
        double r2144043 = r2144042 - r2144039;
        double r2144044 = fma(r2144035, r2144041, r2144043);
        double r2144045 = r2144038 * r2144044;
        double r2144046 = cbrt(r2144045);
        double r2144047 = expm1(r2144046);
        double r2144048 = log1p(r2144047);
        double r2144049 = r2144046 * r2144048;
        double r2144050 = r2144049 * r2144046;
        double r2144051 = eps;
        double r2144052 = -1.0;
        double r2144053 = r2144051 + r2144052;
        double r2144054 = r2144035 * r2144053;
        double r2144055 = exp(r2144054);
        double r2144056 = 1.0;
        double r2144057 = r2144056 / r2144051;
        double r2144058 = fma(r2144055, r2144057, r2144055);
        double r2144059 = fma(r2144051, r2144035, r2144035);
        double r2144060 = exp(r2144059);
        double r2144061 = r2144057 / r2144060;
        double r2144062 = r2144058 - r2144061;
        double r2144063 = r2144056 / r2144060;
        double r2144064 = r2144062 + r2144063;
        double r2144065 = r2144038 * r2144064;
        double r2144066 = r2144037 ? r2144050 : r2144065;
        return r2144066;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.8329629569573715

    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

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    6. Applied add-cube-cbrt1.1

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

    8. Applied log1p-expm1-u1.1

      \[\leadsto \left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\right)\right)}\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\]

    if 1.8329629569573715 < 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

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

    4. Applied div-sub0.5

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

    5. Applied associate--r-0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.8329629569573715:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\right)\right)\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{2}{3}, 2 - x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\mathsf{fma}\left(e^{x \cdot \left(\varepsilon + -1\right)}, \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}\right) - \frac{\frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) + \frac{1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))