Average Error: 29.6 → 0.9
Time: 6.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.80482577976084757:\\ \;\;\;\;\frac{\frac{\left(2.666666666666667 \cdot {x}^{3} + 4\right) - 1 \cdot {x}^{4}}{{x}^{2} \cdot \left(0.66666666666666674 \cdot x + 1\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}{2}\\ \end{array}\]
\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 2.80482577976084757:\\
\;\;\;\;\frac{\frac{\left(2.666666666666667 \cdot {x}^{3} + 4\right) - 1 \cdot {x}^{4}}{{x}^{2} \cdot \left(0.66666666666666674 \cdot x + 1\right) + 2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r42001 = 1.0;
        double r42002 = eps;
        double r42003 = r42001 / r42002;
        double r42004 = r42001 + r42003;
        double r42005 = r42001 - r42002;
        double r42006 = x;
        double r42007 = r42005 * r42006;
        double r42008 = -r42007;
        double r42009 = exp(r42008);
        double r42010 = r42004 * r42009;
        double r42011 = r42003 - r42001;
        double r42012 = r42001 + r42002;
        double r42013 = r42012 * r42006;
        double r42014 = -r42013;
        double r42015 = exp(r42014);
        double r42016 = r42011 * r42015;
        double r42017 = r42010 - r42016;
        double r42018 = 2.0;
        double r42019 = r42017 / r42018;
        return r42019;
}
↓
double f(double x, double eps) {
        double r42020 = x;
        double r42021 = 2.8048257797608476;
        bool r42022 = r42020 <= r42021;
        double r42023 = 2.666666666666667;
        double r42024 = 3.0;
        double r42025 = pow(r42020, r42024);
        double r42026 = r42023 * r42025;
        double r42027 = 4.0;
        double r42028 = r42026 + r42027;
        double r42029 = 1.0;
        double r42030 = 4.0;
        double r42031 = pow(r42020, r42030);
        double r42032 = r42029 * r42031;
        double r42033 = r42028 - r42032;
        double r42034 = 2.0;
        double r42035 = pow(r42020, r42034);
        double r42036 = 0.6666666666666667;
        double r42037 = r42036 * r42020;
        double r42038 = r42037 + r42029;
        double r42039 = r42035 * r42038;
        double r42040 = 2.0;
        double r42041 = r42039 + r42040;
        double r42042 = r42033 / r42041;
        double r42043 = r42042 / r42040;
        double r42044 = eps;
        double r42045 = r42029 / r42044;
        double r42046 = r42029 + r42045;
        double r42047 = r42029 - r42044;
        double r42048 = r42047 * r42020;
        double r42049 = -r42048;
        double r42050 = exp(r42049);
        double r42051 = r42046 * r42050;
        double r42052 = r42045 - r42029;
        double r42053 = exp(1.0);
        double r42054 = r42029 + r42044;
        double r42055 = r42054 * r42020;
        double r42056 = -r42055;
        double r42057 = pow(r42053, r42056);
        double r42058 = r42052 * r42057;
        double r42059 = r42051 - r42058;
        double r42060 = r42059 / r42040;
        double r42061 = r42022 ? r42043 : r42060;
        return r42061;
}

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 < 2.8048257797608476

    1. Initial program 39.5

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

    2. Taylor expanded around 0 1.1

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

    4. Applied flip--1.1

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

    5. Simplified1.1

      \[\leadsto \frac{\frac{\color{blue}{\left(-1 \cdot 1\right) \cdot {x}^{4} + \left(0.66666666666666674 \cdot {x}^{3} + 2\right) \cdot \left(0.66666666666666674 \cdot {x}^{3} + 2\right)}}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) + 1 \cdot {x}^{2}}}{2}\]

    6. Simplified1.1

      \[\leadsto \frac{\frac{\left(-1 \cdot 1\right) \cdot {x}^{4} + \left(0.66666666666666674 \cdot {x}^{3} + 2\right) \cdot \left(0.66666666666666674 \cdot {x}^{3} + 2\right)}{\color{blue}{{x}^{2} \cdot \left(0.66666666666666674 \cdot x + 1\right) + 2}}}{2}\]

    7. Taylor expanded around 0 1.1

      \[\leadsto \frac{\frac{\color{blue}{\left(2.666666666666667 \cdot {x}^{3} + 4\right) - 1 \cdot {x}^{4}}}{{x}^{2} \cdot \left(0.66666666666666674 \cdot x + 1\right) + 2}}{2}\]

    if 2.8048257797608476 < x

    1. Initial program 0.4

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

    3. Applied *-un-lft-identity0.4

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

    4. Applied exp-prod0.4

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

    5. Simplified0.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.80482577976084757:\\ \;\;\;\;\frac{\frac{\left(2.666666666666667 \cdot {x}^{3} + 4\right) - 1 \cdot {x}^{4}}{{x}^{2} \cdot \left(0.66666666666666674 \cdot x + 1\right) + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))