Average Error: 29.3 → 1.0
Time: 30.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\varepsilon} + 1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]
\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 183.1502607165994:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1399981 = 1.0;
        double r1399982 = eps;
        double r1399983 = r1399981 / r1399982;
        double r1399984 = r1399981 + r1399983;
        double r1399985 = r1399981 - r1399982;
        double r1399986 = x;
        double r1399987 = r1399985 * r1399986;
        double r1399988 = -r1399987;
        double r1399989 = exp(r1399988);
        double r1399990 = r1399984 * r1399989;
        double r1399991 = r1399983 - r1399981;
        double r1399992 = r1399981 + r1399982;
        double r1399993 = r1399992 * r1399986;
        double r1399994 = -r1399993;
        double r1399995 = exp(r1399994);
        double r1399996 = r1399991 * r1399995;
        double r1399997 = r1399990 - r1399996;
        double r1399998 = 2.0;
        double r1399999 = r1399997 / r1399998;
        return r1399999;
}
↓
double f(double x, double eps) {
        double r1400000 = x;
        double r1400001 = 183.1502607165994;
        bool r1400002 = r1400000 <= r1400001;
        double r1400003 = 0.6666666666666666;
        double r1400004 = r1400000 * r1400000;
        double r1400005 = r1400004 * r1400000;
        double r1400006 = 2.0;
        double r1400007 = fma(r1400003, r1400005, r1400006);
        double r1400008 = r1400007 - r1400004;
        double r1400009 = r1400008 / r1400006;
        double r1400010 = 1.0;
        double r1400011 = eps;
        double r1400012 = r1400010 / r1400011;
        double r1400013 = r1400012 + r1400010;
        double r1400014 = r1400010 - r1400011;
        double r1400015 = r1400000 * r1400014;
        double r1400016 = exp(r1400015);
        double r1400017 = r1400013 / r1400016;
        double r1400018 = fma(r1400000, r1400011, r1400000);
        double r1400019 = exp(r1400018);
        double r1400020 = r1400010 / r1400019;
        double r1400021 = r1400012 - r1400010;
        double r1400022 = r1400020 * r1400021;
        double r1400023 = r1400017 - r1400022;
        double r1400024 = log(r1400023);
        double r1400025 = exp(r1400024);
        double r1400026 = r1400025 / r1400006;
        double r1400027 = r1400002 ? r1400009 : r1400026;
        return r1400027;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 183.1502607165994

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    if 183.1502607165994 < x

    1. Initial program 0.1

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

    3. Applied add-exp-log0.1

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

    4. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\varepsilon} + 1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))