Average Error: 29.5 → 1.1
Time: 27.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.9890182023189888:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(e^{\left(\varepsilon + -1\right) \cdot x} + \left(\left(e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)} - \frac{e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)}}{\varepsilon}\right) + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)\right)\\ \end{array}\]
\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 1.9890182023189888:\\
\;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)\\

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

\end{array}
double f(double x, double eps) {
        double r1846938 = 1.0;
        double r1846939 = eps;
        double r1846940 = r1846938 / r1846939;
        double r1846941 = r1846938 + r1846940;
        double r1846942 = r1846938 - r1846939;
        double r1846943 = x;
        double r1846944 = r1846942 * r1846943;
        double r1846945 = -r1846944;
        double r1846946 = exp(r1846945);
        double r1846947 = r1846941 * r1846946;
        double r1846948 = r1846940 - r1846938;
        double r1846949 = r1846938 + r1846939;
        double r1846950 = r1846949 * r1846943;
        double r1846951 = -r1846950;
        double r1846952 = exp(r1846951);
        double r1846953 = r1846948 * r1846952;
        double r1846954 = r1846947 - r1846953;
        double r1846955 = 2.0;
        double r1846956 = r1846954 / r1846955;
        return r1846956;
}
↓
double f(double x, double eps) {
        double r1846957 = x;
        double r1846958 = 1.9890182023189888;
        bool r1846959 = r1846957 <= r1846958;
        double r1846960 = 0.5;
        double r1846961 = r1846957 * r1846957;
        double r1846962 = r1846961 * r1846957;
        double r1846963 = 0.6666666666666666;
        double r1846964 = 2.0;
        double r1846965 = r1846964 - r1846961;
        double r1846966 = fma(r1846962, r1846963, r1846965);
        double r1846967 = r1846960 * r1846966;
        double r1846968 = eps;
        double r1846969 = -1.0;
        double r1846970 = r1846968 + r1846969;
        double r1846971 = r1846970 * r1846957;
        double r1846972 = exp(r1846971);
        double r1846973 = -r1846957;
        double r1846974 = fma(r1846968, r1846973, r1846973);
        double r1846975 = exp(r1846974);
        double r1846976 = r1846975 / r1846968;
        double r1846977 = r1846975 - r1846976;
        double r1846978 = r1846972 / r1846968;
        double r1846979 = r1846977 + r1846978;
        double r1846980 = r1846972 + r1846979;
        double r1846981 = r1846960 * r1846980;
        double r1846982 = r1846959 ? r1846967 : r1846981;
        return r1846982;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.9890182023189888

    1. Initial program 39.0

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

    2. Simplified39.0

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    if 1.9890182023189888 < x

    1. Initial program 0.7

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

    2. Simplified0.7

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

    3. Taylor expanded around inf 0.7

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

    4. Simplified0.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.9890182023189888:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(e^{\left(\varepsilon + -1\right) \cdot x} + \left(\left(e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)} - \frac{e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)}}{\varepsilon}\right) + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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))