Average Error: 30.1 → 1.1
Time: 27.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.421628640605042459554852030123583972454:\\ \;\;\;\;\frac{2 - \left(x \cdot x\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}\right)}{2}\\ \end{array}\]
\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 1.421628640605042459554852030123583972454:\\
\;\;\;\;\frac{2 - \left(x \cdot x\right) \cdot 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r1574545 = 1.0;
        double r1574546 = eps;
        double r1574547 = r1574545 / r1574546;
        double r1574548 = r1574545 + r1574547;
        double r1574549 = r1574545 - r1574546;
        double r1574550 = x;
        double r1574551 = r1574549 * r1574550;
        double r1574552 = -r1574551;
        double r1574553 = exp(r1574552);
        double r1574554 = r1574548 * r1574553;
        double r1574555 = r1574547 - r1574545;
        double r1574556 = r1574545 + r1574546;
        double r1574557 = r1574556 * r1574550;
        double r1574558 = -r1574557;
        double r1574559 = exp(r1574558);
        double r1574560 = r1574555 * r1574559;
        double r1574561 = r1574554 - r1574560;
        double r1574562 = 2.0;
        double r1574563 = r1574561 / r1574562;
        return r1574563;
}
↓
double f(double x, double eps) {
        double r1574564 = x;
        double r1574565 = 1.4216286406050425;
        bool r1574566 = r1574564 <= r1574565;
        double r1574567 = 2.0;
        double r1574568 = r1574564 * r1574564;
        double r1574569 = 1.0;
        double r1574570 = r1574568 * r1574569;
        double r1574571 = r1574567 - r1574570;
        double r1574572 = r1574571 / r1574567;
        double r1574573 = eps;
        double r1574574 = r1574573 - r1574569;
        double r1574575 = r1574564 * r1574574;
        double r1574576 = exp(r1574575);
        double r1574577 = r1574569 / r1574573;
        double r1574578 = r1574577 + r1574569;
        double r1574579 = r1574569 - r1574577;
        double r1574580 = r1574569 + r1574573;
        double r1574581 = r1574564 * r1574580;
        double r1574582 = cbrt(r1574581);
        double r1574583 = r1574582 * r1574582;
        double r1574584 = r1574582 * r1574583;
        double r1574585 = exp(r1574584);
        double r1574586 = r1574579 / r1574585;
        double r1574587 = fma(r1574576, r1574578, r1574586);
        double r1574588 = r1574587 / r1574567;
        double r1574589 = r1574566 ? r1574572 : r1574588;
        return r1574589;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.4216286406050425

    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. Simplified39.3

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

    3. Taylor expanded around 0 7.1

      \[\leadsto \frac{\color{blue}{2 - \left(1 \cdot {x}^{2} + 2.77555756156289135105907917022705078125 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]

    4. Simplified7.1

      \[\leadsto \frac{\color{blue}{2 - \mathsf{fma}\left(x \cdot x, 1, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{x \cdot \left(x \cdot x\right)}}\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-log-exp1.8

      \[\leadsto \frac{2 - \mathsf{fma}\left(x \cdot x, 1, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\color{blue}{\log \left(e^{x \cdot \left(x \cdot x\right)}\right)}}}\right)}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.7

      \[\leadsto \frac{2 - \mathsf{fma}\left(x \cdot x, 1, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\log \color{blue}{\left(\left(\sqrt[3]{e^{x \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{e^{x \cdot \left(x \cdot x\right)}}\right) \cdot \sqrt[3]{e^{x \cdot \left(x \cdot x\right)}}\right)}}}\right)}{2}\]

    9. Applied log-prod1.7

      \[\leadsto \frac{2 - \mathsf{fma}\left(x \cdot x, 1, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\color{blue}{\log \left(\sqrt[3]{e^{x \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{e^{x \cdot \left(x \cdot x\right)}}\right) + \log \left(\sqrt[3]{e^{x \cdot \left(x \cdot x\right)}}\right)}}}\right)}{2}\]

    10. Taylor expanded around inf 1.3

      \[\leadsto \frac{2 - \color{blue}{1 \cdot {x}^{2}}}{2}\]

    11. Simplified1.3

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

    if 1.4216286406050425 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied add-cube-cbrt0.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.421628640605042459554852030123583972454:\\ \;\;\;\;\frac{2 - \left(x \cdot x\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))