Average Error: 30.0 → 0.9
Time: 1.7m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.2882158525972256:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot x\right), 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + e^{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right)\right) \cdot \left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)}}{2}\\ \end{array}\]
\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 1.2882158525972256:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot x\right), 2\right) - x \cdot x}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + e^{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right)\right) \cdot \left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)}}{2}\\

\end{array}
double f(double x, double eps) {
        double r3711418 = 1.0;
        double r3711419 = eps;
        double r3711420 = r3711418 / r3711419;
        double r3711421 = r3711418 + r3711420;
        double r3711422 = r3711418 - r3711419;
        double r3711423 = x;
        double r3711424 = r3711422 * r3711423;
        double r3711425 = -r3711424;
        double r3711426 = exp(r3711425);
        double r3711427 = r3711421 * r3711426;
        double r3711428 = r3711420 - r3711418;
        double r3711429 = r3711418 + r3711419;
        double r3711430 = r3711429 * r3711423;
        double r3711431 = -r3711430;
        double r3711432 = exp(r3711431);
        double r3711433 = r3711428 * r3711432;
        double r3711434 = r3711427 - r3711433;
        double r3711435 = 2.0;
        double r3711436 = r3711434 / r3711435;
        return r3711436;
}
↓
double f(double x, double eps) {
        double r3711437 = x;
        double r3711438 = 1.2882158525972256;
        bool r3711439 = r3711437 <= r3711438;
        double r3711440 = 0.6666666666666666;
        double r3711441 = r3711437 * r3711437;
        double r3711442 = r3711441 * r3711437;
        double r3711443 = 2.0;
        double r3711444 = fma(r3711440, r3711442, r3711443);
        double r3711445 = r3711444 - r3711441;
        double r3711446 = r3711445 / r3711443;
        double r3711447 = eps;
        double r3711448 = -1.0;
        double r3711449 = r3711447 + r3711448;
        double r3711450 = r3711449 * r3711437;
        double r3711451 = exp(r3711450);
        double r3711452 = r3711451 / r3711447;
        double r3711453 = r3711451 + r3711452;
        double r3711454 = r3711448 - r3711447;
        double r3711455 = r3711437 * r3711454;
        double r3711456 = exp(r3711455);
        double r3711457 = r3711456 / r3711447;
        double r3711458 = r3711453 - r3711457;
        double r3711459 = cbrt(r3711455);
        double r3711460 = expm1(r3711459);
        double r3711461 = log1p(r3711460);
        double r3711462 = r3711459 * r3711459;
        double r3711463 = r3711461 * r3711462;
        double r3711464 = exp(r3711463);
        double r3711465 = r3711458 + r3711464;
        double r3711466 = r3711465 / r3711443;
        double r3711467 = r3711439 ? r3711446 : r3711466;
        return r3711467;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.2882158525972256

    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. Simplified39.5

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 1.2882158525972256 < 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

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

    4. Applied associate--r-0.6

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

    6. Applied add-cube-cbrt0.6

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

    8. Applied log1p-expm1-u0.6

      \[\leadsto \frac{\left(\left(\frac{e^{x \cdot \left(-1 + \varepsilon\right)}}{\varepsilon} + e^{x \cdot \left(-1 + \varepsilon\right)}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + e^{\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right)\right)}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.2882158525972256:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot x\right), 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + e^{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right)\right) \cdot \left(\sqrt[3]{x \cdot \left(-1 - \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(-1 - \varepsilon\right)}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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))