Average Error: 28.9 → 1.0
Time: 1.3m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.971491489555398:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot x\right), 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(\varepsilon - 1\right) \cdot x} + \left(\left(e^{x \cdot \left(-1 - \varepsilon\right)} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right)}\right)}{2}\\ \end{array}\]
\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 1.971491489555398:\\
\;\;\;\;\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{\log \left(e^{e^{\left(\varepsilon - 1\right) \cdot x} + \left(\left(e^{x \cdot \left(-1 - \varepsilon\right)} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right)}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r2321381 = 1.0;
        double r2321382 = eps;
        double r2321383 = r2321381 / r2321382;
        double r2321384 = r2321381 + r2321383;
        double r2321385 = r2321381 - r2321382;
        double r2321386 = x;
        double r2321387 = r2321385 * r2321386;
        double r2321388 = -r2321387;
        double r2321389 = exp(r2321388);
        double r2321390 = r2321384 * r2321389;
        double r2321391 = r2321383 - r2321381;
        double r2321392 = r2321381 + r2321382;
        double r2321393 = r2321392 * r2321386;
        double r2321394 = -r2321393;
        double r2321395 = exp(r2321394);
        double r2321396 = r2321391 * r2321395;
        double r2321397 = r2321390 - r2321396;
        double r2321398 = 2.0;
        double r2321399 = r2321397 / r2321398;
        return r2321399;
}
↓
double f(double x, double eps) {
        double r2321400 = x;
        double r2321401 = 1.971491489555398;
        bool r2321402 = r2321400 <= r2321401;
        double r2321403 = 0.6666666666666666;
        double r2321404 = r2321400 * r2321400;
        double r2321405 = r2321404 * r2321400;
        double r2321406 = 2.0;
        double r2321407 = fma(r2321403, r2321405, r2321406);
        double r2321408 = r2321407 - r2321404;
        double r2321409 = r2321408 / r2321406;
        double r2321410 = eps;
        double r2321411 = 1.0;
        double r2321412 = r2321410 - r2321411;
        double r2321413 = r2321412 * r2321400;
        double r2321414 = exp(r2321413);
        double r2321415 = -1.0;
        double r2321416 = r2321415 - r2321410;
        double r2321417 = r2321400 * r2321416;
        double r2321418 = exp(r2321417);
        double r2321419 = r2321418 / r2321410;
        double r2321420 = r2321418 - r2321419;
        double r2321421 = r2321414 / r2321410;
        double r2321422 = r2321420 + r2321421;
        double r2321423 = r2321414 + r2321422;
        double r2321424 = exp(r2321423);
        double r2321425 = log(r2321424);
        double r2321426 = r2321425 / r2321406;
        double r2321427 = r2321402 ? r2321409 : r2321426;
        return r2321427;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.971491489555398

    1. Initial program 38.8

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

    2. Simplified38.8

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    if 1.971491489555398 < 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

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

    4. Applied add-log-exp0.6

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

    5. Applied add-log-exp0.6

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

    6. Applied add-log-exp0.6

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

    7. Applied diff-log0.6

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

    8. Applied sum-log0.6

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

    9. Simplified0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.971491489555398:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot x\right), 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{e^{\left(\varepsilon - 1\right) \cdot x} + \left(\left(e^{x \cdot \left(-1 - \varepsilon\right)} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +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))