Average Error: 30.1 → 1.1
Time: 27.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.421628640605042459554852030123583972454:\\ \;\;\;\;\frac{2 - 1 \cdot \left(x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \frac{1 - \frac{1}{\varepsilon}}{e^{\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}\right)}{2}\\ \end{array}\]
\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 - 1 \cdot \left(x \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \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 r2013402 = 1.0;
        double r2013403 = eps;
        double r2013404 = r2013402 / r2013403;
        double r2013405 = r2013402 + r2013404;
        double r2013406 = r2013402 - r2013403;
        double r2013407 = x;
        double r2013408 = r2013406 * r2013407;
        double r2013409 = -r2013408;
        double r2013410 = exp(r2013409);
        double r2013411 = r2013405 * r2013410;
        double r2013412 = r2013404 - r2013402;
        double r2013413 = r2013402 + r2013403;
        double r2013414 = r2013413 * r2013407;
        double r2013415 = -r2013414;
        double r2013416 = exp(r2013415);
        double r2013417 = r2013412 * r2013416;
        double r2013418 = r2013411 - r2013417;
        double r2013419 = 2.0;
        double r2013420 = r2013418 / r2013419;
        return r2013420;
}
↓
double f(double x, double eps) {
        double r2013421 = x;
        double r2013422 = 1.4216286406050425;
        bool r2013423 = r2013421 <= r2013422;
        double r2013424 = 2.0;
        double r2013425 = 1.0;
        double r2013426 = r2013421 * r2013421;
        double r2013427 = r2013425 * r2013426;
        double r2013428 = r2013424 - r2013427;
        double r2013429 = r2013428 / r2013424;
        double r2013430 = eps;
        double r2013431 = r2013430 - r2013425;
        double r2013432 = r2013421 * r2013431;
        double r2013433 = exp(r2013432);
        double r2013434 = r2013425 / r2013430;
        double r2013435 = r2013425 + r2013434;
        double r2013436 = r2013425 - r2013434;
        double r2013437 = r2013425 + r2013430;
        double r2013438 = r2013421 * r2013437;
        double r2013439 = cbrt(r2013438);
        double r2013440 = r2013439 * r2013439;
        double r2013441 = r2013439 * r2013440;
        double r2013442 = exp(r2013441);
        double r2013443 = r2013436 / r2013442;
        double r2013444 = fma(r2013433, r2013435, r2013443);
        double r2013445 = r2013444 / r2013424;
        double r2013446 = r2013423 ? r2013429 : r2013445;
        return r2013446;
}

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

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

    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

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

    6. Applied add-log-exp1.8

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

    8. Applied add-cube-cbrt1.7

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

    9. Applied log-prod1.7

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

    10. Taylor expanded around inf 1.3

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

    11. Simplified1.3

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

    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

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

    4. Applied add-cube-cbrt0.6

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

  4. Final simplification1.1

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

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))