Average Error: 30.1 → 4.3
Time: 24.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4.59599973293413345 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}} \cdot \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}{2}\\ \end{array}\]
\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 4.59599973293413345 \cdot 10^{-7}:\\
\;\;\;\;\frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r39401 = 1.0;
        double r39402 = eps;
        double r39403 = r39401 / r39402;
        double r39404 = r39401 + r39403;
        double r39405 = r39401 - r39402;
        double r39406 = x;
        double r39407 = r39405 * r39406;
        double r39408 = -r39407;
        double r39409 = exp(r39408);
        double r39410 = r39404 * r39409;
        double r39411 = r39403 - r39401;
        double r39412 = r39401 + r39402;
        double r39413 = r39412 * r39406;
        double r39414 = -r39413;
        double r39415 = exp(r39414);
        double r39416 = r39411 * r39415;
        double r39417 = r39410 - r39416;
        double r39418 = 2.0;
        double r39419 = r39417 / r39418;
        return r39419;
}
↓
double f(double x, double eps) {
        double r39420 = x;
        double r39421 = 4.5959997329341335e-07;
        bool r39422 = r39420 <= r39421;
        double r39423 = 2.0;
        double r39424 = 1.0;
        double r39425 = r39420 * r39420;
        double r39426 = 2.7755575615628914e-17;
        double r39427 = cbrt(r39420);
        double r39428 = 2.0;
        double r39429 = exp(r39427);
        double r39430 = cbrt(r39429);
        double r39431 = log(r39430);
        double r39432 = r39428 * r39431;
        double r39433 = r39432 + r39431;
        double r39434 = r39427 * r39433;
        double r39435 = 3.0;
        double r39436 = pow(r39434, r39435);
        double r39437 = eps;
        double r39438 = r39437 / r39420;
        double r39439 = r39436 / r39438;
        double r39440 = r39426 * r39439;
        double r39441 = fma(r39424, r39425, r39440);
        double r39442 = r39423 - r39441;
        double r39443 = r39442 / r39423;
        double r39444 = r39437 - r39424;
        double r39445 = r39420 * r39444;
        double r39446 = exp(r39445);
        double r39447 = r39424 / r39437;
        double r39448 = r39424 + r39447;
        double r39449 = r39424 - r39447;
        double r39450 = cbrt(r39449);
        double r39451 = r39450 * r39450;
        double r39452 = r39424 + r39437;
        double r39453 = r39452 * r39420;
        double r39454 = exp(r39453);
        double r39455 = cbrt(r39454);
        double r39456 = r39455 * r39455;
        double r39457 = r39451 / r39456;
        double r39458 = r39450 / r39455;
        double r39459 = r39457 * r39458;
        double r39460 = fma(r39446, r39448, r39459);
        double r39461 = r39460 / r39423;
        double r39462 = r39422 ? r39443 : r39461;
        return r39462;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 4.5959997329341335e-07

    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

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

    3. Taylor expanded around 0 7.0

      \[\leadsto \frac{\color{blue}{2 - \left(1 \cdot {x}^{2} + 2.77556 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]

    4. Simplified7.0

      \[\leadsto \frac{\color{blue}{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.0

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}{\varepsilon}\right)}{2}\]

    7. Applied unpow-prod-down7.0

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}{\varepsilon}\right)}{2}\]

    8. Applied associate-/l*7.0

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{{\left(\sqrt[3]{x}\right)}^{3}}}}\right)}{2}\]

    9. Simplified7.0

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\color{blue}{\frac{\varepsilon}{x}}}\right)}{2}\]
    10. Using strategy rm

    11. Applied add-log-exp5.3

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \color{blue}{\log \left(e^{\sqrt[3]{x}}\right)}\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt5.1

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt[3]{x}}} \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right) \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right)}\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\]

    14. Applied log-prod5.1

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt[3]{x}}} \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)}\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\]

    15. Simplified5.1

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\]

    if 4.5959997329341335e-07 < x

    1. Initial program 1.8

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

    2. Simplified1.7

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

    4. Applied add-cube-cbrt1.8

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

    5. Applied add-cube-cbrt1.8

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

    6. Applied times-frac1.8

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4.59599973293413345 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - \mathsf{fma}\left(1, x \cdot x, 2.77556 \cdot 10^{-17} \cdot \frac{{\left(\sqrt[3]{x} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}} \cdot \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +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))