Average Error: 29.6 → 4.1
Time: 8.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.00337018905975299205:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5, \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)} \cdot \varepsilon}, \frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right)\\ \end{array}\]
\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 0.00337018905975299205:\\
\;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5, \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)} \cdot \varepsilon}, \frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r61473 = 1.0;
        double r61474 = eps;
        double r61475 = r61473 / r61474;
        double r61476 = r61473 + r61475;
        double r61477 = r61473 - r61474;
        double r61478 = x;
        double r61479 = r61477 * r61478;
        double r61480 = -r61479;
        double r61481 = exp(r61480);
        double r61482 = r61476 * r61481;
        double r61483 = r61475 - r61473;
        double r61484 = r61473 + r61474;
        double r61485 = r61484 * r61478;
        double r61486 = -r61485;
        double r61487 = exp(r61486);
        double r61488 = r61483 * r61487;
        double r61489 = r61482 - r61488;
        double r61490 = 2.0;
        double r61491 = r61489 / r61490;
        return r61491;
}
↓
double f(double x, double eps) {
        double r61492 = x;
        double r61493 = 0.003370189059752992;
        bool r61494 = r61492 <= r61493;
        double r61495 = 1.3877787807814457e-17;
        double r61496 = 2.0;
        double r61497 = cbrt(r61492);
        double r61498 = exp(r61497);
        double r61499 = cbrt(r61498);
        double r61500 = log(r61499);
        double r61501 = r61496 * r61500;
        double r61502 = r61501 + r61500;
        double r61503 = r61502 * r61497;
        double r61504 = 3.0;
        double r61505 = pow(r61503, r61504);
        double r61506 = eps;
        double r61507 = r61506 / r61492;
        double r61508 = r61505 / r61507;
        double r61509 = 1.0;
        double r61510 = 0.5;
        double r61511 = pow(r61492, r61496);
        double r61512 = r61510 * r61511;
        double r61513 = r61509 - r61512;
        double r61514 = fma(r61495, r61508, r61513);
        double r61515 = r61509 + r61506;
        double r61516 = r61515 * r61492;
        double r61517 = -r61516;
        double r61518 = exp(r61517);
        double r61519 = 2.0;
        double r61520 = r61518 / r61519;
        double r61521 = r61509 / r61506;
        double r61522 = r61509 - r61521;
        double r61523 = 1.0;
        double r61524 = r61509 - r61506;
        double r61525 = r61492 * r61524;
        double r61526 = exp(r61525);
        double r61527 = r61526 * r61506;
        double r61528 = r61523 / r61527;
        double r61529 = r61510 / r61526;
        double r61530 = fma(r61510, r61528, r61529);
        double r61531 = fma(r61520, r61522, r61530);
        double r61532 = r61494 ? r61514 : r61531;
        return r61532;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 0.003370189059752992

    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

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

    3. Taylor expanded around 0 6.8

      \[\leadsto \color{blue}{\left(1.38778 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified6.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt6.8

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    7. Applied unpow-prod-down6.8

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    8. Applied associate-/l*6.8

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{{\left(\sqrt[3]{x}\right)}^{3}}}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    9. Simplified6.8

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\color{blue}{\frac{\varepsilon}{x}}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    10. Using strategy rm

    11. Applied add-log-exp5.4

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\color{blue}{\log \left(e^{\sqrt[3]{x}}\right)} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt5.1

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt[3]{x}}} \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right) \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right)} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    14. Applied log-prod5.1

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt[3]{x}}} \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    15. Simplified5.1

      \[\leadsto \mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    if 0.003370189059752992 < x

    1. Initial program 1.1

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

    2. Simplified1.1

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

    3. Taylor expanded around inf 1.1

      \[\leadsto \mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \color{blue}{0.5 \cdot \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)} \cdot \varepsilon} + 0.5 \cdot \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}}}\right)\]

    4. Simplified1.1

      \[\leadsto \mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \color{blue}{\mathsf{fma}\left(0.5, \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)} \cdot \varepsilon}, \frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00337018905975299205:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5, \frac{1}{e^{x \cdot \left(1 - \varepsilon\right)} \cdot \varepsilon}, \frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))