Average Error: 29.6 → 4.3
Time: 7.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.0640289263225317 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\log \left(e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]
\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 1.0640289263225317 \cdot 10^{-25}:\\
\;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r48322 = 1.0;
        double r48323 = eps;
        double r48324 = r48322 / r48323;
        double r48325 = r48322 + r48324;
        double r48326 = r48322 - r48323;
        double r48327 = x;
        double r48328 = r48326 * r48327;
        double r48329 = -r48328;
        double r48330 = exp(r48329);
        double r48331 = r48325 * r48330;
        double r48332 = r48324 - r48322;
        double r48333 = r48322 + r48323;
        double r48334 = r48333 * r48327;
        double r48335 = -r48334;
        double r48336 = exp(r48335);
        double r48337 = r48332 * r48336;
        double r48338 = r48331 - r48337;
        double r48339 = 2.0;
        double r48340 = r48338 / r48339;
        return r48340;
}
↓
double f(double x, double eps) {
        double r48341 = x;
        double r48342 = 1.0640289263225317e-25;
        bool r48343 = r48341 <= r48342;
        double r48344 = 1.3877787807814457e-17;
        double r48345 = cbrt(r48341);
        double r48346 = exp(r48345);
        double r48347 = sqrt(r48346);
        double r48348 = log(r48347);
        double r48349 = r48348 + r48348;
        double r48350 = r48345 * r48349;
        double r48351 = 3.0;
        double r48352 = pow(r48350, r48351);
        double r48353 = eps;
        double r48354 = r48353 / r48341;
        double r48355 = r48352 / r48354;
        double r48356 = 1.0;
        double r48357 = 0.5;
        double r48358 = 2.0;
        double r48359 = pow(r48341, r48358);
        double r48360 = r48357 * r48359;
        double r48361 = r48356 - r48360;
        double r48362 = fma(r48344, r48355, r48361);
        double r48363 = r48356 + r48353;
        double r48364 = r48363 * r48341;
        double r48365 = exp(r48364);
        double r48366 = log(r48365);
        double r48367 = -r48366;
        double r48368 = exp(r48367);
        double r48369 = 2.0;
        double r48370 = r48368 / r48369;
        double r48371 = r48356 / r48353;
        double r48372 = r48356 - r48371;
        double r48373 = r48356 + r48371;
        double r48374 = r48356 - r48353;
        double r48375 = r48374 * r48341;
        double r48376 = exp(r48375);
        double r48377 = r48369 * r48376;
        double r48378 = r48373 / r48377;
        double r48379 = fma(r48370, r48372, r48378);
        double r48380 = r48343 ? r48362 : r48379;
        return r48380;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.0640289263225317e-25

    1. Initial program 38.3

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

    2. Simplified38.3

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

    3. Taylor expanded around 0 6.1

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

    4. Simplified6.1

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

    6. Applied add-cube-cbrt6.1

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

    7. Applied unpow-prod-down6.1

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

    8. Applied associate-/l*6.1

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

    9. Simplified6.1

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

    11. Applied add-log-exp4.2

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

    13. Applied add-sqr-sqrt4.2

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

    14. Applied log-prod4.2

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

    if 1.0640289263225317e-25 < x

    1. Initial program 4.6

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

    2. Simplified4.7

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

    4. Applied add-log-exp4.6

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.0640289263225317 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\log \left(e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))