Average Error: 29.5 → 1.0
Time: 6.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 358.465922844080694:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - \left(12 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon}, 1 \cdot e^{x \cdot \varepsilon - 1 \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 358.465922844080694:\\
\;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - \left(12 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r38308 = 1.0;
        double r38309 = eps;
        double r38310 = r38308 / r38309;
        double r38311 = r38308 + r38310;
        double r38312 = r38308 - r38309;
        double r38313 = x;
        double r38314 = r38312 * r38313;
        double r38315 = -r38314;
        double r38316 = exp(r38315);
        double r38317 = r38311 * r38316;
        double r38318 = r38310 - r38308;
        double r38319 = r38308 + r38309;
        double r38320 = r38319 * r38313;
        double r38321 = -r38320;
        double r38322 = exp(r38321);
        double r38323 = r38318 * r38322;
        double r38324 = r38317 - r38323;
        double r38325 = 2.0;
        double r38326 = r38324 / r38325;
        return r38326;
}
↓
double f(double x, double eps) {
        double r38327 = x;
        double r38328 = 358.4659228440807;
        bool r38329 = r38327 <= r38328;
        double r38330 = 3.0;
        double r38331 = pow(r38327, r38330);
        double r38332 = 8.0;
        double r38333 = 12.0;
        double r38334 = cbrt(r38327);
        double r38335 = r38334 * r38334;
        double r38336 = 2.0;
        double r38337 = pow(r38335, r38336);
        double r38338 = r38333 * r38337;
        double r38339 = pow(r38334, r38336);
        double r38340 = r38338 * r38339;
        double r38341 = r38332 - r38340;
        double r38342 = fma(r38331, r38332, r38341);
        double r38343 = cbrt(r38342);
        double r38344 = 2.0;
        double r38345 = r38343 / r38344;
        double r38346 = 1.0;
        double r38347 = eps;
        double r38348 = r38327 * r38347;
        double r38349 = r38346 * r38327;
        double r38350 = r38348 - r38349;
        double r38351 = exp(r38350);
        double r38352 = r38351 / r38347;
        double r38353 = r38346 * r38351;
        double r38354 = fma(r38346, r38352, r38353);
        double r38355 = r38346 / r38347;
        double r38356 = r38355 - r38346;
        double r38357 = r38346 + r38347;
        double r38358 = r38357 * r38327;
        double r38359 = -r38358;
        double r38360 = exp(r38359);
        double r38361 = r38356 * r38360;
        double r38362 = r38354 - r38361;
        double r38363 = r38362 / r38344;
        double r38364 = r38329 ? r38345 : r38363;
        return r38364;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 358.4659228440807

    1. Initial program 39.6

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right) \cdot \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)\right) \cdot \mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}}{2}\]

    6. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)\right)}^{3}}}}{2}\]

    7. Taylor expanded around 0 1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {x}^{2}}}}{2}\]

    8. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot {x}^{2}\right)}}}{2}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}\right)}}{2}\]

    11. Applied unpow-prod-down1.3

      \[\leadsto \frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - 12 \cdot \color{blue}{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)}\right)}}{2}\]

    12. Applied associate-*r*1.3

      \[\leadsto \frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - \color{blue}{\left(12 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}}\right)}}{2}\]

    if 358.4659228440807 < x

    1. Initial program 0.1

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 358.465922844080694:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left({x}^{3}, 8, 8 - \left(12 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon}, 1 \cdot e^{x \cdot \varepsilon - 1 \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +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))