Average Error: 29.8 → 2.3
Time: 6.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.556503067005827745106216752901673316956:\\ \;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)\right)\\ \end{array}\]
\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 3.556503067005827745106216752901673316956:\\
\;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\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)\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r46258 = 1.0;
        double r46259 = eps;
        double r46260 = r46258 / r46259;
        double r46261 = r46258 + r46260;
        double r46262 = r46258 - r46259;
        double r46263 = x;
        double r46264 = r46262 * r46263;
        double r46265 = -r46264;
        double r46266 = exp(r46265);
        double r46267 = r46261 * r46266;
        double r46268 = r46260 - r46258;
        double r46269 = r46258 + r46259;
        double r46270 = r46269 * r46263;
        double r46271 = -r46270;
        double r46272 = exp(r46271);
        double r46273 = r46268 * r46272;
        double r46274 = r46267 - r46273;
        double r46275 = 2.0;
        double r46276 = r46274 / r46275;
        return r46276;
}
↓
double f(double x, double eps) {
        double r46277 = x;
        double r46278 = 3.5565030670058277;
        bool r46279 = r46277 <= r46278;
        double r46280 = 1.3877787807814457e-17;
        double r46281 = 3.0;
        double r46282 = pow(r46277, r46281);
        double r46283 = cbrt(r46282);
        double r46284 = 2.0;
        double r46285 = exp(r46277);
        double r46286 = cbrt(r46285);
        double r46287 = log(r46286);
        double r46288 = r46284 * r46287;
        double r46289 = r46288 + r46287;
        double r46290 = r46283 * r46289;
        double r46291 = eps;
        double r46292 = r46291 / r46277;
        double r46293 = r46290 / r46292;
        double r46294 = 1.0;
        double r46295 = 0.5;
        double r46296 = pow(r46277, r46284);
        double r46297 = r46295 * r46296;
        double r46298 = r46294 - r46297;
        double r46299 = fma(r46280, r46293, r46298);
        double r46300 = r46294 + r46291;
        double r46301 = r46300 * r46277;
        double r46302 = -r46301;
        double r46303 = exp(r46302);
        double r46304 = 2.0;
        double r46305 = r46303 / r46304;
        double r46306 = r46294 / r46291;
        double r46307 = r46294 - r46306;
        double r46308 = r46294 + r46306;
        double r46309 = r46294 - r46291;
        double r46310 = r46309 * r46277;
        double r46311 = exp(r46310);
        double r46312 = r46304 * r46311;
        double r46313 = r46308 / r46312;
        double r46314 = fma(r46305, r46307, r46313);
        double r46315 = log1p(r46314);
        double r46316 = expm1(r46315);
        double r46317 = r46279 ? r46299 : r46316;
        return r46317;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 3.5565030670058277

    1. Initial program 38.9

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

    2. Simplified38.9

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

    3. Taylor expanded around 0 7.3

      \[\leadsto \color{blue}{\left(1.387778780781445675529539585113525390625 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.3

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\color{blue}{\left(\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}\right) \cdot \sqrt[3]{{x}^{3}}}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    7. Applied associate-/l*7.3

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \color{blue}{\frac{\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}}{\frac{\varepsilon}{\sqrt[3]{{x}^{3}}}}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    8. Simplified7.3

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}}{\color{blue}{\frac{\varepsilon}{x}}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    9. Using strategy rm

    10. Applied add-log-exp2.9

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \color{blue}{\log \left(e^{\sqrt[3]{{x}^{3}}}\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    11. Simplified2.9

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \log \color{blue}{\left(e^{x}\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt2.9

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    14. Applied log-prod2.9

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    15. Simplified2.9

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{x}}\right)} + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    if 3.5565030670058277 < x

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied expm1-log1p-u0.4

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.556503067005827745106216752901673316956:\\ \;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\sqrt[3]{{x}^{3}} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{x}}\right) + \log \left(\sqrt[3]{e^{x}}\right)\right)}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))