Average Error: 29.4 → 4.5
Time: 26.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.512941299653822035866518325164520319959 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)}{\varepsilon} \cdot \left(\sqrt[3]{x} \cdot \left(x \cdot x\right)\right), 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon - 1\right) \cdot x}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\varepsilon + 1\right) \cdot x}\right)\right)}\right)}{2}\\ \end{array}\]
\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 5.512941299653822035866518325164520319959 \cdot 10^{-5}:\\
\;\;\;\;\frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)}{\varepsilon} \cdot \left(\sqrt[3]{x} \cdot \left(x \cdot x\right)\right), 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon - 1\right) \cdot x}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\varepsilon + 1\right) \cdot x}\right)\right)}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r1893466 = 1.0;
        double r1893467 = eps;
        double r1893468 = r1893466 / r1893467;
        double r1893469 = r1893466 + r1893468;
        double r1893470 = r1893466 - r1893467;
        double r1893471 = x;
        double r1893472 = r1893470 * r1893471;
        double r1893473 = -r1893472;
        double r1893474 = exp(r1893473);
        double r1893475 = r1893469 * r1893474;
        double r1893476 = r1893468 - r1893466;
        double r1893477 = r1893466 + r1893467;
        double r1893478 = r1893477 * r1893471;
        double r1893479 = -r1893478;
        double r1893480 = exp(r1893479);
        double r1893481 = r1893476 * r1893480;
        double r1893482 = r1893475 - r1893481;
        double r1893483 = 2.0;
        double r1893484 = r1893482 / r1893483;
        return r1893484;
}
↓
double f(double x, double eps) {
        double r1893485 = x;
        double r1893486 = 5.512941299653822e-05;
        bool r1893487 = r1893485 <= r1893486;
        double r1893488 = 2.0;
        double r1893489 = cbrt(r1893485);
        double r1893490 = exp(r1893489);
        double r1893491 = sqrt(r1893490);
        double r1893492 = log(r1893491);
        double r1893493 = r1893492 + r1893492;
        double r1893494 = r1893489 * r1893493;
        double r1893495 = eps;
        double r1893496 = r1893494 / r1893495;
        double r1893497 = r1893485 * r1893485;
        double r1893498 = r1893489 * r1893497;
        double r1893499 = r1893496 * r1893498;
        double r1893500 = 2.7755575615628914e-17;
        double r1893501 = 1.0;
        double r1893502 = r1893497 * r1893501;
        double r1893503 = fma(r1893499, r1893500, r1893502);
        double r1893504 = r1893488 - r1893503;
        double r1893505 = r1893504 / r1893488;
        double r1893506 = r1893495 - r1893501;
        double r1893507 = r1893506 * r1893485;
        double r1893508 = exp(r1893507);
        double r1893509 = r1893501 / r1893495;
        double r1893510 = r1893509 + r1893501;
        double r1893511 = r1893501 - r1893509;
        double r1893512 = r1893495 + r1893501;
        double r1893513 = r1893512 * r1893485;
        double r1893514 = exp(r1893513);
        double r1893515 = log1p(r1893514);
        double r1893516 = expm1(r1893515);
        double r1893517 = r1893511 / r1893516;
        double r1893518 = fma(r1893508, r1893510, r1893517);
        double r1893519 = r1893518 / r1893488;
        double r1893520 = r1893487 ? r1893505 : r1893519;
        return r1893520;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 5.512941299653822e-05

    1. Initial program 39.1

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

    2. Simplified39.1

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

    3. Taylor expanded around 0 7.5

      \[\leadsto \frac{\color{blue}{2 - \left(1 \cdot {x}^{2} + 2.77555756156289135105907917022705078125 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]

    4. Simplified7.5

      \[\leadsto \frac{\color{blue}{2 - \mathsf{fma}\left(\frac{x}{\frac{\varepsilon}{x \cdot x}}, 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied div-inv7.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(\frac{x}{\color{blue}{\varepsilon \cdot \frac{1}{x \cdot x}}}, 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]

    7. Applied add-cube-cbrt7.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\varepsilon \cdot \frac{1}{x \cdot x}}, 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]

    8. Applied times-frac7.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\varepsilon} \cdot \frac{\sqrt[3]{x}}{\frac{1}{x \cdot x}}}, 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]

    9. Simplified7.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\varepsilon} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x}\right)}, 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]
    10. Using strategy rm

    11. Applied add-log-exp5.6

      \[\leadsto \frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \color{blue}{\log \left(e^{\sqrt[3]{x}}\right)}}{\varepsilon} \cdot \left(\left(x \cdot x\right) \cdot \sqrt[3]{x}\right), 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt5.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \log \color{blue}{\left(\sqrt{e^{\sqrt[3]{x}}} \cdot \sqrt{e^{\sqrt[3]{x}}}\right)}}{\varepsilon} \cdot \left(\left(x \cdot x\right) \cdot \sqrt[3]{x}\right), 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]

    14. Applied log-prod5.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)}}{\varepsilon} \cdot \left(\left(x \cdot x\right) \cdot \sqrt[3]{x}\right), 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\]

    if 5.512941299653822e-05 < x

    1. Initial program 1.6

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

    2. Simplified1.6

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

    4. Applied expm1-log1p-u1.6

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

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.512941299653822035866518325164520319959 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - \mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)}{\varepsilon} \cdot \left(\sqrt[3]{x} \cdot \left(x \cdot x\right)\right), 2.77555756156289135105907917022705078125 \cdot 10^{-17}, \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{\left(\varepsilon - 1\right) \cdot x}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\left(\varepsilon + 1\right) \cdot x}\right)\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))