Average Error: 29.6 → 1.1
Time: 28.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 356.3553531120810475840698927640914916992:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{2}, x \cdot 8 - 12, 8\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon}, 1, 1 \cdot \left(\left(e^{x \cdot \left(\varepsilon - 1\right)} + {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)}\right) - \frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{\varepsilon}\right)\right)}{2}\\ \end{array}\]
\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 356.3553531120810475840698927640914916992:\\
\;\;\;\;\frac{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{2}, x \cdot 8 - 12, 8\right)\right)\right)}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r32537 = 1.0;
        double r32538 = eps;
        double r32539 = r32537 / r32538;
        double r32540 = r32537 + r32539;
        double r32541 = r32537 - r32538;
        double r32542 = x;
        double r32543 = r32541 * r32542;
        double r32544 = -r32543;
        double r32545 = exp(r32544);
        double r32546 = r32540 * r32545;
        double r32547 = r32539 - r32537;
        double r32548 = r32537 + r32538;
        double r32549 = r32548 * r32542;
        double r32550 = -r32549;
        double r32551 = exp(r32550);
        double r32552 = r32547 * r32551;
        double r32553 = r32546 - r32552;
        double r32554 = 2.0;
        double r32555 = r32553 / r32554;
        return r32555;
}
↓
double f(double x, double eps) {
        double r32556 = x;
        double r32557 = 356.35535311208105;
        bool r32558 = r32556 <= r32557;
        double r32559 = 2.0;
        double r32560 = pow(r32556, r32559);
        double r32561 = 8.0;
        double r32562 = r32556 * r32561;
        double r32563 = 12.0;
        double r32564 = r32562 - r32563;
        double r32565 = fma(r32560, r32564, r32561);
        double r32566 = log1p(r32565);
        double r32567 = expm1(r32566);
        double r32568 = cbrt(r32567);
        double r32569 = 2.0;
        double r32570 = r32568 / r32569;
        double r32571 = eps;
        double r32572 = 1.0;
        double r32573 = r32571 - r32572;
        double r32574 = r32556 * r32573;
        double r32575 = exp(r32574);
        double r32576 = r32575 / r32571;
        double r32577 = r32572 + r32571;
        double r32578 = exp(r32577);
        double r32579 = -r32556;
        double r32580 = pow(r32578, r32579);
        double r32581 = r32575 + r32580;
        double r32582 = r32577 * r32556;
        double r32583 = -r32582;
        double r32584 = exp(r32583);
        double r32585 = r32584 / r32571;
        double r32586 = r32581 - r32585;
        double r32587 = r32572 * r32586;
        double r32588 = fma(r32576, r32572, r32587);
        double r32589 = r32588 / r32569;
        double r32590 = r32558 ? r32570 : r32589;
        return r32590;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 356.35535311208105

    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. Taylor expanded around 0 1.4

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}\right)\right) \cdot \left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}\right)}}}{2}\]

    6. Simplified1.4

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left({x}^{2}, 0.6666666666666667406815349750104360282421 \cdot x - 1, 2\right)\right)}^{3}}}}{2}\]

    7. Taylor expanded around 0 1.4

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

    8. Simplified1.4

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

    10. Applied expm1-log1p-u1.4

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{2}, x \cdot 8 - 12, 8\right)\right)\right)}}}{2}\]

    if 356.35535311208105 < 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 0.0

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

    3. Simplified0.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 356.3553531120810475840698927640914916992:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left({x}^{2}, x \cdot 8 - 12, 8\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon}, 1, 1 \cdot \left(\left(e^{x \cdot \left(\varepsilon - 1\right)} + {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)}\right) - \frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{\varepsilon}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))