Average Error: 29.6 → 1.1
Time: 28.8s
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 r35609 = 1.0;
        double r35610 = eps;
        double r35611 = r35609 / r35610;
        double r35612 = r35609 + r35611;
        double r35613 = r35609 - r35610;
        double r35614 = x;
        double r35615 = r35613 * r35614;
        double r35616 = -r35615;
        double r35617 = exp(r35616);
        double r35618 = r35612 * r35617;
        double r35619 = r35611 - r35609;
        double r35620 = r35609 + r35610;
        double r35621 = r35620 * r35614;
        double r35622 = -r35621;
        double r35623 = exp(r35622);
        double r35624 = r35619 * r35623;
        double r35625 = r35618 - r35624;
        double r35626 = 2.0;
        double r35627 = r35625 / r35626;
        return r35627;
}
↓
double f(double x, double eps) {
        double r35628 = x;
        double r35629 = 356.35535311208105;
        bool r35630 = r35628 <= r35629;
        double r35631 = 2.0;
        double r35632 = pow(r35628, r35631);
        double r35633 = 8.0;
        double r35634 = r35628 * r35633;
        double r35635 = 12.0;
        double r35636 = r35634 - r35635;
        double r35637 = fma(r35632, r35636, r35633);
        double r35638 = log1p(r35637);
        double r35639 = expm1(r35638);
        double r35640 = cbrt(r35639);
        double r35641 = 2.0;
        double r35642 = r35640 / r35641;
        double r35643 = eps;
        double r35644 = 1.0;
        double r35645 = r35643 - r35644;
        double r35646 = r35628 * r35645;
        double r35647 = exp(r35646);
        double r35648 = r35647 / r35643;
        double r35649 = r35644 + r35643;
        double r35650 = exp(r35649);
        double r35651 = -r35628;
        double r35652 = pow(r35650, r35651);
        double r35653 = r35647 + r35652;
        double r35654 = r35649 * r35628;
        double r35655 = -r35654;
        double r35656 = exp(r35655);
        double r35657 = r35656 / r35643;
        double r35658 = r35653 - r35657;
        double r35659 = r35644 * r35658;
        double r35660 = fma(r35648, r35644, r35659);
        double r35661 = r35660 / r35641;
        double r35662 = r35630 ? r35642 : r35661;
        return r35662;
}

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))