Average Error: 29.6 → 1.1
Time: 29.7s
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 r35647 = 1.0;
        double r35648 = eps;
        double r35649 = r35647 / r35648;
        double r35650 = r35647 + r35649;
        double r35651 = r35647 - r35648;
        double r35652 = x;
        double r35653 = r35651 * r35652;
        double r35654 = -r35653;
        double r35655 = exp(r35654);
        double r35656 = r35650 * r35655;
        double r35657 = r35649 - r35647;
        double r35658 = r35647 + r35648;
        double r35659 = r35658 * r35652;
        double r35660 = -r35659;
        double r35661 = exp(r35660);
        double r35662 = r35657 * r35661;
        double r35663 = r35656 - r35662;
        double r35664 = 2.0;
        double r35665 = r35663 / r35664;
        return r35665;
}
↓
double f(double x, double eps) {
        double r35666 = x;
        double r35667 = 356.35535311208105;
        bool r35668 = r35666 <= r35667;
        double r35669 = 2.0;
        double r35670 = pow(r35666, r35669);
        double r35671 = 8.0;
        double r35672 = r35666 * r35671;
        double r35673 = 12.0;
        double r35674 = r35672 - r35673;
        double r35675 = fma(r35670, r35674, r35671);
        double r35676 = log1p(r35675);
        double r35677 = expm1(r35676);
        double r35678 = cbrt(r35677);
        double r35679 = 2.0;
        double r35680 = r35678 / r35679;
        double r35681 = eps;
        double r35682 = 1.0;
        double r35683 = r35681 - r35682;
        double r35684 = r35666 * r35683;
        double r35685 = exp(r35684);
        double r35686 = r35685 / r35681;
        double r35687 = r35682 + r35681;
        double r35688 = exp(r35687);
        double r35689 = -r35666;
        double r35690 = pow(r35688, r35689);
        double r35691 = r35685 + r35690;
        double r35692 = r35687 * r35666;
        double r35693 = -r35692;
        double r35694 = exp(r35693);
        double r35695 = r35694 / r35681;
        double r35696 = r35691 - r35695;
        double r35697 = r35682 * r35696;
        double r35698 = fma(r35686, r35682, r35697);
        double r35699 = r35698 / r35679;
        double r35700 = r35668 ? r35680 : r35699;
        return r35700;
}

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))