Average Error: 29.8 → 7.5
Time: 17.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(e^{2}\right)}^{\left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2}, \left(x \cdot {a}^{3}\right) \cdot \frac{1}{6}\right), a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\
\;\;\;\;\sqrt[3]{\frac{{\left(e^{2}\right)}^{\left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2}, \left(x \cdot {a}^{3}\right) \cdot \frac{1}{6}\right), a \cdot x\right)\\

\end{array}
double f(double a, double x) {
        double r65740 = a;
        double r65741 = x;
        double r65742 = r65740 * r65741;
        double r65743 = exp(r65742);
        double r65744 = 1.0;
        double r65745 = r65743 - r65744;
        return r65745;
}
↓
double f(double a, double x) {
        double r65746 = a;
        double r65747 = x;
        double r65748 = r65746 * r65747;
        double r65749 = -8.293963492047655e-20;
        bool r65750 = r65748 <= r65749;
        double r65751 = 2.0;
        double r65752 = exp(r65751);
        double r65753 = pow(r65752, r65748);
        double r65754 = 1.0;
        double r65755 = r65754 * r65754;
        double r65756 = r65753 - r65755;
        double r65757 = exp(r65748);
        double r65758 = r65757 + r65754;
        double r65759 = r65756 / r65758;
        double r65760 = cbrt(r65759);
        double r65761 = r65757 - r65754;
        double r65762 = cbrt(r65761);
        double r65763 = r65751 * r65748;
        double r65764 = exp(r65763);
        double r65765 = r65764 - r65755;
        double r65766 = cbrt(r65765);
        double r65767 = cbrt(r65758);
        double r65768 = r65766 / r65767;
        double r65769 = r65762 * r65768;
        double r65770 = r65760 * r65769;
        double r65771 = pow(r65747, r65751);
        double r65772 = 0.5;
        double r65773 = pow(r65746, r65751);
        double r65774 = 3.0;
        double r65775 = pow(r65746, r65774);
        double r65776 = r65747 * r65775;
        double r65777 = 0.16666666666666666;
        double r65778 = r65776 * r65777;
        double r65779 = fma(r65772, r65773, r65778);
        double r65780 = fma(r65771, r65779, r65748);
        double r65781 = r65750 ? r65770 : r65780;
        return r65781;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.8
Target0.1
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.293963492047655e-20

    1. Initial program 1.9

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-log-exp1.9

      \[\leadsto e^{a \cdot x} - \color{blue}{\log \left(e^{1}\right)}\]

    4. Applied add-log-exp1.9

      \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x}}\right)} - \log \left(e^{1}\right)\]

    5. Applied diff-log2.0

      \[\leadsto \color{blue}{\log \left(\frac{e^{e^{a \cdot x}}}{e^{1}}\right)}\]

    6. Simplified1.9

      \[\leadsto \log \color{blue}{\left(e^{e^{a \cdot x} - 1}\right)}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.9

      \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}}\right)\]

    9. Applied exp-prod1.9

      \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right)}^{\left(\sqrt[3]{e^{a \cdot x} - 1}\right)}\right)}\]

    10. Applied log-pow1.9

      \[\leadsto \color{blue}{\sqrt[3]{e^{a \cdot x} - 1} \cdot \log \left(e^{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}}\right)}\]

    11. Simplified1.9

      \[\leadsto \sqrt[3]{e^{a \cdot x} - 1} \cdot \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right)}\]
    12. Using strategy rm

    13. Applied flip--1.9

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

    14. Applied cbrt-div1.9

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

    15. Simplified1.9

      \[\leadsto \sqrt[3]{e^{a \cdot x} - 1} \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\color{blue}{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}}{\sqrt[3]{e^{a \cdot x} + 1}}\right)\]
    16. Using strategy rm

    17. Applied flip--1.9

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

    18. Simplified1.8

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

    if -8.293963492047655e-20 < (* a x)

    1. Initial program 45.3

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-log-exp45.3

      \[\leadsto e^{a \cdot x} - \color{blue}{\log \left(e^{1}\right)}\]

    4. Applied add-log-exp45.5

      \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x}}\right)} - \log \left(e^{1}\right)\]

    5. Applied diff-log45.5

      \[\leadsto \color{blue}{\log \left(\frac{e^{e^{a \cdot x}}}{e^{1}}\right)}\]

    6. Simplified45.5

      \[\leadsto \log \color{blue}{\left(e^{e^{a \cdot x} - 1}\right)}\]

    7. Taylor expanded around 0 14.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    8. Simplified10.6

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2}, \left(x \cdot {a}^{3}\right) \cdot \frac{1}{6}\right), a \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.293963492047654929060613942291516367562 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(e^{2}\right)}^{\left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}} \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2}, \left(x \cdot {a}^{3}\right) \cdot \frac{1}{6}\right), a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))