Average Error: 29.6 → 5.3
Time: 16.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\ \;\;\;\;\sqrt[3]{{\left({\left(e^{a}\right)}^{x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot \left(\sqrt[3]{a} \cdot x\right)\right) \cdot {\left(\sqrt[3]{a}\right)}^{2}, a\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\
\;\;\;\;\sqrt[3]{{\left({\left(e^{a}\right)}^{x} - 1\right)}^{3}}\\

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

\end{array}
double f(double a, double x) {
        double r98688 = a;
        double r98689 = x;
        double r98690 = r98688 * r98689;
        double r98691 = exp(r98690);
        double r98692 = 1.0;
        double r98693 = r98691 - r98692;
        return r98693;
}
↓
double f(double a, double x) {
        double r98694 = a;
        double r98695 = x;
        double r98696 = r98694 * r98695;
        double r98697 = -173.65326083361478;
        bool r98698 = r98696 <= r98697;
        double r98699 = exp(r98694);
        double r98700 = pow(r98699, r98695);
        double r98701 = 1.0;
        double r98702 = r98700 - r98701;
        double r98703 = 3.0;
        double r98704 = pow(r98702, r98703);
        double r98705 = cbrt(r98704);
        double r98706 = 0.5;
        double r98707 = cbrt(r98694);
        double r98708 = r98707 * r98695;
        double r98709 = r98694 * r98708;
        double r98710 = 2.0;
        double r98711 = pow(r98707, r98710);
        double r98712 = r98709 * r98711;
        double r98713 = fma(r98706, r98712, r98694);
        double r98714 = r98695 * r98713;
        double r98715 = r98698 ? r98705 : r98714;
        return r98715;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.6
Target0.2
Herbie5.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -173.65326083361478

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Simplified0

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

    6. Applied add-log-exp14.3

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

    7. Applied exp-to-pow14.3

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

    if -173.65326083361478 < (* a x)

    1. Initial program 44.1

      \[e^{a \cdot x} - 1\]

    2. Taylor expanded around 0 14.4

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

    3. Simplified11.7

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

    4. Taylor expanded around 0 8.9

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

    5. Simplified5.1

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

    7. Applied add-cube-cbrt5.1

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

    8. Applied unpow-prod-down5.1

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

    9. Applied associate-*r*2.0

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

    10. Simplified1.0

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -173.653260833614780267453170381486415863:\\ \;\;\;\;\sqrt[3]{{\left({\left(e^{a}\right)}^{x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{1}{2}, \left(a \cdot \left(\sqrt[3]{a} \cdot x\right)\right) \cdot {\left(\sqrt[3]{a}\right)}^{2}, a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))