Average Error: 29.7 → 0.5
Time: 20.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.7296069399112494746617585406056605279446:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right)\right) \cdot x + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.7296069399112494746617585406056605279446:\\
\;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\

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

\end{array}
double f(double a, double x) {
        double r4790833 = a;
        double r4790834 = x;
        double r4790835 = r4790833 * r4790834;
        double r4790836 = exp(r4790835);
        double r4790837 = 1.0;
        double r4790838 = r4790836 - r4790837;
        return r4790838;
}
↓
double f(double a, double x) {
        double r4790839 = a;
        double r4790840 = x;
        double r4790841 = r4790839 * r4790840;
        double r4790842 = -0.7296069399112495;
        bool r4790843 = r4790841 <= r4790842;
        double r4790844 = exp(r4790841);
        double r4790845 = 1.0;
        double r4790846 = r4790844 - r4790845;
        double r4790847 = cbrt(r4790846);
        double r4790848 = r4790847 * r4790847;
        double r4790849 = cbrt(r4790848);
        double r4790850 = cbrt(r4790847);
        double r4790851 = r4790849 * r4790850;
        double r4790852 = r4790848 * r4790851;
        double r4790853 = 0.5;
        double r4790854 = r4790841 * r4790841;
        double r4790855 = 0.16666666666666666;
        double r4790856 = r4790839 * r4790855;
        double r4790857 = r4790854 * r4790856;
        double r4790858 = r4790857 * r4790840;
        double r4790859 = r4790858 + r4790841;
        double r4790860 = fma(r4790853, r4790854, r4790859);
        double r4790861 = r4790843 ? r4790852 : r4790860;
        return r4790861;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.7
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.7296069399112495

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    5. Applied add-cube-cbrt0.0

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

    6. Applied cbrt-prod0.0

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

    if -0.7296069399112495 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 14.9

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

    3. Simplified0.8

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

    5. Applied distribute-lft-in0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.7296069399112494746617585406056605279446:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left(a \cdot x\right) \cdot \left(a \cdot x\right), \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right)\right) \cdot x + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))