Average Error: 29.4 → 0.6
Time: 15.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.3564141224269422036918797402904601767659:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a + \frac{1}{2} \cdot \left(a \cdot \left(a \cdot x\right)\right)\right) \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.3564141224269422036918797402904601767659:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r37853 = a;
        double r37854 = x;
        double r37855 = r37853 * r37854;
        double r37856 = exp(r37855);
        double r37857 = 1.0;
        double r37858 = r37856 - r37857;
        return r37858;
}
↓
double f(double a, double x) {
        double r37859 = a;
        double r37860 = x;
        double r37861 = r37859 * r37860;
        double r37862 = -0.3564141224269422;
        bool r37863 = r37861 <= r37862;
        double r37864 = exp(r37861);
        double r37865 = 1.0;
        double r37866 = r37864 - r37865;
        double r37867 = 0.5;
        double r37868 = r37859 * r37861;
        double r37869 = r37867 * r37868;
        double r37870 = r37859 + r37869;
        double r37871 = r37870 * r37860;
        double r37872 = r37863 ? r37866 : r37871;
        return r37872;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.3564141224269422

    1. Initial program 0.0

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

    if -0.3564141224269422 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified14.5

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

    4. Taylor expanded around 0 8.4

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

    5. Simplified8.4

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

    7. Applied associate-*l*4.8

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

    8. Simplified1.0

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

    9. Taylor expanded around inf 8.4

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

    10. Simplified1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.3564141224269422036918797402904601767659:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a + \frac{1}{2} \cdot \left(a \cdot \left(a \cdot x\right)\right)\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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))