Average Error: 29.2 → 0.5
Time: 1.1m
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.441676283005617:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + a \cdot x\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.441676283005617:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r11660900 = a;
        double r11660901 = x;
        double r11660902 = r11660900 * r11660901;
        double r11660903 = exp(r11660902);
        double r11660904 = 1.0;
        double r11660905 = r11660903 - r11660904;
        return r11660905;
}
↓
double f(double a, double x) {
        double r11660906 = a;
        double r11660907 = x;
        double r11660908 = r11660906 * r11660907;
        double r11660909 = -3.441676283005617;
        bool r11660910 = r11660908 <= r11660909;
        double r11660911 = exp(r11660908);
        double r11660912 = 1.0;
        double r11660913 = r11660911 - r11660912;
        double r11660914 = 0.16666666666666666;
        double r11660915 = r11660906 * r11660914;
        double r11660916 = r11660908 * r11660908;
        double r11660917 = r11660915 * r11660916;
        double r11660918 = r11660917 * r11660907;
        double r11660919 = r11660918 + r11660908;
        double r11660920 = 0.5;
        double r11660921 = r11660908 * r11660920;
        double r11660922 = r11660921 * r11660908;
        double r11660923 = r11660919 + r11660922;
        double r11660924 = r11660910 ? r11660913 : r11660923;
        return r11660924;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -3.441676283005617

    1. Initial program 0.0

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

    if -3.441676283005617 < (* a x)

    1. Initial program 43.7

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

    2. Taylor expanded around 0 14.0

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

    3. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.441676283005617:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + a \cdot x\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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