Average Error: 29.8 → 0.4
Time: 33.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.227271540662068:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + a \cdot x\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.227271540662068:\\
\;\;\;\;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 r8091583 = a;
        double r8091584 = x;
        double r8091585 = r8091583 * r8091584;
        double r8091586 = exp(r8091585);
        double r8091587 = 1.0;
        double r8091588 = r8091586 - r8091587;
        return r8091588;
}
↓
double f(double a, double x) {
        double r8091589 = a;
        double r8091590 = x;
        double r8091591 = r8091589 * r8091590;
        double r8091592 = -1.227271540662068;
        bool r8091593 = r8091591 <= r8091592;
        double r8091594 = exp(r8091591);
        double r8091595 = 1.0;
        double r8091596 = r8091594 - r8091595;
        double r8091597 = 0.16666666666666666;
        double r8091598 = r8091589 * r8091597;
        double r8091599 = r8091591 * r8091591;
        double r8091600 = r8091598 * r8091599;
        double r8091601 = r8091600 * r8091590;
        double r8091602 = r8091601 + r8091591;
        double r8091603 = 0.5;
        double r8091604 = r8091591 * r8091603;
        double r8091605 = r8091604 * r8091591;
        double r8091606 = r8091602 + r8091605;
        double r8091607 = r8091593 ? r8091596 : r8091606;
        return r8091607;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -1.227271540662068

    1. Initial program 0

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

    if -1.227271540662068 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 14.6

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

    3. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.227271540662068:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + a \cdot x\right) + \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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))