Average Error: 28.8 → 0.4
Time: 22.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01568834547608203:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + \left(\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.01568834547608203:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r2218715 = a;
        double r2218716 = x;
        double r2218717 = r2218715 * r2218716;
        double r2218718 = exp(r2218717);
        double r2218719 = 1.0;
        double r2218720 = r2218718 - r2218719;
        return r2218720;
}
↓
double f(double a, double x) {
        double r2218721 = a;
        double r2218722 = x;
        double r2218723 = r2218721 * r2218722;
        double r2218724 = -0.01568834547608203;
        bool r2218725 = r2218723 <= r2218724;
        double r2218726 = exp(r2218723);
        double r2218727 = 1.0;
        double r2218728 = r2218726 - r2218727;
        double r2218729 = exp(r2218728);
        double r2218730 = log(r2218729);
        double r2218731 = r2218723 * r2218723;
        double r2218732 = 0.5;
        double r2218733 = r2218731 * r2218732;
        double r2218734 = r2218731 * r2218723;
        double r2218735 = 0.16666666666666666;
        double r2218736 = r2218734 * r2218735;
        double r2218737 = r2218736 + r2218723;
        double r2218738 = r2218733 + r2218737;
        double r2218739 = r2218725 ? r2218730 : r2218738;
        return r2218739;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.8
Target0.2
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) < -0.01568834547608203

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.01568834547608203 < (* a x)

    1. Initial program 43.4

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

    2. Taylor expanded around 0 13.7

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

    3. Simplified0.6

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

    5. Applied distribute-rgt-in0.6

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

    6. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01568834547608203:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + \left(\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))