Average Error: 29.6 → 0.3
Time: 1.7m
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0041328414439843995:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0041328414439843995:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\

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

\end{array}
double f(double a, double x) {
        double r15455659 = a;
        double r15455660 = x;
        double r15455661 = r15455659 * r15455660;
        double r15455662 = exp(r15455661);
        double r15455663 = 1.0;
        double r15455664 = r15455662 - r15455663;
        return r15455664;
}
↓
double f(double a, double x) {
        double r15455665 = a;
        double r15455666 = x;
        double r15455667 = r15455665 * r15455666;
        double r15455668 = -0.0041328414439843995;
        bool r15455669 = r15455667 <= r15455668;
        double r15455670 = exp(r15455667);
        double r15455671 = sqrt(r15455670);
        double r15455672 = 1.0;
        double r15455673 = r15455671 - r15455672;
        double r15455674 = r15455672 + r15455671;
        double r15455675 = r15455673 * r15455674;
        double r15455676 = r15455667 * r15455667;
        double r15455677 = 0.16666666666666666;
        double r15455678 = r15455677 * r15455665;
        double r15455679 = r15455676 * r15455678;
        double r15455680 = r15455666 * r15455679;
        double r15455681 = r15455667 + r15455680;
        double r15455682 = 0.5;
        double r15455683 = r15455682 * r15455667;
        double r15455684 = r15455667 * r15455683;
        double r15455685 = r15455681 + r15455684;
        double r15455686 = r15455669 ? r15455675 : r15455685;
        return r15455686;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0041328414439843995

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied add-sqr-sqrt0.0

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1 \cdot 1\]

    5. Applied difference-of-squares0.0

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

    if -0.0041328414439843995 < (* a x)

    1. Initial program 43.7

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

    2. Taylor expanded around 0 13.9

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0041328414439843995:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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))