Average Error: 29.3 → 0.3
Time: 22.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0008570408630770138:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\right) \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\frac{e^{a \cdot \left(\left(x + x\right) + x\right)} + -1}{\left(1 + e^{a \cdot x} \cdot e^{a \cdot x}\right) + e^{a \cdot x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0008570408630770138:\\
\;\;\;\;\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\right) \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\frac{e^{a \cdot \left(\left(x + x\right) + x\right)} + -1}{\left(1 + e^{a \cdot x} \cdot e^{a \cdot x}\right) + e^{a \cdot x}}}\right)\\

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

\end{array}
double f(double a, double x) {
        double r3567685 = a;
        double r3567686 = x;
        double r3567687 = r3567685 * r3567686;
        double r3567688 = exp(r3567687);
        double r3567689 = 1.0;
        double r3567690 = r3567688 - r3567689;
        return r3567690;
}
↓
double f(double a, double x) {
        double r3567691 = a;
        double r3567692 = x;
        double r3567693 = r3567691 * r3567692;
        double r3567694 = -0.0008570408630770138;
        bool r3567695 = r3567693 <= r3567694;
        double r3567696 = exp(r3567693);
        double r3567697 = 1.0;
        double r3567698 = r3567696 - r3567697;
        double r3567699 = cbrt(r3567698);
        double r3567700 = cbrt(r3567699);
        double r3567701 = r3567700 * r3567700;
        double r3567702 = r3567700 * r3567701;
        double r3567703 = r3567692 + r3567692;
        double r3567704 = r3567703 + r3567692;
        double r3567705 = r3567691 * r3567704;
        double r3567706 = exp(r3567705);
        double r3567707 = -1.0;
        double r3567708 = r3567706 + r3567707;
        double r3567709 = r3567696 * r3567696;
        double r3567710 = r3567697 + r3567709;
        double r3567711 = r3567710 + r3567696;
        double r3567712 = r3567708 / r3567711;
        double r3567713 = cbrt(r3567712);
        double r3567714 = r3567699 * r3567713;
        double r3567715 = r3567702 * r3567714;
        double r3567716 = 0.16666666666666666;
        double r3567717 = r3567693 * r3567693;
        double r3567718 = r3567693 * r3567717;
        double r3567719 = r3567716 * r3567718;
        double r3567720 = r3567719 + r3567693;
        double r3567721 = 0.5;
        double r3567722 = r3567721 * r3567693;
        double r3567723 = r3567722 * r3567693;
        double r3567724 = r3567720 + r3567723;
        double r3567725 = r3567695 ? r3567715 : r3567724;
        return r3567725;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
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) < -0.0008570408630770138

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.1

      \[\leadsto \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)}\]
    6. Using strategy rm

    7. Applied flip3--0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

    if -0.0008570408630770138 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0008570408630770138:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\right) \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\frac{e^{a \cdot \left(\left(x + x\right) + x\right)} + -1}{\left(1 + e^{a \cdot x} \cdot e^{a \cdot x}\right) + e^{a \cdot x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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))