Average Error: 29.7 → 0.4
Time: 17.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00025504826120803785:\\ \;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00025504826120803785:\\
\;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \left(e^{a \cdot x} + 1\right) + 1}\\

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

\end{array}
double f(double a, double x) {
        double r4943833 = a;
        double r4943834 = x;
        double r4943835 = r4943833 * r4943834;
        double r4943836 = exp(r4943835);
        double r4943837 = 1.0;
        double r4943838 = r4943836 - r4943837;
        return r4943838;
}
↓
double f(double a, double x) {
        double r4943839 = a;
        double r4943840 = x;
        double r4943841 = r4943839 * r4943840;
        double r4943842 = -0.00025504826120803785;
        bool r4943843 = r4943841 <= r4943842;
        double r4943844 = -1.0;
        double r4943845 = 3.0;
        double r4943846 = r4943839 * r4943845;
        double r4943847 = r4943840 * r4943846;
        double r4943848 = exp(r4943847);
        double r4943849 = r4943844 + r4943848;
        double r4943850 = exp(r4943841);
        double r4943851 = cbrt(r4943850);
        double r4943852 = r4943851 * r4943851;
        double r4943853 = r4943852 * r4943851;
        double r4943854 = 1.0;
        double r4943855 = r4943850 + r4943854;
        double r4943856 = r4943853 * r4943855;
        double r4943857 = r4943856 + r4943854;
        double r4943858 = r4943849 / r4943857;
        double r4943859 = 0.5;
        double r4943860 = r4943841 * r4943841;
        double r4943861 = r4943859 * r4943860;
        double r4943862 = r4943841 + r4943861;
        double r4943863 = r4943841 * r4943860;
        double r4943864 = 0.16666666666666666;
        double r4943865 = r4943863 * r4943864;
        double r4943866 = r4943862 + r4943865;
        double r4943867 = r4943843 ? r4943858 : r4943866;
        return r4943867;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
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) < -0.00025504826120803785

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-cube-cbrt0.0

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

    if -0.00025504826120803785 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00025504826120803785:\\ \;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \left(e^{a \cdot x} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))