Average Error: 29.3 → 0.9
Time: 3.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.1346218148553372 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.1346218148553372 \cdot 10^{-7}:\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;x \cdot a\\

\end{array}
double f(double a, double x) {
        double r112161 = a;
        double r112162 = x;
        double r112163 = r112161 * r112162;
        double r112164 = exp(r112163);
        double r112165 = 1.0;
        double r112166 = r112164 - r112165;
        return r112166;
}
↓
double f(double a, double x) {
        double r112167 = a;
        double r112168 = x;
        double r112169 = r112167 * r112168;
        double r112170 = -2.1346218148553372e-07;
        bool r112171 = r112169 <= r112170;
        double r112172 = exp(r112169);
        double r112173 = 1.0;
        double r112174 = r112172 - r112173;
        double r112175 = cbrt(r112174);
        double r112176 = r112175 * r112175;
        double r112177 = r112176 * r112175;
        double r112178 = r112168 * r112167;
        double r112179 = r112171 ? r112177 : r112178;
        return r112179;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2.1346218148553372e-07

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    if -2.1346218148553372e-07 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified14.2

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

    4. Taylor expanded around 0 8.2

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

    5. Simplified4.6

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

    6. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{a \cdot x}\]

    7. Simplified1.3

      \[\leadsto \color{blue}{x \cdot a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.1346218148553372 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))