Average Error: 29.6 → 0.7
Time: 1.6m
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot a \le -649331.22512804926373064517974853515625:\\ \;\;\;\;\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(x \cdot a\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \cdot a \le -649331.22512804926373064517974853515625:\\
\;\;\;\;\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\\

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

\end{array}
double f(double a, double x) {
        double r4911044 = a;
        double r4911045 = x;
        double r4911046 = r4911044 * r4911045;
        double r4911047 = exp(r4911046);
        double r4911048 = 1.0;
        double r4911049 = r4911047 - r4911048;
        return r4911049;
}
↓
double f(double a, double x) {
        double r4911050 = x;
        double r4911051 = a;
        double r4911052 = r4911050 * r4911051;
        double r4911053 = -649331.2251280493;
        bool r4911054 = r4911052 <= r4911053;
        double r4911055 = exp(r4911052);
        double r4911056 = 1.0;
        double r4911057 = r4911055 - r4911056;
        double r4911058 = cbrt(r4911057);
        double r4911059 = r4911058 * r4911058;
        double r4911060 = r4911059 * r4911058;
        double r4911061 = r4911052 * r4911052;
        double r4911062 = 0.5;
        double r4911063 = 0.16666666666666666;
        double r4911064 = r4911063 * r4911052;
        double r4911065 = r4911062 + r4911064;
        double r4911066 = r4911061 * r4911065;
        double r4911067 = r4911052 + r4911066;
        double r4911068 = r4911054 ? r4911060 : r4911067;
        return r4911068;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -649331.2251280493

    1. Initial program 0

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

    3. Applied add-cube-cbrt0

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

    if -649331.2251280493 < (* a x)

    1. Initial program 43.8

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

    2. Taylor expanded around 0 14.9

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

    3. Simplified1.0

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

    4. Taylor expanded around inf 14.9

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

    5. Simplified1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot a \le -649331.22512804926373064517974853515625:\\ \;\;\;\;\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \left(x \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))