Average Error: 28.6 → 0.4
Time: 13.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00014300690794423777:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00014300690794423777:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r3816585 = a;
        double r3816586 = x;
        double r3816587 = r3816585 * r3816586;
        double r3816588 = exp(r3816587);
        double r3816589 = 1.0;
        double r3816590 = r3816588 - r3816589;
        return r3816590;
}
↓
double f(double a, double x) {
        double r3816591 = a;
        double r3816592 = x;
        double r3816593 = r3816591 * r3816592;
        double r3816594 = -0.00014300690794423777;
        bool r3816595 = r3816593 <= r3816594;
        double r3816596 = exp(r3816593);
        double r3816597 = 1.0;
        double r3816598 = r3816596 - r3816597;
        double r3816599 = r3816593 * r3816593;
        double r3816600 = r3816593 * r3816599;
        double r3816601 = 0.16666666666666666;
        double r3816602 = r3816600 * r3816601;
        double r3816603 = 0.5;
        double r3816604 = r3816599 * r3816603;
        double r3816605 = r3816602 + r3816604;
        double r3816606 = r3816605 + r3816593;
        double r3816607 = r3816595 ? r3816598 : r3816606;
        return r3816607;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.6
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) < -0.00014300690794423777

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

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

    if -0.00014300690794423777 < (* a x)

    1. Initial program 43.8

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

    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. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00014300690794423777:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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))