Average Error: 28.8 → 0.3
Time: 46.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00032158523493504607:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00032158523493504607:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r7843543 = a;
        double r7843544 = x;
        double r7843545 = r7843543 * r7843544;
        double r7843546 = exp(r7843545);
        double r7843547 = 1.0;
        double r7843548 = r7843546 - r7843547;
        return r7843548;
}
↓
double f(double a, double x) {
        double r7843549 = a;
        double r7843550 = x;
        double r7843551 = r7843549 * r7843550;
        double r7843552 = -0.00032158523493504607;
        bool r7843553 = r7843551 <= r7843552;
        double r7843554 = exp(r7843551);
        double r7843555 = 1.0;
        double r7843556 = r7843554 - r7843555;
        double r7843557 = exp(r7843556);
        double r7843558 = log(r7843557);
        double r7843559 = 0.5;
        double r7843560 = r7843551 * r7843559;
        double r7843561 = r7843560 * r7843551;
        double r7843562 = 0.16666666666666666;
        double r7843563 = r7843549 * r7843562;
        double r7843564 = r7843551 * r7843551;
        double r7843565 = r7843563 * r7843564;
        double r7843566 = r7843550 * r7843565;
        double r7843567 = r7843566 + r7843551;
        double r7843568 = r7843561 + r7843567;
        double r7843569 = r7843553 ? r7843558 : r7843568;
        return r7843569;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.8
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) < -0.00032158523493504607

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.00032158523493504607 < (* a x)

    1. Initial program 43.9

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00032158523493504607:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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))