Average Error: 29.1 → 9.6
Time: 13.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -7.166980254436657516383787675125359952696 \cdot 10^{-23}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -7.166980254436657516383787675125359952696 \cdot 10^{-23}:\\
\;\;\;\;e^{a \cdot x} - 1\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double a, double x) {
        double r89324 = a;
        double r89325 = x;
        double r89326 = r89324 * r89325;
        double r89327 = exp(r89326);
        double r89328 = 1.0;
        double r89329 = r89327 - r89328;
        return r89329;
}
↓
double f(double a, double x) {
        double r89330 = a;
        double r89331 = x;
        double r89332 = r89330 * r89331;
        double r89333 = -7.166980254436658e-23;
        bool r89334 = r89332 <= r89333;
        double r89335 = exp(r89332);
        double r89336 = 1.0;
        double r89337 = r89335 - r89336;
        double r89338 = 0.5;
        double r89339 = 2.0;
        double r89340 = pow(r89330, r89339);
        double r89341 = r89338 * r89340;
        double r89342 = r89341 * r89331;
        double r89343 = r89330 + r89342;
        double r89344 = r89331 * r89343;
        double r89345 = 0.16666666666666666;
        double r89346 = 3.0;
        double r89347 = pow(r89330, r89346);
        double r89348 = pow(r89331, r89346);
        double r89349 = r89347 * r89348;
        double r89350 = r89345 * r89349;
        double r89351 = r89344 + r89350;
        double r89352 = r89334 ? r89337 : r89351;
        return r89352;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.1
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -7.166980254436658e-23

    1. Initial program 2.1

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

    if -7.166980254436658e-23 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified13.8

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -7.166980254436657516383787675125359952696 \cdot 10^{-23}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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