Average Error: 29.3 → 0.3
Time: 39.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00030732624574047915:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00030732624574047915:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\

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

\end{array}
double f(double a, double x) {
        double r7636533 = a;
        double r7636534 = x;
        double r7636535 = r7636533 * r7636534;
        double r7636536 = exp(r7636535);
        double r7636537 = 1.0;
        double r7636538 = r7636536 - r7636537;
        return r7636538;
}
↓
double f(double a, double x) {
        double r7636539 = a;
        double r7636540 = x;
        double r7636541 = r7636539 * r7636540;
        double r7636542 = -0.00030732624574047915;
        bool r7636543 = r7636541 <= r7636542;
        double r7636544 = exp(r7636541);
        double r7636545 = sqrt(r7636544);
        double r7636546 = 1.0;
        double r7636547 = r7636545 - r7636546;
        double r7636548 = r7636546 + r7636545;
        double r7636549 = r7636547 * r7636548;
        double r7636550 = r7636541 * r7636541;
        double r7636551 = 0.16666666666666666;
        double r7636552 = r7636551 * r7636539;
        double r7636553 = r7636550 * r7636552;
        double r7636554 = r7636540 * r7636553;
        double r7636555 = r7636541 + r7636554;
        double r7636556 = 0.5;
        double r7636557 = r7636556 * r7636541;
        double r7636558 = r7636541 * r7636557;
        double r7636559 = r7636555 + r7636558;
        double r7636560 = r7636543 ? r7636549 : r7636559;
        return r7636560;
}

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
Target0.1
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) < -0.00030732624574047915

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{e^{a \cdot x} - 1}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied difference-of-squares0.0

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

    if -0.00030732624574047915 < (* a x)

    1. Initial program 43.9

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

    2. Taylor expanded around 0 14.1

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00030732624574047915:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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))