Average Error: 29.1 → 13.6
Time: 21.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.768723182707655568906929830418420445558 \cdot 10^{113}:\\ \;\;\;\;\frac{e^{3 \cdot \left(a \cdot x\right)} - \left(1 \cdot 1\right) \cdot 1}{\left(e^{a \cdot x} + 1\right) \cdot e^{a \cdot x} + 1 \cdot 1}\\ \mathbf{elif}\;x \le 2.432251098386136333102546791981940133481 \cdot 10^{61}:\\ \;\;\;\;x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right) + a\right) + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{3 \cdot \left(a \cdot x\right)} - \left(1 \cdot 1\right) \cdot 1}{\left(e^{a \cdot x} + 1\right) \cdot e^{a \cdot x} + 1 \cdot 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.768723182707655568906929830418420445558 \cdot 10^{113}:\\
\;\;\;\;\frac{e^{3 \cdot \left(a \cdot x\right)} - \left(1 \cdot 1\right) \cdot 1}{\left(e^{a \cdot x} + 1\right) \cdot e^{a \cdot x} + 1 \cdot 1}\\

\mathbf{elif}\;x \le 2.432251098386136333102546791981940133481 \cdot 10^{61}:\\
\;\;\;\;x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right) + a\right) + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{3 \cdot \left(a \cdot x\right)} - \left(1 \cdot 1\right) \cdot 1}{\left(e^{a \cdot x} + 1\right) \cdot e^{a \cdot x} + 1 \cdot 1}\\

\end{array}
double f(double a, double x) {
        double r5778556 = a;
        double r5778557 = x;
        double r5778558 = r5778556 * r5778557;
        double r5778559 = exp(r5778558);
        double r5778560 = 1.0;
        double r5778561 = r5778559 - r5778560;
        return r5778561;
}
↓
double f(double a, double x) {
        double r5778562 = x;
        double r5778563 = -1.7687231827076556e+113;
        bool r5778564 = r5778562 <= r5778563;
        double r5778565 = 3.0;
        double r5778566 = a;
        double r5778567 = r5778566 * r5778562;
        double r5778568 = r5778565 * r5778567;
        double r5778569 = exp(r5778568);
        double r5778570 = 1.0;
        double r5778571 = r5778570 * r5778570;
        double r5778572 = r5778571 * r5778570;
        double r5778573 = r5778569 - r5778572;
        double r5778574 = exp(r5778567);
        double r5778575 = r5778574 + r5778570;
        double r5778576 = r5778575 * r5778574;
        double r5778577 = r5778576 + r5778571;
        double r5778578 = r5778573 / r5778577;
        double r5778579 = 2.4322510983861363e+61;
        bool r5778580 = r5778562 <= r5778579;
        double r5778581 = r5778567 * r5778567;
        double r5778582 = 0.16666666666666666;
        double r5778583 = r5778566 * r5778582;
        double r5778584 = r5778581 * r5778583;
        double r5778585 = r5778584 + r5778566;
        double r5778586 = r5778562 * r5778585;
        double r5778587 = 0.5;
        double r5778588 = r5778581 * r5778587;
        double r5778589 = r5778586 + r5778588;
        double r5778590 = r5778580 ? r5778589 : r5778578;
        double r5778591 = r5778564 ? r5778578 : r5778590;
        return r5778591;
}

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
Target0.3
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.7687231827076556e+113 or 2.4322510983861363e+61 < x

    1. Initial program 16.8

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

    3. Applied flip3--16.9

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

    4. Simplified16.8

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

    5. Simplified16.8

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

    if -1.7687231827076556e+113 < x < 2.4322510983861363e+61

    1. Initial program 34.0

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

    2. Taylor expanded around 0 19.7

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

    3. Simplified12.4

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.768723182707655568906929830418420445558 \cdot 10^{113}:\\ \;\;\;\;\frac{e^{3 \cdot \left(a \cdot x\right)} - \left(1 \cdot 1\right) \cdot 1}{\left(e^{a \cdot x} + 1\right) \cdot e^{a \cdot x} + 1 \cdot 1}\\ \mathbf{elif}\;x \le 2.432251098386136333102546791981940133481 \cdot 10^{61}:\\ \;\;\;\;x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right) + a\right) + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{3 \cdot \left(a \cdot x\right)} - \left(1 \cdot 1\right) \cdot 1}{\left(e^{a \cdot x} + 1\right) \cdot e^{a \cdot x} + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))