Average Error: 29.9 → 0.3
Time: 28.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00015296354407290798:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot x} \cdot e^{a \cdot x} - 1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00015296354407290798:\\
\;\;\;\;\frac{\log \left(e^{e^{a \cdot x} \cdot e^{a \cdot x} - 1}\right)}{e^{a \cdot x} + 1}\\

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

\end{array}
double f(double a, double x) {
        double r4727501 = a;
        double r4727502 = x;
        double r4727503 = r4727501 * r4727502;
        double r4727504 = exp(r4727503);
        double r4727505 = 1.0;
        double r4727506 = r4727504 - r4727505;
        return r4727506;
}
↓
double f(double a, double x) {
        double r4727507 = a;
        double r4727508 = x;
        double r4727509 = r4727507 * r4727508;
        double r4727510 = -0.00015296354407290798;
        bool r4727511 = r4727509 <= r4727510;
        double r4727512 = exp(r4727509);
        double r4727513 = r4727512 * r4727512;
        double r4727514 = 1.0;
        double r4727515 = r4727513 - r4727514;
        double r4727516 = exp(r4727515);
        double r4727517 = log(r4727516);
        double r4727518 = r4727512 + r4727514;
        double r4727519 = r4727517 / r4727518;
        double r4727520 = r4727509 * r4727509;
        double r4727521 = r4727520 * r4727509;
        double r4727522 = 0.16666666666666666;
        double r4727523 = r4727521 * r4727522;
        double r4727524 = 0.5;
        double r4727525 = r4727509 * r4727524;
        double r4727526 = r4727509 * r4727525;
        double r4727527 = r4727509 + r4727526;
        double r4727528 = r4727523 + r4727527;
        double r4727529 = r4727511 ? r4727519 : r4727528;
        return r4727529;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
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) < -0.00015296354407290798

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Simplified0.1

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

    6. Applied add-log-exp0.1

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

    if -0.00015296354407290798 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00015296354407290798:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot x} \cdot e^{a \cdot x} - 1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6} + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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))