Average Error: 29.4 → 0.4
Time: 34.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00015435000108057162:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot x} \cdot e^{a \cdot x} - 1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00015435000108057162:\\
\;\;\;\;\frac{\log \left(e^{e^{a \cdot x} \cdot e^{a \cdot x} - 1}\right)}{e^{a \cdot x} + 1}\\

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

\end{array}
double f(double a, double x) {
        double r4496680 = a;
        double r4496681 = x;
        double r4496682 = r4496680 * r4496681;
        double r4496683 = exp(r4496682);
        double r4496684 = 1.0;
        double r4496685 = r4496683 - r4496684;
        return r4496685;
}
↓
double f(double a, double x) {
        double r4496686 = a;
        double r4496687 = x;
        double r4496688 = r4496686 * r4496687;
        double r4496689 = -0.00015435000108057162;
        bool r4496690 = r4496688 <= r4496689;
        double r4496691 = exp(r4496688);
        double r4496692 = r4496691 * r4496691;
        double r4496693 = 1.0;
        double r4496694 = r4496692 - r4496693;
        double r4496695 = exp(r4496694);
        double r4496696 = log(r4496695);
        double r4496697 = r4496691 + r4496693;
        double r4496698 = r4496696 / r4496697;
        double r4496699 = 0.16666666666666666;
        double r4496700 = r4496688 * r4496688;
        double r4496701 = r4496688 * r4496700;
        double r4496702 = r4496699 * r4496701;
        double r4496703 = 0.5;
        double r4496704 = r4496703 * r4496688;
        double r4496705 = r4496688 * r4496704;
        double r4496706 = r4496702 + r4496705;
        double r4496707 = r4496706 + r4496688;
        double r4496708 = r4496690 ? r4496698 : r4496707;
        return r4496708;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00015435000108057162

    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 -0.00015435000108057162 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00015435000108057162:\\ \;\;\;\;\frac{\log \left(e^{e^{a \cdot x} \cdot e^{a \cdot x} - 1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))