Average Error: 28.9 → 0.3
Time: 17.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.101958155932286257490046454954324417486 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(x \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.101958155932286257490046454954324417486 \cdot 10^{-4}:\\
\;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\

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

\end{array}
double f(double a, double x) {
        double r45724 = a;
        double r45725 = x;
        double r45726 = r45724 * r45725;
        double r45727 = exp(r45726);
        double r45728 = 1.0;
        double r45729 = r45727 - r45728;
        return r45729;
}
↓
double f(double a, double x) {
        double r45730 = a;
        double r45731 = x;
        double r45732 = r45730 * r45731;
        double r45733 = -0.00011019581559322863;
        bool r45734 = r45732 <= r45733;
        double r45735 = exp(r45732);
        double r45736 = 1.0;
        double r45737 = r45735 - r45736;
        double r45738 = 3.0;
        double r45739 = pow(r45737, r45738);
        double r45740 = cbrt(r45739);
        double r45741 = r45731 * r45730;
        double r45742 = r45741 * r45730;
        double r45743 = 0.16666666666666666;
        double r45744 = r45732 * r45743;
        double r45745 = 0.5;
        double r45746 = r45744 + r45745;
        double r45747 = r45742 * r45746;
        double r45748 = r45731 * r45747;
        double r45749 = r45732 + r45748;
        double r45750 = r45734 ? r45740 : r45749;
        return r45750;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.9
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00011019581559322863

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    if -0.00011019581559322863 < (* a x)

    1. Initial program 43.7

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified4.3

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left({a}^{2} \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied associate-*r*4.3

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

    6. Simplified0.4

      \[\leadsto x \cdot \left(a + \color{blue}{\left(\left(x \cdot a\right) \cdot a\right)} \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\]
    7. Using strategy rm

    8. Applied distribute-lft-in0.4

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

    9. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.101958155932286257490046454954324417486 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(x \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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