Average Error: 28.9 → 0.3
Time: 29.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.101958155932286257490046454954324417486 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(a \cdot x\right) + 1\right) \cdot \left(a \cdot x\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -1.101958155932286257490046454954324417486 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \left(a \cdot x\right) + 1\right) \cdot \left(a \cdot x\right) + \frac{1}{6} \cdot {\left(a \cdot x\right)}^{3}\\

\end{array}
double f(double a, double x) {
        double r70334 = a;
        double r70335 = x;
        double r70336 = r70334 * r70335;
        double r70337 = exp(r70336);
        double r70338 = 1.0;
        double r70339 = r70337 - r70338;
        return r70339;
}
↓
double f(double a, double x) {
        double r70340 = a;
        double r70341 = x;
        double r70342 = r70340 * r70341;
        double r70343 = -0.00011019581559322863;
        bool r70344 = r70342 <= r70343;
        double r70345 = exp(r70342);
        double r70346 = 1.0;
        double r70347 = r70345 - r70346;
        double r70348 = cbrt(r70347);
        double r70349 = r70348 * r70348;
        double r70350 = cbrt(r70348);
        double r70351 = r70350 * r70350;
        double r70352 = r70351 * r70350;
        double r70353 = r70349 * r70352;
        double r70354 = 0.5;
        double r70355 = r70354 * r70342;
        double r70356 = 1.0;
        double r70357 = r70355 + r70356;
        double r70358 = r70357 * r70342;
        double r70359 = 0.16666666666666666;
        double r70360 = 3.0;
        double r70361 = pow(r70342, r70360);
        double r70362 = r70359 * r70361;
        double r70363 = r70358 + r70362;
        double r70364 = r70344 ? r70353 : r70363;
        return r70364;
}

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-cube-cbrt0.1

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

    5. Applied add-cube-cbrt0.1

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

    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. Simplified0.4

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

  4. Final simplification0.3

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

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))