Average Error: 29.5 → 0.4
Time: 16.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.3967409376004402132842585615435382351279:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\left(x \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.3967409376004402132842585615435382351279:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot \left(e^{a \cdot x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + \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 r47439 = a;
        double r47440 = x;
        double r47441 = r47439 * r47440;
        double r47442 = exp(r47441);
        double r47443 = 1.0;
        double r47444 = r47442 - r47443;
        return r47444;
}
↓
double f(double a, double x) {
        double r47445 = a;
        double r47446 = x;
        double r47447 = r47445 * r47446;
        double r47448 = -0.3967409376004402;
        bool r47449 = r47447 <= r47448;
        double r47450 = exp(r47447);
        double r47451 = 3.0;
        double r47452 = pow(r47450, r47451);
        double r47453 = 1.0;
        double r47454 = pow(r47453, r47451);
        double r47455 = r47452 - r47454;
        double r47456 = 2.0;
        double r47457 = r47456 * r47447;
        double r47458 = exp(r47457);
        double r47459 = r47450 + r47453;
        double r47460 = r47453 * r47459;
        double r47461 = r47458 + r47460;
        double r47462 = r47455 / r47461;
        double r47463 = r47446 * r47445;
        double r47464 = r47463 * r47445;
        double r47465 = 0.16666666666666666;
        double r47466 = r47447 * r47465;
        double r47467 = 0.5;
        double r47468 = r47466 + r47467;
        double r47469 = r47464 * r47468;
        double r47470 = r47445 + r47469;
        double r47471 = r47446 * r47470;
        double r47472 = r47449 ? r47462 : r47471;
        return r47472;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.3967409376004402

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    if -0.3967409376004402 < (* a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified4.7

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

    5. Applied associate-*r*4.7

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

    6. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.3967409376004402132842585615435382351279:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\left(x \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))