Average Error: 29.6 → 0.3
Time: 20.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00014440966996837057:\\ \;\;\;\;\sqrt[3]{\left(\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) + \left(\left(a \cdot x\right) \cdot \log \left(e^{\left(a \cdot x\right) \cdot \left(a \cdot x\right)}\right)\right) \cdot \frac{1}{6}\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00014440966996837057:\\
\;\;\;\;\sqrt[3]{\left(\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\

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

\end{array}
double f(double a, double x) {
        double r3389271 = a;
        double r3389272 = x;
        double r3389273 = r3389271 * r3389272;
        double r3389274 = exp(r3389273);
        double r3389275 = 1.0;
        double r3389276 = r3389274 - r3389275;
        return r3389276;
}
↓
double f(double a, double x) {
        double r3389277 = a;
        double r3389278 = x;
        double r3389279 = r3389277 * r3389278;
        double r3389280 = -0.00014440966996837057;
        bool r3389281 = r3389279 <= r3389280;
        double r3389282 = 1.0;
        double r3389283 = exp(r3389279);
        double r3389284 = sqrt(r3389283);
        double r3389285 = r3389282 + r3389284;
        double r3389286 = r3389284 - r3389282;
        double r3389287 = r3389285 * r3389286;
        double r3389288 = r3389283 - r3389282;
        double r3389289 = r3389288 * r3389288;
        double r3389290 = r3389287 * r3389289;
        double r3389291 = cbrt(r3389290);
        double r3389292 = 0.5;
        double r3389293 = r3389292 * r3389279;
        double r3389294 = r3389279 * r3389293;
        double r3389295 = r3389279 * r3389279;
        double r3389296 = exp(r3389295);
        double r3389297 = log(r3389296);
        double r3389298 = r3389279 * r3389297;
        double r3389299 = 0.16666666666666666;
        double r3389300 = r3389298 * r3389299;
        double r3389301 = r3389294 + r3389300;
        double r3389302 = r3389301 + r3389279;
        double r3389303 = r3389281 ? r3389291 : r3389302;
        return r3389303;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
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) < -0.00014440966996837057

    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. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied difference-of-sqr-10.1

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

    if -0.00014440966996837057 < (* a x)

    1. Initial program 44.3

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

    2. Taylor expanded around 0 13.5

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

    3. Simplified0.4

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

    5. Applied add-log-exp0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00014440966996837057:\\ \;\;\;\;\sqrt[3]{\left(\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) + \left(\left(a \cdot x\right) \cdot \log \left(e^{\left(a \cdot x\right) \cdot \left(a \cdot x\right)}\right)\right) \cdot \frac{1}{6}\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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))