Average Error: 29.6 → 0.4
Time: 17.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01718427484698143575814199834894679952413:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.01718427484698143575814199834894679952413:\\
\;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\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 r89285 = a;
        double r89286 = x;
        double r89287 = r89285 * r89286;
        double r89288 = exp(r89287);
        double r89289 = 1.0;
        double r89290 = r89288 - r89289;
        return r89290;
}
↓
double f(double a, double x) {
        double r89291 = a;
        double r89292 = x;
        double r89293 = r89291 * r89292;
        double r89294 = -0.017184274846981436;
        bool r89295 = r89293 <= r89294;
        double r89296 = 2.0;
        double r89297 = r89296 * r89293;
        double r89298 = exp(r89297);
        double r89299 = 1.0;
        double r89300 = r89299 * r89299;
        double r89301 = r89298 - r89300;
        double r89302 = exp(r89293);
        double r89303 = r89302 + r89299;
        double r89304 = r89301 / r89303;
        double r89305 = r89293 * r89291;
        double r89306 = 0.16666666666666666;
        double r89307 = r89293 * r89306;
        double r89308 = 0.5;
        double r89309 = r89307 + r89308;
        double r89310 = r89305 * r89309;
        double r89311 = r89292 * r89310;
        double r89312 = r89293 + r89311;
        double r89313 = r89295 ? r89304 : r89312;
        return r89313;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.017184274846981436

    1. Initial program 0.0

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

    3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

    4. Simplified0.0

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

    if -0.017184274846981436 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 15.0

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

    3. Simplified4.8

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

    5. Applied associate-*r*4.8

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

    6. Simplified0.7

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

    8. Applied distribute-lft-in0.7

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

    9. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01718427484698143575814199834894679952413:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))