Average Error: 29.2 → 1.1
Time: 35.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -11614523509.551191:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\left(a \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + a\right) \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -11614523509.551191:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\

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

\end{array}
double f(double a, double x) {
        double r5643300 = a;
        double r5643301 = x;
        double r5643302 = r5643300 * r5643301;
        double r5643303 = exp(r5643302);
        double r5643304 = 1.0;
        double r5643305 = r5643303 - r5643304;
        return r5643305;
}
↓
double f(double a, double x) {
        double r5643306 = a;
        double r5643307 = x;
        double r5643308 = r5643306 * r5643307;
        double r5643309 = -11614523509.551191;
        bool r5643310 = r5643308 <= r5643309;
        double r5643311 = exp(r5643308);
        double r5643312 = sqrt(r5643311);
        double r5643313 = 1.0;
        double r5643314 = r5643312 - r5643313;
        double r5643315 = r5643313 + r5643312;
        double r5643316 = r5643314 * r5643315;
        double r5643317 = 0.16666666666666666;
        double r5643318 = r5643308 * r5643308;
        double r5643319 = r5643308 * r5643318;
        double r5643320 = r5643317 * r5643319;
        double r5643321 = r5643306 * r5643308;
        double r5643322 = 0.5;
        double r5643323 = r5643321 * r5643322;
        double r5643324 = r5643323 + r5643306;
        double r5643325 = r5643324 * r5643307;
        double r5643326 = r5643320 + r5643325;
        double r5643327 = r5643310 ? r5643316 : r5643326;
        return r5643327;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.2
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -11614523509.551191

    1. Initial program 0

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

    3. Applied *-un-lft-identity0

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

    4. Applied add-sqr-sqrt0

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

    5. Applied difference-of-squares0

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

    if -11614523509.551191 < (* a x)

    1. Initial program 43.0

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified1.7

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

    5. Applied *-un-lft-identity1.7

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

    6. Applied distribute-rgt-out1.7

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

    8. Applied *-commutative1.7

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

    9. Applied associate-*l*1.7

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

    10. Simplified1.7

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -11614523509.551191:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} - 1\right) \cdot \left(1 + \sqrt{e^{a \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\left(a \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + a\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))