Average Error: 29.4 → 0.4
Time: 10.2s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r63041 = a;
        double r63042 = x;
        double r63043 = r63041 * r63042;
        double r63044 = exp(r63043);
        double r63045 = 1.0;
        double r63046 = r63044 - r63045;
        return r63046;
}
↓
double f(double a, double x) {
        double r63047 = a;
        double r63048 = x;
        double r63049 = r63047 * r63048;
        double r63050 = -0.0009427648001274835;
        bool r63051 = r63049 <= r63050;
        double r63052 = exp(r63049);
        double r63053 = sqrt(r63052);
        double r63054 = 1.0;
        double r63055 = sqrt(r63054);
        double r63056 = r63053 + r63055;
        double r63057 = r63053 - r63055;
        double r63058 = r63056 * r63057;
        double r63059 = 0.5;
        double r63060 = 2.0;
        double r63061 = r63060 / r63060;
        double r63062 = pow(r63047, r63061);
        double r63063 = r63048 * r63049;
        double r63064 = r63062 * r63063;
        double r63065 = r63059 * r63064;
        double r63066 = r63065 + r63049;
        double r63067 = r63051 ? r63058 : r63066;
        return r63067;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0009427648001274835

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    if -0.0009427648001274835 < (* a x)

    1. Initial program 44.3

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified7.5

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

    4. Taylor expanded around 0 8.4

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

    6. Applied sqr-pow8.4

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

    7. Applied associate-*l*4.9

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

    8. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left({a}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))