Average Error: 29.5 → 0.3
Time: 18.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0009957798500582489:\\ \;\;\;\;\log \left(\frac{e^{e^{a \cdot x}}}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + \left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0009957798500582489:\\
\;\;\;\;\log \left(\frac{e^{e^{a \cdot x}}}{e}\right)\\

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

\end{array}
double f(double a, double x) {
        double r4585080 = a;
        double r4585081 = x;
        double r4585082 = r4585080 * r4585081;
        double r4585083 = exp(r4585082);
        double r4585084 = 1.0;
        double r4585085 = r4585083 - r4585084;
        return r4585085;
}
↓
double f(double a, double x) {
        double r4585086 = a;
        double r4585087 = x;
        double r4585088 = r4585086 * r4585087;
        double r4585089 = -0.0009957798500582489;
        bool r4585090 = r4585088 <= r4585089;
        double r4585091 = exp(r4585088);
        double r4585092 = exp(r4585091);
        double r4585093 = exp(1.0);
        double r4585094 = r4585092 / r4585093;
        double r4585095 = log(r4585094);
        double r4585096 = 0.16666666666666666;
        double r4585097 = r4585086 * r4585096;
        double r4585098 = r4585088 * r4585088;
        double r4585099 = r4585097 * r4585098;
        double r4585100 = r4585099 * r4585087;
        double r4585101 = 0.5;
        double r4585102 = r4585088 * r4585101;
        double r4585103 = r4585102 * r4585088;
        double r4585104 = r4585103 + r4585088;
        double r4585105 = r4585100 + r4585104;
        double r4585106 = r4585090 ? r4585095 : r4585105;
        return r4585106;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0009957798500582489

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
    4. Using strategy rm

    5. Applied exp-diff0.0

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

    6. Simplified0.0

      \[\leadsto \log \left(\frac{e^{e^{a \cdot x}}}{\color{blue}{e}}\right)\]

    if -0.0009957798500582489 < (* a x)

    1. Initial program 44.8

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

    3. Applied add-log-exp44.9

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

    4. Taylor expanded around 0 14.3

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

    5. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0009957798500582489:\\ \;\;\;\;\log \left(\frac{e^{e^{a \cdot x}}}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot x + \left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot x\right) + a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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))