Average Error: 29.8 → 0.5
Time: 14.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot a\right) \cdot x\right) \cdot a + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\
\;\;\;\;e^{a \cdot x} - 1\\

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

\end{array}
double f(double a, double x) {
        double r94895 = a;
        double r94896 = x;
        double r94897 = r94895 * r94896;
        double r94898 = exp(r94897);
        double r94899 = 1.0;
        double r94900 = r94898 - r94899;
        return r94900;
}
↓
double f(double a, double x) {
        double r94901 = a;
        double r94902 = x;
        double r94903 = r94901 * r94902;
        double r94904 = -0.012897902566370275;
        bool r94905 = r94903 <= r94904;
        double r94906 = exp(r94903);
        double r94907 = 1.0;
        double r94908 = r94906 - r94907;
        double r94909 = 0.5;
        double r94910 = r94909 * r94902;
        double r94911 = r94910 * r94901;
        double r94912 = r94911 * r94902;
        double r94913 = r94912 * r94901;
        double r94914 = r94913 + r94903;
        double r94915 = r94905 ? r94908 : r94914;
        return r94915;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.012897902566370275

    1. Initial program 0.0

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

    if -0.012897902566370275 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 15.5

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

    3. Simplified15.5

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

    4. Taylor expanded around 0 8.6

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

    5. Simplified8.6

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

    7. Applied associate-*l*4.6

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

    8. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot a\right) \cdot x\right) \cdot a + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))