Average Error: 29.4 → 0.3
Time: 54.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0003811817474161898:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) + \left(a \cdot x + x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.0003811817474161898:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}\\

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

\end{array}
double f(double a, double x) {
        double r11305920 = a;
        double r11305921 = x;
        double r11305922 = r11305920 * r11305921;
        double r11305923 = exp(r11305922);
        double r11305924 = 1.0;
        double r11305925 = r11305923 - r11305924;
        return r11305925;
}
↓
double f(double a, double x) {
        double r11305926 = a;
        double r11305927 = x;
        double r11305928 = r11305926 * r11305927;
        double r11305929 = -0.0003811817474161898;
        bool r11305930 = r11305928 <= r11305929;
        double r11305931 = exp(r11305928);
        double r11305932 = 3.0;
        double r11305933 = pow(r11305931, r11305932);
        double r11305934 = 1.0;
        double r11305935 = r11305933 - r11305934;
        double r11305936 = r11305931 * r11305931;
        double r11305937 = r11305934 + r11305931;
        double r11305938 = r11305936 + r11305937;
        double r11305939 = r11305935 / r11305938;
        double r11305940 = 0.5;
        double r11305941 = r11305940 * r11305928;
        double r11305942 = r11305928 * r11305941;
        double r11305943 = 0.16666666666666666;
        double r11305944 = r11305926 * r11305943;
        double r11305945 = r11305928 * r11305928;
        double r11305946 = r11305944 * r11305945;
        double r11305947 = r11305927 * r11305946;
        double r11305948 = r11305928 + r11305947;
        double r11305949 = r11305942 + r11305948;
        double r11305950 = r11305930 ? r11305939 : r11305949;
        return r11305950;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0003811817474161898

    1. Initial program 0.0

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

    3. Applied flip3--0.1

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

    if -0.0003811817474161898 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 14.1

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0003811817474161898:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) + \left(a \cdot x + x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a x)
  :name "expax (section 3.5)"

  :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))