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 r11305952 = a;
        double r11305953 = x;
        double r11305954 = r11305952 * r11305953;
        double r11305955 = exp(r11305954);
        double r11305956 = 1.0;
        double r11305957 = r11305955 - r11305956;
        return r11305957;
}
↓
double f(double a, double x) {
        double r11305958 = a;
        double r11305959 = x;
        double r11305960 = r11305958 * r11305959;
        double r11305961 = -0.0003811817474161898;
        bool r11305962 = r11305960 <= r11305961;
        double r11305963 = exp(r11305960);
        double r11305964 = 3.0;
        double r11305965 = pow(r11305963, r11305964);
        double r11305966 = 1.0;
        double r11305967 = r11305965 - r11305966;
        double r11305968 = r11305963 * r11305963;
        double r11305969 = r11305966 + r11305963;
        double r11305970 = r11305968 + r11305969;
        double r11305971 = r11305967 / r11305970;
        double r11305972 = 0.5;
        double r11305973 = r11305972 * r11305960;
        double r11305974 = r11305960 * r11305973;
        double r11305975 = 0.16666666666666666;
        double r11305976 = r11305958 * r11305975;
        double r11305977 = r11305960 * r11305960;
        double r11305978 = r11305976 * r11305977;
        double r11305979 = r11305959 * r11305978;
        double r11305980 = r11305960 + r11305979;
        double r11305981 = r11305974 + r11305980;
        double r11305982 = r11305962 ? r11305971 : r11305981;
        return r11305982;
}

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))