Average Error: 29.4 → 1.0
Time: 2.3m
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2762069740.2828617:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2762069740.2828617:\\
\;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\

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

\end{array}
double f(double a, double x) {
        double r19023977 = a;
        double r19023978 = x;
        double r19023979 = r19023977 * r19023978;
        double r19023980 = exp(r19023979);
        double r19023981 = 1.0;
        double r19023982 = r19023980 - r19023981;
        return r19023982;
}
↓
double f(double a, double x) {
        double r19023983 = a;
        double r19023984 = x;
        double r19023985 = r19023983 * r19023984;
        double r19023986 = -2762069740.2828617;
        bool r19023987 = r19023985 <= r19023986;
        double r19023988 = exp(r19023985);
        double r19023989 = 1.0;
        double r19023990 = r19023988 - r19023989;
        double r19023991 = r19023990 * r19023990;
        double r19023992 = r19023990 * r19023991;
        double r19023993 = cbrt(r19023992);
        double r19023994 = r19023985 * r19023985;
        double r19023995 = 0.16666666666666666;
        double r19023996 = r19023995 * r19023983;
        double r19023997 = r19023994 * r19023996;
        double r19023998 = r19023984 * r19023997;
        double r19023999 = r19023985 + r19023998;
        double r19024000 = 0.5;
        double r19024001 = r19024000 * r19023985;
        double r19024002 = r19023985 * r19024001;
        double r19024003 = r19023999 + r19024002;
        double r19024004 = r19023987 ? r19023993 : r19024003;
        return r19024004;
}

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
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2762069740.2828617

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    if -2762069740.2828617 < (* a x)

    1. Initial program 43.3

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

    2. Taylor expanded around 0 15.1

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

    3. Simplified1.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2762069740.2828617:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{6} \cdot a\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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))