Average Error: 29.9 → 0.5
Time: 14.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.403878346472134:\\ \;\;\;\;e^{a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.403878346472134:\\
\;\;\;\;e^{a \cdot x} + -1\\

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

\end{array}
double f(double a, double x) {
        double r3797667 = a;
        double r3797668 = x;
        double r3797669 = r3797667 * r3797668;
        double r3797670 = exp(r3797669);
        double r3797671 = 1.0;
        double r3797672 = r3797670 - r3797671;
        return r3797672;
}
↓
double f(double a, double x) {
        double r3797673 = a;
        double r3797674 = x;
        double r3797675 = r3797673 * r3797674;
        double r3797676 = -3.403878346472134;
        bool r3797677 = r3797675 <= r3797676;
        double r3797678 = exp(r3797675);
        double r3797679 = -1.0;
        double r3797680 = r3797678 + r3797679;
        double r3797681 = 0.16666666666666666;
        double r3797682 = r3797681 * r3797674;
        double r3797683 = r3797682 * r3797673;
        double r3797684 = 0.5;
        double r3797685 = r3797683 + r3797684;
        double r3797686 = r3797675 * r3797675;
        double r3797687 = r3797685 * r3797686;
        double r3797688 = r3797687 + r3797675;
        double r3797689 = r3797677 ? r3797680 : r3797688;
        return r3797689;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -3.403878346472134

    1. Initial program 0

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

    3. Applied sub-neg0

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

    4. Simplified0

      \[\leadsto e^{a \cdot x} + \color{blue}{-1}\]

    if -3.403878346472134 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.403878346472134:\\ \;\;\;\;e^{a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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))