Average Error: 29.1 → 0.3
Time: 10.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001614619952117295458710044542272044054698:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.001614619952117295458710044542272044054698:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r131794 = a;
        double r131795 = x;
        double r131796 = r131794 * r131795;
        double r131797 = exp(r131796);
        double r131798 = 1.0;
        double r131799 = r131797 - r131798;
        return r131799;
}
↓
double f(double a, double x) {
        double r131800 = a;
        double r131801 = x;
        double r131802 = r131800 * r131801;
        double r131803 = -0.0016146199521172955;
        bool r131804 = r131802 <= r131803;
        double r131805 = exp(r131802);
        double r131806 = 1.0;
        double r131807 = r131805 - r131806;
        double r131808 = exp(r131807);
        double r131809 = log(r131808);
        double r131810 = r131802 * r131800;
        double r131811 = 0.16666666666666666;
        double r131812 = r131802 * r131811;
        double r131813 = 0.5;
        double r131814 = r131812 + r131813;
        double r131815 = r131810 * r131814;
        double r131816 = r131801 * r131815;
        double r131817 = r131802 + r131816;
        double r131818 = r131804 ? r131809 : r131817;
        return r131818;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0016146199521172955

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    if -0.0016146199521172955 < (* a x)

    1. Initial program 43.6

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

    2. Taylor expanded around 0 13.7

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

    3. Simplified4.4

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

    5. Applied associate-*r*4.4

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

    6. Simplified0.5

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

    8. Applied distribute-lft-in0.5

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

    9. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001614619952117295458710044542272044054698:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + x \cdot \left(\left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))