Average Error: 28.8 → 0.4
Time: 47.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00014871104845393037:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \log \left(\sqrt{e^{a \cdot x}}\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00014871104845393037:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \log \left(\sqrt{e^{a \cdot x}}\right) \cdot \left(a \cdot x\right)\right)\\

\end{array}
double f(double a, double x) {
        double r4019994 = a;
        double r4019995 = x;
        double r4019996 = r4019994 * r4019995;
        double r4019997 = exp(r4019996);
        double r4019998 = 1.0;
        double r4019999 = r4019997 - r4019998;
        return r4019999;
}
↓
double f(double a, double x) {
        double r4020000 = a;
        double r4020001 = x;
        double r4020002 = r4020000 * r4020001;
        double r4020003 = -0.00014871104845393037;
        bool r4020004 = r4020002 <= r4020003;
        double r4020005 = exp(r4020002);
        double r4020006 = 1.0;
        double r4020007 = r4020005 - r4020006;
        double r4020008 = exp(r4020007);
        double r4020009 = log(r4020008);
        double r4020010 = r4020002 * r4020002;
        double r4020011 = r4020002 * r4020010;
        double r4020012 = 0.16666666666666666;
        double r4020013 = r4020011 * r4020012;
        double r4020014 = sqrt(r4020005);
        double r4020015 = log(r4020014);
        double r4020016 = r4020015 * r4020002;
        double r4020017 = r4020013 + r4020016;
        double r4020018 = r4020002 + r4020017;
        double r4020019 = r4020004 ? r4020009 : r4020018;
        return r4020019;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.8
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00014871104845393037

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.00014871104845393037 < (* a x)

    1. Initial program 43.5

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

    2. Taylor expanded around 0 13.6

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

    3. Simplified0.5

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

    5. Applied add-log-exp0.5

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

    7. Applied add-log-exp0.5

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

    8. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00014871104845393037:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \log \left(\sqrt{e^{a \cdot x}}\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))