Average Error: 29.3 → 3.2
Time: 17.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2847.816859463585387857165187597274780273:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left({a}^{2} \cdot x\right) \cdot \left(\frac{1}{2} + \left(\frac{1}{6} \cdot a\right) \cdot x\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2847.816859463585387857165187597274780273:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r72703 = a;
        double r72704 = x;
        double r72705 = r72703 * r72704;
        double r72706 = exp(r72705);
        double r72707 = 1.0;
        double r72708 = r72706 - r72707;
        return r72708;
}
↓
double f(double a, double x) {
        double r72709 = a;
        double r72710 = x;
        double r72711 = r72709 * r72710;
        double r72712 = -2847.8168594635854;
        bool r72713 = r72711 <= r72712;
        double r72714 = exp(r72711);
        double r72715 = 1.0;
        double r72716 = r72714 - r72715;
        double r72717 = exp(r72716);
        double r72718 = log(r72717);
        double r72719 = 2.0;
        double r72720 = pow(r72709, r72719);
        double r72721 = r72720 * r72710;
        double r72722 = 0.5;
        double r72723 = 0.16666666666666666;
        double r72724 = r72723 * r72709;
        double r72725 = r72724 * r72710;
        double r72726 = r72722 + r72725;
        double r72727 = r72721 * r72726;
        double r72728 = r72709 + r72727;
        double r72729 = r72710 * r72728;
        double r72730 = r72713 ? r72718 : r72729;
        return r72730;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -2847.8168594635854

    1. Initial program 0

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

    3. Applied add-log-exp0

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

    4. Applied add-log-exp0

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

    5. Applied diff-log0

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

    6. Simplified0

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

    if -2847.8168594635854 < (* a x)

    1. Initial program 43.7

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

    3. Applied add-log-exp43.7

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

    4. Applied add-log-exp43.9

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

    5. Applied diff-log43.9

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

    6. Simplified43.9

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

    7. Taylor expanded around 0 14.3

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

    8. Simplified4.7

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2847.816859463585387857165187597274780273:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left({a}^{2} \cdot x\right) \cdot \left(\frac{1}{2} + \left(\frac{1}{6} \cdot a\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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