Average Error: 29.8 → 0.3
Time: 15.8s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001397225479405418777784086969973031955305:\\ \;\;\;\;\log \left(e^{\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.001397225479405418777784086969973031955305:\\
\;\;\;\;\log \left(e^{\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + \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 r98792 = a;
        double r98793 = x;
        double r98794 = r98792 * r98793;
        double r98795 = exp(r98794);
        double r98796 = 1.0;
        double r98797 = r98795 - r98796;
        return r98797;
}
↓
double f(double a, double x) {
        double r98798 = a;
        double r98799 = x;
        double r98800 = r98798 * r98799;
        double r98801 = -0.0013972254794054188;
        bool r98802 = r98800 <= r98801;
        double r98803 = exp(r98800);
        double r98804 = 1.0;
        double r98805 = r98803 - r98804;
        double r98806 = 3.0;
        double r98807 = pow(r98805, r98806);
        double r98808 = cbrt(r98807);
        double r98809 = exp(r98808);
        double r98810 = log(r98809);
        double r98811 = r98800 * r98798;
        double r98812 = 0.16666666666666666;
        double r98813 = r98800 * r98812;
        double r98814 = 0.5;
        double r98815 = r98813 + r98814;
        double r98816 = r98811 * r98815;
        double r98817 = r98798 + r98816;
        double r98818 = r98799 * r98817;
        double r98819 = r98802 ? r98810 : r98818;
        return r98819;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.1
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) < -0.0013972254794054188

    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

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

    8. Applied add-cbrt-cube0.0

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

    9. Simplified0.0

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

    if -0.0013972254794054188 < (* a x)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 15.0

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

    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. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001397225479405418777784086969973031955305:\\ \;\;\;\;\log \left(e^{\sqrt[3]{{\left(e^{a \cdot x} - 1\right)}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\left(a \cdot x\right) \cdot a\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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))