Average Error: 29.2 → 9.4
Time: 4.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001683951068471057754133424211318015295547:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \frac{{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} - {\left(\sqrt{{1}^{3}}\right)}^{3}}{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right) + \sqrt{{\left(e^{a \cdot x}\right)}^{3}} \cdot \sqrt{{1}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.001683951068471057754133424211318015295547:\\
\;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \frac{{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} - {\left(\sqrt{{1}^{3}}\right)}^{3}}{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right) + \sqrt{{\left(e^{a \cdot x}\right)}^{3}} \cdot \sqrt{{1}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\

\end{array}
double f(double a, double x) {
        double r98837 = a;
        double r98838 = x;
        double r98839 = r98837 * r98838;
        double r98840 = exp(r98839);
        double r98841 = 1.0;
        double r98842 = r98840 - r98841;
        return r98842;
}
↓
double f(double a, double x) {
        double r98843 = a;
        double r98844 = x;
        double r98845 = r98843 * r98844;
        double r98846 = -0.0016839510684710578;
        bool r98847 = r98845 <= r98846;
        double r98848 = exp(r98845);
        double r98849 = 3.0;
        double r98850 = pow(r98848, r98849);
        double r98851 = sqrt(r98850);
        double r98852 = 1.0;
        double r98853 = pow(r98852, r98849);
        double r98854 = sqrt(r98853);
        double r98855 = r98851 + r98854;
        double r98856 = pow(r98851, r98849);
        double r98857 = pow(r98854, r98849);
        double r98858 = r98856 - r98857;
        double r98859 = r98850 + r98853;
        double r98860 = r98851 * r98854;
        double r98861 = r98859 + r98860;
        double r98862 = r98858 / r98861;
        double r98863 = r98855 * r98862;
        double r98864 = r98848 + r98852;
        double r98865 = r98848 * r98864;
        double r98866 = r98852 * r98852;
        double r98867 = r98865 + r98866;
        double r98868 = r98863 / r98867;
        double r98869 = 0.5;
        double r98870 = 2.0;
        double r98871 = pow(r98843, r98870);
        double r98872 = r98869 * r98871;
        double r98873 = r98872 * r98844;
        double r98874 = r98843 + r98873;
        double r98875 = r98844 * r98874;
        double r98876 = 0.16666666666666666;
        double r98877 = pow(r98843, r98849);
        double r98878 = pow(r98844, r98849);
        double r98879 = r98877 * r98878;
        double r98880 = r98876 * r98879;
        double r98881 = r98875 + r98880;
        double r98882 = r98847 ? r98868 : r98881;
        return r98882;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.2
Herbie9.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0016839510684710578

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

      \[\leadsto \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{\color{blue}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied difference-of-squares0.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} - \sqrt{{1}^{3}}\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\]
    9. Using strategy rm

    10. Applied flip3--0.0

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

    11. Simplified0.0

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

    if -0.0016839510684710578 < (* a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.3

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

    3. Simplified14.3

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001683951068471057754133424211318015295547:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \frac{{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}}\right)}^{3} - {\left(\sqrt{{1}^{3}}\right)}^{3}}{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right) + \sqrt{{\left(e^{a \cdot x}\right)}^{3}} \cdot \sqrt{{1}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))