Average Error: 4.5 → 0.8
Time: 13.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.381341772123064465938767048839104413151 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{{1}^{3} + {\left(e^{x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.381341772123064465938767048839104413151 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{{1}^{3} + {\left(e^{x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\

\end{array}
double f(double x) {
        double r19861 = 2.0;
        double r19862 = x;
        double r19863 = r19861 * r19862;
        double r19864 = exp(r19863);
        double r19865 = 1.0;
        double r19866 = r19864 - r19865;
        double r19867 = exp(r19862);
        double r19868 = r19867 - r19865;
        double r19869 = r19866 / r19868;
        double r19870 = sqrt(r19869);
        return r19870;
}
↓
double f(double x) {
        double r19871 = x;
        double r19872 = -1.3813417721230645e-16;
        bool r19873 = r19871 <= r19872;
        double r19874 = 2.0;
        double r19875 = r19874 * r19871;
        double r19876 = exp(r19875);
        double r19877 = 1.0;
        double r19878 = r19876 - r19877;
        double r19879 = r19871 + r19871;
        double r19880 = exp(r19879);
        double r19881 = r19877 * r19877;
        double r19882 = r19880 - r19881;
        double r19883 = 3.0;
        double r19884 = pow(r19877, r19883);
        double r19885 = exp(r19871);
        double r19886 = pow(r19885, r19883);
        double r19887 = r19884 + r19886;
        double r19888 = r19882 / r19887;
        double r19889 = r19885 * r19885;
        double r19890 = r19877 * r19885;
        double r19891 = r19889 - r19890;
        double r19892 = r19881 + r19891;
        double r19893 = r19888 * r19892;
        double r19894 = r19878 / r19893;
        double r19895 = sqrt(r19894);
        double r19896 = 2.0;
        double r19897 = pow(r19871, r19896);
        double r19898 = sqrt(r19874);
        double r19899 = r19897 / r19898;
        double r19900 = 0.25;
        double r19901 = 0.125;
        double r19902 = r19901 / r19874;
        double r19903 = r19900 - r19902;
        double r19904 = r19899 * r19903;
        double r19905 = 0.5;
        double r19906 = r19871 / r19898;
        double r19907 = r19905 * r19906;
        double r19908 = r19898 + r19907;
        double r19909 = r19904 + r19908;
        double r19910 = r19873 ? r19895 : r19909;
        return r19910;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3813417721230645e-16

    1. Initial program 0.9

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

    3. Applied flip--0.6

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

    4. Simplified0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}}}\]

    5. Simplified0.0

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{\color{blue}{1 + e^{x}}}}}\]
    6. Using strategy rm

    7. Applied flip3-+0.0

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

    8. Applied associate-/r/0.0

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

    if -1.3813417721230645e-16 < x

    1. Initial program 37.8

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

    2. Taylor expanded around 0 8.3

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    3. Simplified8.3

      \[\leadsto \color{blue}{\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.381341772123064465938767048839104413151 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{{1}^{3} + {\left(e^{x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))