Average Error: 4.5 → 0.9
Time: 5.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3855421290877288 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left|\sqrt[3]{e^{2 \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3855421290877288 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\left(\left|\sqrt[3]{e^{2 \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\end{array}
double f(double x) {
        double r11812 = 2.0;
        double r11813 = x;
        double r11814 = r11812 * r11813;
        double r11815 = exp(r11814);
        double r11816 = 1.0;
        double r11817 = r11815 - r11816;
        double r11818 = exp(r11813);
        double r11819 = r11818 - r11816;
        double r11820 = r11817 / r11819;
        double r11821 = sqrt(r11820);
        return r11821;
}
↓
double f(double x) {
        double r11822 = x;
        double r11823 = -1.3855421290877288e-05;
        bool r11824 = r11822 <= r11823;
        double r11825 = 2.0;
        double r11826 = r11825 * r11822;
        double r11827 = exp(r11826);
        double r11828 = cbrt(r11827);
        double r11829 = fabs(r11828);
        double r11830 = sqrt(r11828);
        double r11831 = r11829 * r11830;
        double r11832 = 1.0;
        double r11833 = sqrt(r11832);
        double r11834 = r11831 + r11833;
        double r11835 = sqrt(r11827);
        double r11836 = r11835 - r11833;
        double r11837 = exp(r11822);
        double r11838 = r11837 - r11832;
        double r11839 = r11836 / r11838;
        double r11840 = r11834 * r11839;
        double r11841 = sqrt(r11840);
        double r11842 = 0.5;
        double r11843 = r11842 * r11822;
        double r11844 = r11832 + r11843;
        double r11845 = r11822 * r11844;
        double r11846 = r11845 + r11825;
        double r11847 = sqrt(r11846);
        double r11848 = r11824 ? r11841 : r11847;
        return r11848;
}

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 < -1.3855421290877288e-05

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied difference-of-squares0.0

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

    7. Applied times-frac0.0

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

    8. Simplified0.0

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

    10. Applied add-cube-cbrt0.0

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

    11. Applied sqrt-prod0.0

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

    12. Simplified0.0

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

    if -1.3855421290877288e-05 < x

    1. Initial program 34.7

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

    2. Taylor expanded around 0 7.0

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified7.0

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3855421290877288 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left|\sqrt[3]{e^{2 \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))