Average Error: 4.2 → 0.8
Time: 25.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.517756586909433448818844716085219221681 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{e^{1 \cdot x} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.517756586909433448818844716085219221681 \cdot 10^{-12}:\\
\;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{e^{1 \cdot x} - \sqrt{1}}}}\\

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

\end{array}
double f(double x) {
        double r1250771 = 2.0;
        double r1250772 = x;
        double r1250773 = r1250771 * r1250772;
        double r1250774 = exp(r1250773);
        double r1250775 = 1.0;
        double r1250776 = r1250774 - r1250775;
        double r1250777 = exp(r1250772);
        double r1250778 = r1250777 - r1250775;
        double r1250779 = r1250776 / r1250778;
        double r1250780 = sqrt(r1250779);
        return r1250780;
}
↓
double f(double x) {
        double r1250781 = x;
        double r1250782 = -2.5177565869094334e-12;
        bool r1250783 = r1250781 <= r1250782;
        double r1250784 = 1.0;
        double r1250785 = sqrt(r1250784);
        double r1250786 = 2.0;
        double r1250787 = r1250786 * r1250781;
        double r1250788 = exp(r1250787);
        double r1250789 = sqrt(r1250788);
        double r1250790 = r1250785 + r1250789;
        double r1250791 = exp(r1250781);
        double r1250792 = r1250791 - r1250784;
        double r1250793 = r1250784 * r1250781;
        double r1250794 = exp(r1250793);
        double r1250795 = r1250794 - r1250785;
        double r1250796 = r1250792 / r1250795;
        double r1250797 = r1250790 / r1250796;
        double r1250798 = sqrt(r1250797);
        double r1250799 = 0.5;
        double r1250800 = r1250781 * r1250799;
        double r1250801 = r1250784 + r1250800;
        double r1250802 = r1250781 * r1250801;
        double r1250803 = r1250786 + r1250802;
        double r1250804 = sqrt(r1250803);
        double r1250805 = r1250783 ? r1250798 : r1250804;
        return r1250805;
}

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 < -2.5177565869094334e-12

    1. Initial program 0.5

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

    3. Applied add-sqr-sqrt0.5

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

    4. Applied add-sqr-sqrt0.5

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

    5. Applied difference-of-squares0.2

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

    6. Applied associate-/l*0.1

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

    8. Applied add-log-exp0.1

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

    9. Applied exp-to-pow0.1

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

    10. Applied sqrt-pow10.0

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

    11. Taylor expanded around inf 0.0

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

    if -2.5177565869094334e-12 < x

    1. Initial program 35.3

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

    2. Taylor expanded around 0 7.4

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

    3. Simplified7.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.517756586909433448818844716085219221681 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{e^{2 \cdot x}}}{\frac{e^{x} - 1}{e^{1 \cdot x} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))