Average Error: 4.4 → 0.8
Time: 47.0s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.356135868947112289167877818840679537971 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.356135868947112289167877818840679537971 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\

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

\end{array}
double f(double x) {
        double r24892 = 2.0;
        double r24893 = x;
        double r24894 = r24892 * r24893;
        double r24895 = exp(r24894);
        double r24896 = 1.0;
        double r24897 = r24895 - r24896;
        double r24898 = exp(r24893);
        double r24899 = r24898 - r24896;
        double r24900 = r24897 / r24899;
        double r24901 = sqrt(r24900);
        return r24901;
}
↓
double f(double x) {
        double r24902 = x;
        double r24903 = -3.356135868947112e-05;
        bool r24904 = r24902 <= r24903;
        double r24905 = 2.0;
        double r24906 = r24905 * r24902;
        double r24907 = exp(r24906);
        double r24908 = sqrt(r24907);
        double r24909 = 1.0;
        double r24910 = sqrt(r24909);
        double r24911 = r24908 + r24910;
        double r24912 = r24908 - r24910;
        double r24913 = r24911 * r24912;
        double r24914 = exp(r24902);
        double r24915 = r24914 - r24909;
        double r24916 = r24913 / r24915;
        double r24917 = sqrt(r24916);
        double r24918 = 0.5;
        double r24919 = r24918 * r24902;
        double r24920 = r24919 + r24909;
        double r24921 = r24902 * r24920;
        double r24922 = r24905 + r24921;
        double r24923 = sqrt(r24922);
        double r24924 = r24904 ? r24917 : r24923;
        return r24924;
}

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 < -3.356135868947112e-05

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.0

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

    if -3.356135868947112e-05 < x

    1. Initial program 34.2

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

    2. Taylor expanded around 0 6.2

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

    3. Simplified6.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.356135868947112289167877818840679537971 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))