Average Error: 4.6 → 0.9
Time: 20.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.485235983952412026150476098695918381054 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.485235983952412026150476098695918381054 \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{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\end{array}
double f(double x) {
        double r24153 = 2.0;
        double r24154 = x;
        double r24155 = r24153 * r24154;
        double r24156 = exp(r24155);
        double r24157 = 1.0;
        double r24158 = r24156 - r24157;
        double r24159 = exp(r24154);
        double r24160 = r24159 - r24157;
        double r24161 = r24158 / r24160;
        double r24162 = sqrt(r24161);
        return r24162;
}
↓
double f(double x) {
        double r24163 = x;
        double r24164 = -1.485235983952412e-05;
        bool r24165 = r24163 <= r24164;
        double r24166 = 2.0;
        double r24167 = r24166 * r24163;
        double r24168 = exp(r24167);
        double r24169 = sqrt(r24168);
        double r24170 = 1.0;
        double r24171 = sqrt(r24170);
        double r24172 = r24169 + r24171;
        double r24173 = r24169 - r24171;
        double r24174 = r24172 * r24173;
        double r24175 = exp(r24163);
        double r24176 = r24175 - r24170;
        double r24177 = r24174 / r24176;
        double r24178 = sqrt(r24177);
        double r24179 = 0.5;
        double r24180 = r24179 * r24163;
        double r24181 = r24170 + r24180;
        double r24182 = r24163 * r24181;
        double r24183 = r24182 + r24166;
        double r24184 = sqrt(r24183);
        double r24185 = r24165 ? r24178 : r24184;
        return r24185;
}

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 < -1.485235983952412e-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 -1.485235983952412e-05 < x

    1. Initial program 34.8

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

    2. Taylor expanded around 0 6.6

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

    3. Simplified6.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.485235983952412026150476098695918381054 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))