Average Error: 4.9 → 0.8
Time: 25.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.639709207461163649661474439600539199091 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.639709207461163649661474439600539199091 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}\right)}^{3}}}}\\

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

\end{array}
double f(double x) {
        double r23169 = 2.0;
        double r23170 = x;
        double r23171 = r23169 * r23170;
        double r23172 = exp(r23171);
        double r23173 = 1.0;
        double r23174 = r23172 - r23173;
        double r23175 = exp(r23170);
        double r23176 = r23175 - r23173;
        double r23177 = r23174 / r23176;
        double r23178 = sqrt(r23177);
        return r23178;
}
↓
double f(double x) {
        double r23179 = x;
        double r23180 = -1.6397092074611636e-05;
        bool r23181 = r23179 <= r23180;
        double r23182 = 2.0;
        double r23183 = r23182 * r23179;
        double r23184 = exp(r23183);
        double r23185 = 1.0;
        double r23186 = r23184 - r23185;
        double r23187 = r23179 + r23179;
        double r23188 = exp(r23187);
        double r23189 = r23185 * r23185;
        double r23190 = r23188 - r23189;
        double r23191 = exp(r23179);
        double r23192 = r23185 + r23191;
        double r23193 = r23190 / r23192;
        double r23194 = 3.0;
        double r23195 = pow(r23193, r23194);
        double r23196 = cbrt(r23195);
        double r23197 = r23186 / r23196;
        double r23198 = sqrt(r23197);
        double r23199 = 0.5;
        double r23200 = r23199 * r23179;
        double r23201 = r23200 + r23185;
        double r23202 = r23179 * r23201;
        double r23203 = r23182 + r23202;
        double r23204 = sqrt(r23203);
        double r23205 = r23181 ? r23198 : r23204;
        return r23205;
}

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 < -1.6397092074611636e-05

    1. Initial program 0.1

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

    3. Applied flip--0.0

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

    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 add-cbrt-cube0.0

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

    8. Applied add-cbrt-cube0.0

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

    9. Applied cbrt-undiv0.0

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

    10. Simplified0.0

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

    if -1.6397092074611636e-05 < x

    1. Initial program 35.2

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

    2. Taylor expanded around 0 5.9

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

    3. Simplified5.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.639709207461163649661474439600539199091 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))