Average Error: 4.7 → 1.0
Time: 5.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.25354992509482999 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.25354992509482999 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}{e^{x} - 1}}\\

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

\end{array}
double f(double x) {
        double r16302 = 2.0;
        double r16303 = x;
        double r16304 = r16302 * r16303;
        double r16305 = exp(r16304);
        double r16306 = 1.0;
        double r16307 = r16305 - r16306;
        double r16308 = exp(r16303);
        double r16309 = r16308 - r16306;
        double r16310 = r16307 / r16309;
        double r16311 = sqrt(r16310);
        return r16311;
}
↓
double f(double x) {
        double r16312 = x;
        double r16313 = -8.25354992509483e-06;
        bool r16314 = r16312 <= r16313;
        double r16315 = 2.0;
        double r16316 = r16315 * r16312;
        double r16317 = exp(r16316);
        double r16318 = sqrt(r16317);
        double r16319 = 1.0;
        double r16320 = sqrt(r16319);
        double r16321 = r16318 + r16320;
        double r16322 = r16318 - r16320;
        double r16323 = 3.0;
        double r16324 = pow(r16322, r16323);
        double r16325 = cbrt(r16324);
        double r16326 = exp(r16312);
        double r16327 = r16326 - r16319;
        double r16328 = r16325 / r16327;
        double r16329 = r16321 * r16328;
        double r16330 = sqrt(r16329);
        double r16331 = 0.5;
        double r16332 = r16331 * r16312;
        double r16333 = r16319 + r16332;
        double r16334 = r16312 * r16333;
        double r16335 = r16334 + r16315;
        double r16336 = sqrt(r16335);
        double r16337 = r16314 ? r16330 : r16336;
        return r16337;
}

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 < -8.25354992509483e-06

    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-cbrt-cube0.0

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

    11. Simplified0.0

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

    if -8.25354992509483e-06 < x

    1. Initial program 33.9

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

    2. Taylor expanded around 0 7.1

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

    3. Simplified7.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.25354992509482999 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}^{3}}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))