Average Error: 4.4 → 0.8
Time: 21.4s
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 r25302 = 2.0;
        double r25303 = x;
        double r25304 = r25302 * r25303;
        double r25305 = exp(r25304);
        double r25306 = 1.0;
        double r25307 = r25305 - r25306;
        double r25308 = exp(r25303);
        double r25309 = r25308 - r25306;
        double r25310 = r25307 / r25309;
        double r25311 = sqrt(r25310);
        return r25311;
}
↓
double f(double x) {
        double r25312 = x;
        double r25313 = -3.356135868947112e-05;
        bool r25314 = r25312 <= r25313;
        double r25315 = 2.0;
        double r25316 = r25315 * r25312;
        double r25317 = exp(r25316);
        double r25318 = sqrt(r25317);
        double r25319 = 1.0;
        double r25320 = sqrt(r25319);
        double r25321 = r25318 + r25320;
        double r25322 = r25318 - r25320;
        double r25323 = r25321 * r25322;
        double r25324 = exp(r25312);
        double r25325 = r25324 - r25319;
        double r25326 = r25323 / r25325;
        double r25327 = sqrt(r25326);
        double r25328 = 0.5;
        double r25329 = r25328 * r25312;
        double r25330 = r25329 + r25319;
        double r25331 = r25312 * r25330;
        double r25332 = r25315 + r25331;
        double r25333 = sqrt(r25332);
        double r25334 = r25314 ? r25327 : r25333;
        return r25334;
}

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))))