Average Error: 4.5 → 0.9
Time: 7.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.87791414643454749 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.87791414643454749 \cdot 10^{-6}:\\
\;\;\;\;\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 r23425 = 2.0;
        double r23426 = x;
        double r23427 = r23425 * r23426;
        double r23428 = exp(r23427);
        double r23429 = 1.0;
        double r23430 = r23428 - r23429;
        double r23431 = exp(r23426);
        double r23432 = r23431 - r23429;
        double r23433 = r23430 / r23432;
        double r23434 = sqrt(r23433);
        return r23434;
}
↓
double f(double x) {
        double r23435 = x;
        double r23436 = -6.8779141464345475e-06;
        bool r23437 = r23435 <= r23436;
        double r23438 = 2.0;
        double r23439 = r23438 * r23435;
        double r23440 = exp(r23439);
        double r23441 = sqrt(r23440);
        double r23442 = 1.0;
        double r23443 = sqrt(r23442);
        double r23444 = r23441 + r23443;
        double r23445 = r23441 - r23443;
        double r23446 = r23444 * r23445;
        double r23447 = exp(r23435);
        double r23448 = r23447 - r23442;
        double r23449 = r23446 / r23448;
        double r23450 = sqrt(r23449);
        double r23451 = 0.5;
        double r23452 = r23451 * r23435;
        double r23453 = r23442 + r23452;
        double r23454 = r23435 * r23453;
        double r23455 = r23454 + r23438;
        double r23456 = sqrt(r23455);
        double r23457 = r23437 ? r23450 : r23456;
        return r23457;
}

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 < -6.8779141464345475e-06

    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 -6.8779141464345475e-06 < x

    1. Initial program 34.1

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

    2. Taylor expanded around 0 6.5

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

    3. Simplified6.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.87791414643454749 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))