Average Error: 4.4 → 0.7
Time: 23.6s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.43227249335068785122152625098948530713 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{1} + \sqrt{e^{2 \cdot x}}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 - \frac{0.125}{2}\right) \cdot \frac{x}{\frac{\sqrt{2}}{x}} + \frac{0.5}{\frac{\sqrt{2}}{x}}\right) + \sqrt{2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.43227249335068785122152625098948530713 \cdot 10^{-8}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{1} + \sqrt{e^{2 \cdot x}}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(0.25 - \frac{0.125}{2}\right) \cdot \frac{x}{\frac{\sqrt{2}}{x}} + \frac{0.5}{\frac{\sqrt{2}}{x}}\right) + \sqrt{2}\\

\end{array}
double f(double x) {
        double r2291547 = 2.0;
        double r2291548 = x;
        double r2291549 = r2291547 * r2291548;
        double r2291550 = exp(r2291549);
        double r2291551 = 1.0;
        double r2291552 = r2291550 - r2291551;
        double r2291553 = exp(r2291548);
        double r2291554 = r2291553 - r2291551;
        double r2291555 = r2291552 / r2291554;
        double r2291556 = sqrt(r2291555);
        return r2291556;
}
↓
double f(double x) {
        double r2291557 = x;
        double r2291558 = -2.432272493350688e-08;
        bool r2291559 = r2291557 <= r2291558;
        double r2291560 = 1.0;
        double r2291561 = sqrt(r2291560);
        double r2291562 = 2.0;
        double r2291563 = r2291562 * r2291557;
        double r2291564 = exp(r2291563);
        double r2291565 = sqrt(r2291564);
        double r2291566 = r2291561 + r2291565;
        double r2291567 = r2291565 - r2291561;
        double r2291568 = r2291566 * r2291567;
        double r2291569 = exp(r2291557);
        double r2291570 = r2291569 - r2291560;
        double r2291571 = r2291568 / r2291570;
        double r2291572 = sqrt(r2291571);
        double r2291573 = 0.25;
        double r2291574 = 0.125;
        double r2291575 = r2291574 / r2291562;
        double r2291576 = r2291573 - r2291575;
        double r2291577 = sqrt(r2291562);
        double r2291578 = r2291577 / r2291557;
        double r2291579 = r2291557 / r2291578;
        double r2291580 = r2291576 * r2291579;
        double r2291581 = 0.5;
        double r2291582 = r2291581 / r2291578;
        double r2291583 = r2291580 + r2291582;
        double r2291584 = r2291583 + r2291577;
        double r2291585 = r2291559 ? r2291572 : r2291584;
        return r2291585;
}

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 < -2.432272493350688e-08

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied add-sqr-sqrt0.2

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

    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 -2.432272493350688e-08 < x

    1. Initial program 35.6

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

    2. Taylor expanded around 0 6.0

      \[\leadsto \color{blue}{\left(\sqrt{2} + \left(0.5 \cdot \frac{x}{\sqrt{2}} + 0.25 \cdot \frac{{x}^{2}}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    3. Simplified6.0

      \[\leadsto \color{blue}{\sqrt{2} + \left(\frac{0.5}{\frac{\sqrt{2}}{x}} + \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.43227249335068785122152625098948530713 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{1} + \sqrt{e^{2 \cdot x}}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 - \frac{0.125}{2}\right) \cdot \frac{x}{\frac{\sqrt{2}}{x}} + \frac{0.5}{\frac{\sqrt{2}}{x}}\right) + \sqrt{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))