Average Error: 4.4 → 0.8
Time: 6.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.9771617419782324 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.9771617419782324 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}\right)}{e^{x} - 1}}\\

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

\end{array}
double f(double x) {
        double r15390 = 2.0;
        double r15391 = x;
        double r15392 = r15390 * r15391;
        double r15393 = exp(r15392);
        double r15394 = 1.0;
        double r15395 = r15393 - r15394;
        double r15396 = exp(r15391);
        double r15397 = r15396 - r15394;
        double r15398 = r15395 / r15397;
        double r15399 = sqrt(r15398);
        return r15399;
}
↓
double f(double x) {
        double r15400 = x;
        double r15401 = -6.977161741978232e-16;
        bool r15402 = r15400 <= r15401;
        double r15403 = 2.0;
        double r15404 = r15403 * r15400;
        double r15405 = exp(r15404);
        double r15406 = sqrt(r15405);
        double r15407 = 1.0;
        double r15408 = sqrt(r15407);
        double r15409 = r15406 + r15408;
        double r15410 = exp(r15403);
        double r15411 = 2.0;
        double r15412 = r15400 / r15411;
        double r15413 = pow(r15410, r15412);
        double r15414 = r15413 - r15408;
        double r15415 = r15409 * r15414;
        double r15416 = exp(r15400);
        double r15417 = r15416 - r15407;
        double r15418 = r15415 / r15417;
        double r15419 = sqrt(r15418);
        double r15420 = 0.5;
        double r15421 = sqrt(r15403);
        double r15422 = r15400 / r15421;
        double r15423 = r15420 * r15422;
        double r15424 = pow(r15400, r15411);
        double r15425 = r15424 / r15421;
        double r15426 = 0.25;
        double r15427 = 0.125;
        double r15428 = r15427 / r15403;
        double r15429 = r15426 - r15428;
        double r15430 = r15425 * r15429;
        double r15431 = r15421 + r15430;
        double r15432 = r15423 + r15431;
        double r15433 = r15402 ? r15419 : r15432;
        return r15433;
}

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 < -6.977161741978232e-16

    1. Initial program 0.8

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

    3. Applied add-sqr-sqrt0.8

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

    4. Applied add-sqr-sqrt0.7

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

    5. Applied difference-of-squares0.2

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{e^{x} - 1}}\]
    6. Using strategy rm

    7. Applied add-log-exp0.2

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

    8. Applied exp-to-pow0.2

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

    9. Applied sqrt-pow10.0

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

    if -6.977161741978232e-16 < x

    1. Initial program 36.4

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

    2. Taylor expanded around 0 7.9

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

    3. Simplified7.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.9771617419782324 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))