Average Error: 4.4 → 0.9
Time: 21.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.293549828942968312194372434010088052064 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.293549828942968312194372434010088052064 \cdot 10^{-15}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{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 r19387 = 2.0;
        double r19388 = x;
        double r19389 = r19387 * r19388;
        double r19390 = exp(r19389);
        double r19391 = 1.0;
        double r19392 = r19390 - r19391;
        double r19393 = exp(r19388);
        double r19394 = r19393 - r19391;
        double r19395 = r19392 / r19394;
        double r19396 = sqrt(r19395);
        return r19396;
}
↓
double f(double x) {
        double r19397 = x;
        double r19398 = -1.2935498289429683e-15;
        bool r19399 = r19397 <= r19398;
        double r19400 = 2.0;
        double r19401 = r19400 * r19397;
        double r19402 = exp(r19401);
        double r19403 = sqrt(r19402);
        double r19404 = 1.0;
        double r19405 = sqrt(r19404);
        double r19406 = r19403 + r19405;
        double r19407 = exp(r19397);
        double r19408 = r19407 - r19404;
        double r19409 = exp(r19400);
        double r19410 = 2.0;
        double r19411 = r19397 / r19410;
        double r19412 = pow(r19409, r19411);
        double r19413 = r19412 - r19405;
        double r19414 = r19408 / r19413;
        double r19415 = r19406 / r19414;
        double r19416 = sqrt(r19415);
        double r19417 = 0.5;
        double r19418 = sqrt(r19400);
        double r19419 = r19397 / r19418;
        double r19420 = r19417 * r19419;
        double r19421 = pow(r19397, r19410);
        double r19422 = r19421 / r19418;
        double r19423 = 0.25;
        double r19424 = 0.125;
        double r19425 = r19424 / r19400;
        double r19426 = r19423 - r19425;
        double r19427 = r19422 * r19426;
        double r19428 = r19418 + r19427;
        double r19429 = r19420 + r19428;
        double r19430 = r19399 ? r19416 : r19429;
        return r19430;
}

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 < -1.2935498289429683e-15

    1. Initial program 0.7

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied add-sqr-sqrt0.6

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

    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. Applied associate-/l*0.2

      \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}}\]
    7. Using strategy rm

    8. Applied add-log-exp0.2

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

    9. Applied exp-to-pow0.2

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

    10. Applied sqrt-pow10.0

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

    if -1.2935498289429683e-15 < x

    1. Initial program 36.5

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

    2. Taylor expanded around 0 8.4

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

    3. Simplified8.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.293549828942968312194372434010088052064 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))