Average Error: 4.2 → 0.8
Time: 4.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.4779760723373188 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.4779760723373188 \cdot 10^{-11}:\\
\;\;\;\;\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}:\\
\;\;\;\;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 r10507 = 2.0;
        double r10508 = x;
        double r10509 = r10507 * r10508;
        double r10510 = exp(r10509);
        double r10511 = 1.0;
        double r10512 = r10510 - r10511;
        double r10513 = exp(r10508);
        double r10514 = r10513 - r10511;
        double r10515 = r10512 / r10514;
        double r10516 = sqrt(r10515);
        return r10516;
}
↓
double f(double x) {
        double r10517 = x;
        double r10518 = -5.477976072337319e-11;
        bool r10519 = r10517 <= r10518;
        double r10520 = 2.0;
        double r10521 = r10520 * r10517;
        double r10522 = exp(r10521);
        double r10523 = sqrt(r10522);
        double r10524 = 1.0;
        double r10525 = sqrt(r10524);
        double r10526 = r10523 + r10525;
        double r10527 = r10523 - r10525;
        double r10528 = r10526 * r10527;
        double r10529 = exp(r10517);
        double r10530 = r10529 - r10524;
        double r10531 = r10528 / r10530;
        double r10532 = sqrt(r10531);
        double r10533 = 0.5;
        double r10534 = sqrt(r10520);
        double r10535 = r10517 / r10534;
        double r10536 = r10533 * r10535;
        double r10537 = 2.0;
        double r10538 = pow(r10517, r10537);
        double r10539 = r10538 / r10534;
        double r10540 = 0.25;
        double r10541 = 0.125;
        double r10542 = r10541 / r10520;
        double r10543 = r10540 - r10542;
        double r10544 = r10539 * r10543;
        double r10545 = r10534 + r10544;
        double r10546 = r10536 + r10545;
        double r10547 = r10519 ? r10532 : r10546;
        return r10547;
}

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 < -5.477976072337319e-11

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied add-sqr-sqrt0.4

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

    5. Applied difference-of-squares0.1

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

    if -5.477976072337319e-11 < x

    1. Initial program 36.1

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

    2. Taylor expanded around 0 7.3

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

    3. Simplified7.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.4779760723373188 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))