Average Error: 4.3 → 0.6
Time: 4.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{2 \cdot x} - 1}{e^{x} - 1} \le 3.1614723444166669 \cdot 10^{100}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{2 \cdot x} - 1}{e^{x} - 1} \le 3.1614723444166669 \cdot 10^{100}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{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 r10470 = 2.0;
        double r10471 = x;
        double r10472 = r10470 * r10471;
        double r10473 = exp(r10472);
        double r10474 = 1.0;
        double r10475 = r10473 - r10474;
        double r10476 = exp(r10471);
        double r10477 = r10476 - r10474;
        double r10478 = r10475 / r10477;
        double r10479 = sqrt(r10478);
        return r10479;
}
↓
double f(double x) {
        double r10480 = 2.0;
        double r10481 = x;
        double r10482 = r10480 * r10481;
        double r10483 = exp(r10482);
        double r10484 = 1.0;
        double r10485 = r10483 - r10484;
        double r10486 = exp(r10481);
        double r10487 = r10486 - r10484;
        double r10488 = r10485 / r10487;
        double r10489 = 3.161472344416667e+100;
        bool r10490 = r10488 <= r10489;
        double r10491 = sqrt(r10483);
        double r10492 = sqrt(r10484);
        double r10493 = r10491 + r10492;
        double r10494 = 1.0;
        double r10495 = r10493 / r10494;
        double r10496 = sqrt(r10495);
        double r10497 = r10491 - r10492;
        double r10498 = r10497 / r10487;
        double r10499 = sqrt(r10498);
        double r10500 = r10496 * r10499;
        double r10501 = 0.5;
        double r10502 = sqrt(r10480);
        double r10503 = r10481 / r10502;
        double r10504 = r10501 * r10503;
        double r10505 = 2.0;
        double r10506 = pow(r10481, r10505);
        double r10507 = r10506 / r10502;
        double r10508 = 0.25;
        double r10509 = 0.125;
        double r10510 = r10509 / r10480;
        double r10511 = r10508 - r10510;
        double r10512 = r10507 * r10511;
        double r10513 = r10502 + r10512;
        double r10514 = r10504 + r10513;
        double r10515 = r10490 ? r10500 : r10514;
        return r10515;
}

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 (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0)) < 3.161472344416667e+100

    1. Initial program 1.6

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

    3. Applied *-un-lft-identity1.6

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

    4. Applied add-sqr-sqrt1.6

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

    5. Applied add-sqr-sqrt1.3

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

    6. Applied difference-of-squares0.5

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

    7. Applied times-frac0.5

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

    8. Applied sqrt-prod0.5

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

    if 3.161472344416667e+100 < (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))

    1. Initial program 63.8

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

    2. Taylor expanded around 0 2.2

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

    3. Simplified2.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{2 \cdot x} - 1}{e^{x} - 1} \le 3.1614723444166669 \cdot 10^{100}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))