Average Error: 4.6 → 0.9
Time: 17.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.751047260344680493835030354876636238259 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{{\left(e^{x}\right)}^{2}}}{\frac{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + \left(x + x \cdot \left(x \cdot \frac{1}{2}\right)\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.751047260344680493835030354876636238259 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{{\left(e^{x}\right)}^{2}}}{\frac{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 + \left(x + x \cdot \left(x \cdot \frac{1}{2}\right)\right)}\\

\end{array}
double f(double x) {
        double r20505 = 2.0;
        double r20506 = x;
        double r20507 = r20505 * r20506;
        double r20508 = exp(r20507);
        double r20509 = 1.0;
        double r20510 = r20508 - r20509;
        double r20511 = exp(r20506);
        double r20512 = r20511 - r20509;
        double r20513 = r20510 / r20512;
        double r20514 = sqrt(r20513);
        return r20514;
}
↓
double f(double x) {
        double r20515 = x;
        double r20516 = -1.7510472603446805e-05;
        bool r20517 = r20515 <= r20516;
        double r20518 = 1.0;
        double r20519 = sqrt(r20518);
        double r20520 = exp(r20515);
        double r20521 = 2.0;
        double r20522 = pow(r20520, r20521);
        double r20523 = sqrt(r20522);
        double r20524 = r20519 + r20523;
        double r20525 = r20520 - r20518;
        double r20526 = 3.0;
        double r20527 = pow(r20525, r20526);
        double r20528 = cbrt(r20527);
        double r20529 = r20523 - r20519;
        double r20530 = r20528 / r20529;
        double r20531 = r20524 / r20530;
        double r20532 = sqrt(r20531);
        double r20533 = 2.0;
        double r20534 = 0.5;
        double r20535 = r20515 * r20534;
        double r20536 = r20515 * r20535;
        double r20537 = r20515 + r20536;
        double r20538 = r20533 + r20537;
        double r20539 = sqrt(r20538);
        double r20540 = r20517 ? r20532 : r20539;
        return r20540;
}

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 < -1.7510472603446805e-05

    1. Initial program 0.1

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

    2. Simplified0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied difference-of-squares0.0

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

    7. Applied associate-/l*0.0

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

    9. Applied add-cbrt-cube0.0

      \[\leadsto \sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} + \sqrt{1}}{\frac{\color{blue}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}}{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}}}\]

    10. Simplified0.0

      \[\leadsto \sqrt{\frac{\sqrt{{\left(e^{x}\right)}^{2}} + \sqrt{1}}{\frac{\sqrt[3]{\color{blue}{{\left(e^{x} - 1\right)}^{3}}}}{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}}}\]

    if -1.7510472603446805e-05 < x

    1. Initial program 34.1

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

    2. Simplified29.5

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

    3. Taylor expanded around 0 6.7

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

    4. Simplified6.7

      \[\leadsto \sqrt{\color{blue}{2 + \left(\left(\frac{1}{2} \cdot x\right) \cdot x + x\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.751047260344680493835030354876636238259 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\sqrt{1} + \sqrt{{\left(e^{x}\right)}^{2}}}{\frac{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}{\sqrt{{\left(e^{x}\right)}^{2}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + \left(x + x \cdot \left(x \cdot \frac{1}{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))