Average Error: 4.6 → 0.9
Time: 19.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.392149690240268874105180295364103670863 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\sqrt{1} + {\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right)}^{3} + 2 \cdot \sqrt{2}}{\left(2 - \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right) \cdot \sqrt{2}\right) + \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right) \cdot \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.392149690240268874105180295364103670863 \cdot 10^{-7}:\\
\;\;\;\;\sqrt{\sqrt{1} + {\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right)}^{3} + 2 \cdot \sqrt{2}}{\left(2 - \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right) \cdot \sqrt{2}\right) + \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right) \cdot \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right)}\\

\end{array}
double f(double x) {
        double r25587 = 2.0;
        double r25588 = x;
        double r25589 = r25587 * r25588;
        double r25590 = exp(r25589);
        double r25591 = 1.0;
        double r25592 = r25590 - r25591;
        double r25593 = exp(r25588);
        double r25594 = r25593 - r25591;
        double r25595 = r25592 / r25594;
        double r25596 = sqrt(r25595);
        return r25596;
}
↓
double f(double x) {
        double r25597 = x;
        double r25598 = -3.392149690240269e-07;
        bool r25599 = r25597 <= r25598;
        double r25600 = 1.0;
        double r25601 = sqrt(r25600);
        double r25602 = exp(r25597);
        double r25603 = 2.0;
        double r25604 = 2.0;
        double r25605 = r25603 / r25604;
        double r25606 = pow(r25602, r25605);
        double r25607 = r25601 + r25606;
        double r25608 = sqrt(r25607);
        double r25609 = r25606 - r25601;
        double r25610 = r25602 - r25600;
        double r25611 = r25609 / r25610;
        double r25612 = sqrt(r25611);
        double r25613 = r25608 * r25612;
        double r25614 = r25597 * r25597;
        double r25615 = 0.1875;
        double r25616 = r25614 * r25615;
        double r25617 = sqrt(r25604);
        double r25618 = r25616 / r25617;
        double r25619 = 0.5;
        double r25620 = r25619 / r25617;
        double r25621 = r25597 * r25620;
        double r25622 = r25618 + r25621;
        double r25623 = 3.0;
        double r25624 = pow(r25622, r25623);
        double r25625 = r25604 * r25617;
        double r25626 = r25624 + r25625;
        double r25627 = r25622 * r25617;
        double r25628 = r25604 - r25627;
        double r25629 = r25622 * r25622;
        double r25630 = r25628 + r25629;
        double r25631 = r25626 / r25630;
        double r25632 = r25599 ? r25613 : r25631;
        return r25632;
}

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 < -3.392149690240269e-07

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied *-un-lft-identity0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied sqr-pow0.1

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

    7. Applied difference-of-squares0.0

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

    8. Applied times-frac0.0

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

    9. Applied sqrt-prod0.0

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

    10. Simplified0.0

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

    if -3.392149690240269e-07 < x

    1. Initial program 34.7

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

    2. Simplified29.7

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

    3. Taylor expanded around 0 6.8

      \[\leadsto \color{blue}{\left(\sqrt{2} + \left(\frac{1}{4} \cdot \frac{{x}^{2}}{\sqrt{2}} + \frac{1}{2} \cdot \frac{x}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    4. Simplified6.8

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) + \sqrt{2}}\]
    5. Using strategy rm

    6. Applied flip3-+6.7

      \[\leadsto \color{blue}{\frac{{\left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right)}^{3} + {\left(\sqrt{2}\right)}^{3}}{\left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) \cdot \left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) + \left(\sqrt{2} \cdot \sqrt{2} - \left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) \cdot \sqrt{2}\right)}}\]

    7. Simplified7.4

      \[\leadsto \frac{\color{blue}{{\left(\frac{\frac{1}{2}}{\sqrt{2}} \cdot x + \frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}}\right)}^{3} + 2 \cdot \sqrt{2}}}{\left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) \cdot \left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) + \left(\sqrt{2} \cdot \sqrt{2} - \left(\frac{\frac{1}{2} \cdot x}{\sqrt{2}} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{1}{16}\right)\right) \cdot \sqrt{2}\right)}\]

    8. Simplified6.8

      \[\leadsto \frac{{\left(\frac{\frac{1}{2}}{\sqrt{2}} \cdot x + \frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}}\right)}^{3} + 2 \cdot \sqrt{2}}{\color{blue}{\left(\frac{\frac{1}{2}}{\sqrt{2}} \cdot x + \frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}}\right) \cdot \left(\frac{\frac{1}{2}}{\sqrt{2}} \cdot x + \frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}}\right) + \left(2 - \left(\frac{\frac{1}{2}}{\sqrt{2}} \cdot x + \frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}}\right) \cdot \sqrt{2}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.392149690240268874105180295364103670863 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\sqrt{1} + {\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right)}^{3} + 2 \cdot \sqrt{2}}{\left(2 - \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right) \cdot \sqrt{2}\right) + \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right) \cdot \left(\frac{\left(x \cdot x\right) \cdot \frac{3}{16}}{\sqrt{2}} + x \cdot \frac{\frac{1}{2}}{\sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))