Average Error: 4.8 → 0.9
Time: 5.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.5425646206775067 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\frac{e^{2 \cdot x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.5425646206775067 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\frac{e^{2 \cdot x} - 1 \cdot 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 r12560 = 2.0;
        double r12561 = x;
        double r12562 = r12560 * r12561;
        double r12563 = exp(r12562);
        double r12564 = 1.0;
        double r12565 = r12563 - r12564;
        double r12566 = exp(r12561);
        double r12567 = r12566 - r12564;
        double r12568 = r12565 / r12567;
        double r12569 = sqrt(r12568);
        return r12569;
}
↓
double f(double x) {
        double r12570 = x;
        double r12571 = -7.542564620677507e-11;
        bool r12572 = r12570 <= r12571;
        double r12573 = 2.0;
        double r12574 = r12573 * r12570;
        double r12575 = exp(r12574);
        double r12576 = 1.0;
        double r12577 = r12575 - r12576;
        double r12578 = 3.0;
        double r12579 = pow(r12577, r12578);
        double r12580 = cbrt(r12579);
        double r12581 = 2.0;
        double r12582 = r12581 * r12570;
        double r12583 = exp(r12582);
        double r12584 = r12576 * r12576;
        double r12585 = r12583 - r12584;
        double r12586 = exp(r12570);
        double r12587 = r12586 + r12576;
        double r12588 = r12585 / r12587;
        double r12589 = r12580 / r12588;
        double r12590 = sqrt(r12589);
        double r12591 = 0.5;
        double r12592 = sqrt(r12573);
        double r12593 = r12570 / r12592;
        double r12594 = r12591 * r12593;
        double r12595 = pow(r12570, r12581);
        double r12596 = r12595 / r12592;
        double r12597 = 0.25;
        double r12598 = 0.125;
        double r12599 = r12598 / r12573;
        double r12600 = r12597 - r12599;
        double r12601 = r12596 * r12600;
        double r12602 = r12592 + r12601;
        double r12603 = r12594 + r12602;
        double r12604 = r12572 ? r12590 : r12603;
        return r12604;
}

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 < -7.542564620677507e-11

    1. Initial program 0.4

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

    3. Applied flip--0.2

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

    5. Applied prod-exp0.0

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

    6. Simplified0.0

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

    8. Applied add-cbrt-cube0.0

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

    9. Simplified0.0

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

    if -7.542564620677507e-11 < x

    1. Initial program 36.8

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

    2. Taylor expanded around 0 7.5

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

    3. Simplified7.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.5425646206775067 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\frac{e^{2 \cdot x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))