Average Error: 4.2 → 0.1
Time: 4.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.582886078662358144753842525975784383263 \cdot 10^{-5} \lor \neg \left(x \le 3.955846843948518696530453498533264067305 \cdot 10^{-7}\right):\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.582886078662358144753842525975784383263 \cdot 10^{-5} \lor \neg \left(x \le 3.955846843948518696530453498533264067305 \cdot 10^{-7}\right):\\
\;\;\;\;\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}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\end{array}
double f(double x) {
        double r8652 = 2.0;
        double r8653 = x;
        double r8654 = r8652 * r8653;
        double r8655 = exp(r8654);
        double r8656 = 1.0;
        double r8657 = r8655 - r8656;
        double r8658 = exp(r8653);
        double r8659 = r8658 - r8656;
        double r8660 = r8657 / r8659;
        double r8661 = sqrt(r8660);
        return r8661;
}
↓
double f(double x) {
        double r8662 = x;
        double r8663 = -1.582886078662358e-05;
        bool r8664 = r8662 <= r8663;
        double r8665 = 3.9558468439485187e-07;
        bool r8666 = r8662 <= r8665;
        double r8667 = !r8666;
        bool r8668 = r8664 || r8667;
        double r8669 = 2.0;
        double r8670 = r8669 * r8662;
        double r8671 = exp(r8670);
        double r8672 = sqrt(r8671);
        double r8673 = 1.0;
        double r8674 = sqrt(r8673);
        double r8675 = r8672 + r8674;
        double r8676 = 1.0;
        double r8677 = r8675 / r8676;
        double r8678 = sqrt(r8677);
        double r8679 = r8672 - r8674;
        double r8680 = exp(r8662);
        double r8681 = r8680 - r8673;
        double r8682 = r8679 / r8681;
        double r8683 = sqrt(r8682);
        double r8684 = r8678 * r8683;
        double r8685 = 0.5;
        double r8686 = r8685 * r8662;
        double r8687 = r8673 + r8686;
        double r8688 = r8662 * r8687;
        double r8689 = r8688 + r8669;
        double r8690 = sqrt(r8689);
        double r8691 = r8668 ? r8684 : r8690;
        return r8691;
}

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 < -1.582886078662358e-05 or 3.9558468439485187e-07 < x

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied add-sqr-sqrt0.3

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

    6. Applied difference-of-squares0.1

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

    7. Applied times-frac0.1

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

    8. Applied sqrt-prod0.1

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

    if -1.582886078662358e-05 < x < 3.9558468439485187e-07

    1. Initial program 40.4

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified0.1

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.582886078662358144753842525975784383263 \cdot 10^{-5} \lor \neg \left(x \le 3.955846843948518696530453498533264067305 \cdot 10^{-7}\right):\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))