Average Error: 4.4 → 0.8
Time: 6.0s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.15639988716910344 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\frac{\left({\left(\sqrt{e^{2 \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{2 \cdot x}} \cdot \sqrt{1}\right)}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.15639988716910344 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\frac{\left({\left(\sqrt{e^{2 \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{2 \cdot x}} \cdot \sqrt{1}\right)}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\

\end{array}
double f(double x) {
        double r15534 = 2.0;
        double r15535 = x;
        double r15536 = r15534 * r15535;
        double r15537 = exp(r15536);
        double r15538 = 1.0;
        double r15539 = r15537 - r15538;
        double r15540 = exp(r15535);
        double r15541 = r15540 - r15538;
        double r15542 = r15539 / r15541;
        double r15543 = sqrt(r15542);
        return r15543;
}
↓
double f(double x) {
        double r15544 = x;
        double r15545 = -1.1563998871691034e-05;
        bool r15546 = r15544 <= r15545;
        double r15547 = 2.0;
        double r15548 = r15547 * r15544;
        double r15549 = exp(r15548);
        double r15550 = sqrt(r15549);
        double r15551 = 3.0;
        double r15552 = pow(r15550, r15551);
        double r15553 = 1.0;
        double r15554 = sqrt(r15553);
        double r15555 = pow(r15554, r15551);
        double r15556 = r15552 + r15555;
        double r15557 = r15550 - r15554;
        double r15558 = r15556 * r15557;
        double r15559 = r15550 * r15550;
        double r15560 = r15554 * r15554;
        double r15561 = r15550 * r15554;
        double r15562 = r15560 - r15561;
        double r15563 = r15559 + r15562;
        double r15564 = r15558 / r15563;
        double r15565 = exp(r15544);
        double r15566 = r15565 - r15553;
        double r15567 = r15564 / r15566;
        double r15568 = sqrt(r15567);
        double r15569 = 0.5;
        double r15570 = r15569 * r15544;
        double r15571 = r15553 + r15570;
        double r15572 = r15544 * r15571;
        double r15573 = r15572 + r15547;
        double r15574 = sqrt(r15573);
        double r15575 = r15546 ? r15568 : r15574;
        return r15575;
}

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 < -1.1563998871691034e-05

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.0

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

    7. Applied flip3-+0.0

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

    8. Applied associate-*l/0.0

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

    if -1.1563998871691034e-05 < x

    1. Initial program 34.9

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

    2. Taylor expanded around 0 6.4

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

    3. Simplified6.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.15639988716910344 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\frac{\left({\left(\sqrt{e^{2 \cdot x}}\right)}^{3} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}} + \left(\sqrt{1} \cdot \sqrt{1} - \sqrt{e^{2 \cdot x}} \cdot \sqrt{1}\right)}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))