Average Error: 4.6 → 0.8
Time: 11.7s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3409676457456808 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3409676457456808 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\

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

\end{array}
double f(double x) {
        double r14505 = 2.0;
        double r14506 = x;
        double r14507 = r14505 * r14506;
        double r14508 = exp(r14507);
        double r14509 = 1.0;
        double r14510 = r14508 - r14509;
        double r14511 = exp(r14506);
        double r14512 = r14511 - r14509;
        double r14513 = r14510 / r14512;
        double r14514 = sqrt(r14513);
        return r14514;
}
↓
double f(double x) {
        double r14515 = x;
        double r14516 = -1.3409676457456808e-05;
        bool r14517 = r14515 <= r14516;
        double r14518 = 2.0;
        double r14519 = r14518 * r14515;
        double r14520 = exp(r14519);
        double r14521 = 1.0;
        double r14522 = r14520 - r14521;
        double r14523 = r14515 + r14515;
        double r14524 = exp(r14523);
        double r14525 = r14521 * r14521;
        double r14526 = r14524 - r14525;
        double r14527 = exp(r14515);
        double r14528 = r14527 + r14521;
        double r14529 = r14526 / r14528;
        double r14530 = 3.0;
        double r14531 = pow(r14529, r14530);
        double r14532 = cbrt(r14531);
        double r14533 = r14522 / r14532;
        double r14534 = sqrt(r14533);
        double r14535 = 0.5;
        double r14536 = 2.0;
        double r14537 = pow(r14515, r14536);
        double r14538 = r14535 * r14537;
        double r14539 = r14521 * r14515;
        double r14540 = r14539 + r14518;
        double r14541 = r14538 + r14540;
        double r14542 = sqrt(r14541);
        double r14543 = r14517 ? r14534 : r14542;
        return r14543;
}

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 < -1.3409676457456808e-05

    1. Initial program 0.1

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

    3. Applied flip--0.0

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

    4. Simplified0.0

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

    6. Applied add-cbrt-cube0.0

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

    7. Applied add-cbrt-cube0.0

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

    8. Applied cbrt-undiv0.0

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

    9. Simplified0.0

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

    if -1.3409676457456808e-05 < x

    1. Initial program 34.4

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

    3. Applied flip--31.3

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

    4. Simplified21.7

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

    5. Taylor expanded around 0 6.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3409676457456808 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\sqrt[3]{{\left(\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))