Average Error: 4.6 → 0.8
Time: 11.6s
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 r14500 = 2.0;
        double r14501 = x;
        double r14502 = r14500 * r14501;
        double r14503 = exp(r14502);
        double r14504 = 1.0;
        double r14505 = r14503 - r14504;
        double r14506 = exp(r14501);
        double r14507 = r14506 - r14504;
        double r14508 = r14505 / r14507;
        double r14509 = sqrt(r14508);
        return r14509;
}
↓
double f(double x) {
        double r14510 = x;
        double r14511 = -1.3409676457456808e-05;
        bool r14512 = r14510 <= r14511;
        double r14513 = 2.0;
        double r14514 = r14513 * r14510;
        double r14515 = exp(r14514);
        double r14516 = 1.0;
        double r14517 = r14515 - r14516;
        double r14518 = r14510 + r14510;
        double r14519 = exp(r14518);
        double r14520 = r14516 * r14516;
        double r14521 = r14519 - r14520;
        double r14522 = exp(r14510);
        double r14523 = r14522 + r14516;
        double r14524 = r14521 / r14523;
        double r14525 = 3.0;
        double r14526 = pow(r14524, r14525);
        double r14527 = cbrt(r14526);
        double r14528 = r14517 / r14527;
        double r14529 = sqrt(r14528);
        double r14530 = 0.5;
        double r14531 = 2.0;
        double r14532 = pow(r14510, r14531);
        double r14533 = r14530 * r14532;
        double r14534 = r14516 * r14510;
        double r14535 = r14534 + r14513;
        double r14536 = r14533 + r14535;
        double r14537 = sqrt(r14536);
        double r14538 = r14512 ? r14529 : r14537;
        return r14538;
}

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))))