Average Error: 4.5 → 0.8
Time: 7.8s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.204189327707012493312837414505267474851 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\frac{\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}}{\sqrt{e^{x}} - \sqrt{1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.204189327707012493312837414505267474851 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\frac{\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}}{\sqrt{e^{x}} - \sqrt{1}}\right)}\\

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

\end{array}
double f(double x) {
        double r30519 = 2.0;
        double r30520 = x;
        double r30521 = r30519 * r30520;
        double r30522 = exp(r30521);
        double r30523 = 1.0;
        double r30524 = r30522 - r30523;
        double r30525 = exp(r30520);
        double r30526 = r30525 - r30523;
        double r30527 = r30524 / r30526;
        double r30528 = sqrt(r30527);
        return r30528;
}
↓
double f(double x) {
        double r30529 = x;
        double r30530 = -1.2041893277070125e-05;
        bool r30531 = r30529 <= r30530;
        double r30532 = 2.0;
        double r30533 = r30532 * r30529;
        double r30534 = exp(r30533);
        double r30535 = sqrt(r30534);
        double r30536 = 1.0;
        double r30537 = sqrt(r30536);
        double r30538 = r30535 + r30537;
        double r30539 = sqrt(r30535);
        double r30540 = sqrt(r30537);
        double r30541 = r30539 + r30540;
        double r30542 = exp(r30529);
        double r30543 = sqrt(r30542);
        double r30544 = r30543 + r30537;
        double r30545 = r30541 / r30544;
        double r30546 = r30539 - r30540;
        double r30547 = r30543 - r30537;
        double r30548 = r30546 / r30547;
        double r30549 = r30545 * r30548;
        double r30550 = r30538 * r30549;
        double r30551 = sqrt(r30550);
        double r30552 = 0.5;
        double r30553 = r30552 * r30529;
        double r30554 = r30536 + r30553;
        double r30555 = r30529 * r30554;
        double r30556 = r30555 + r30532;
        double r30557 = sqrt(r30556);
        double r30558 = r30531 ? r30551 : r30557;
        return r30558;
}

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 < -1.2041893277070125e-05

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied difference-of-squares0.0

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

    7. Applied times-frac0.0

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

    8. Simplified0.0

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

    10. Applied add-sqr-sqrt0.0

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

    11. Applied add-sqr-sqrt0.1

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

    12. Applied difference-of-squares0.1

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

    13. Applied add-sqr-sqrt0.1

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

    14. Applied sqrt-prod0.1

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

    15. Applied add-sqr-sqrt0.1

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

    16. Applied sqrt-prod0.1

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

    17. Applied difference-of-squares0.0

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

    18. Applied times-frac0.0

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

    if -1.2041893277070125e-05 < x

    1. Initial program 34.6

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

    2. Taylor expanded around 0 5.9

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

    3. Simplified5.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.204189327707012493312837414505267474851 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\frac{\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}}{\sqrt{e^{x}} - \sqrt{1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))