Average Error: 4.7 → 0.9
Time: 22.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-2434735197277449}{147573952589676412928}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 2}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-2434735197277449}{147573952589676412928}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\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}}}\\

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

\end{array}
double f(double x) {
        double r27119 = 2.0;
        double r27120 = x;
        double r27121 = r27119 * r27120;
        double r27122 = exp(r27121);
        double r27123 = 1.0;
        double r27124 = r27122 - r27123;
        double r27125 = exp(r27120);
        double r27126 = r27125 - r27123;
        double r27127 = r27124 / r27126;
        double r27128 = sqrt(r27127);
        return r27128;
}
↓
double f(double x) {
        double r27129 = x;
        double r27130 = -2434735197277449.0;
        double r27131 = 1.4757395258967641e+20;
        double r27132 = r27130 / r27131;
        bool r27133 = r27129 <= r27132;
        double r27134 = 2.0;
        double r27135 = r27134 * r27129;
        double r27136 = exp(r27135);
        double r27137 = sqrt(r27136);
        double r27138 = 1.0;
        double r27139 = sqrt(r27138);
        double r27140 = r27137 + r27139;
        double r27141 = 1.0;
        double r27142 = r27140 / r27141;
        double r27143 = sqrt(r27142);
        double r27144 = sqrt(r27137);
        double r27145 = sqrt(r27139);
        double r27146 = r27144 + r27145;
        double r27147 = exp(r27129);
        double r27148 = sqrt(r27147);
        double r27149 = r27148 + r27139;
        double r27150 = r27146 / r27149;
        double r27151 = r27144 - r27145;
        double r27152 = r27148 - r27139;
        double r27153 = r27151 / r27152;
        double r27154 = r27150 * r27153;
        double r27155 = sqrt(r27154);
        double r27156 = r27143 * r27155;
        double r27157 = r27138 / r27134;
        double r27158 = r27157 * r27129;
        double r27159 = r27138 + r27158;
        double r27160 = r27129 * r27159;
        double r27161 = r27160 + r27134;
        double r27162 = sqrt(r27161);
        double r27163 = r27133 ? r27156 : r27162;
        return r27163;
}

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 < -1.6498407439469584e-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. Applied sqrt-prod0.0

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

    10. Applied add-sqr-sqrt0.0

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

    11. Applied add-sqr-sqrt0.0

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

    12. Applied difference-of-squares0.1

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

    13. Applied add-sqr-sqrt0.1

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

    14. Applied sqrt-prod0.1

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

    15. Applied add-sqr-sqrt0.1

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

    16. Applied sqrt-prod0.1

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

    17. Applied difference-of-squares0.0

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

    18. Applied times-frac0.0

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

    if -1.6498407439469584e-05 < x

    1. Initial program 34.4

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

    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

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-2434735197277449}{147573952589676412928}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))