Average Error: 4.7 → 0.1
Time: 26.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.013702749797138520368410019867925313981 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}}\\ \mathbf{elif}\;x \le 5.535044263275191921405584805585021641155 \cdot 10^{-17}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.013702749797138520368410019867925313981 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}}\\

\mathbf{elif}\;x \le 5.535044263275191921405584805585021641155 \cdot 10^{-17}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}}\\

\end{array}
double f(double x) {
        double r27123 = 2.0;
        double r27124 = x;
        double r27125 = r27123 * r27124;
        double r27126 = exp(r27125);
        double r27127 = 1.0;
        double r27128 = r27126 - r27127;
        double r27129 = exp(r27124);
        double r27130 = r27129 - r27127;
        double r27131 = r27128 / r27130;
        double r27132 = sqrt(r27131);
        return r27132;
}
↓
double f(double x) {
        double r27133 = x;
        double r27134 = -4.0137027497971385e-17;
        bool r27135 = r27133 <= r27134;
        double r27136 = 2.0;
        double r27137 = r27136 * r27133;
        double r27138 = exp(r27137);
        double r27139 = 1.0;
        double r27140 = r27138 - r27139;
        double r27141 = 3.0;
        double r27142 = pow(r27140, r27141);
        double r27143 = cbrt(r27142);
        double r27144 = r27133 + r27133;
        double r27145 = exp(r27144);
        double r27146 = r27139 * r27139;
        double r27147 = r27145 - r27146;
        double r27148 = exp(r27133);
        double r27149 = r27139 + r27148;
        double r27150 = r27147 / r27149;
        double r27151 = r27143 / r27150;
        double r27152 = sqrt(r27151);
        double r27153 = 5.535044263275192e-17;
        bool r27154 = r27133 <= r27153;
        double r27155 = cbrt(r27133);
        double r27156 = r27155 * r27155;
        double r27157 = 2.0;
        double r27158 = pow(r27156, r27157);
        double r27159 = sqrt(r27136);
        double r27160 = sqrt(r27159);
        double r27161 = r27158 / r27160;
        double r27162 = pow(r27155, r27157);
        double r27163 = r27162 / r27160;
        double r27164 = 0.25;
        double r27165 = 0.125;
        double r27166 = r27165 / r27136;
        double r27167 = r27164 - r27166;
        double r27168 = r27163 * r27167;
        double r27169 = r27161 * r27168;
        double r27170 = 0.5;
        double r27171 = r27133 / r27159;
        double r27172 = r27170 * r27171;
        double r27173 = r27159 + r27172;
        double r27174 = r27169 + r27173;
        double r27175 = r27140 / r27150;
        double r27176 = sqrt(r27175);
        double r27177 = r27154 ? r27174 : r27176;
        double r27178 = r27135 ? r27152 : r27177;
        return r27178;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4.0137027497971385e-17

    1. Initial program 0.9

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

    3. Applied flip--0.7

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-cbrt-cube0.0

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

    8. Simplified0.0

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

    if -4.0137027497971385e-17 < x < 5.535044263275192e-17

    1. Initial program 64.0

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

    2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]

    3. Simplified0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0

      \[\leadsto \frac{{x}^{2}}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    6. Applied sqrt-prod0

      \[\leadsto \frac{{x}^{2}}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    7. Applied add-cube-cbrt0

      \[\leadsto \frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    8. Applied unpow-prod-down0

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    9. Applied times-frac0

      \[\leadsto \color{blue}{\left(\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}}\right)} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    10. Applied associate-*l*0

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\]

    if 5.535044263275192e-17 < x

    1. Initial program 17.2

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

    3. Applied flip--13.1

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

    4. Simplified1.7

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

    5. Simplified1.7

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{\color{blue}{1 + e^{x}}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.013702749797138520368410019867925313981 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}}\\ \mathbf{elif}\;x \le 5.535044263275191921405584805585021641155 \cdot 10^{-17}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt{\sqrt{2}}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))