Average Error: 4.2 → 0.7
Time: 11.4s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.5679454952614643 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.5679454952614643 \cdot 10^{-15}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\

\end{array}
double f(double x) {
        double r18168 = 2.0;
        double r18169 = x;
        double r18170 = r18168 * r18169;
        double r18171 = exp(r18170);
        double r18172 = 1.0;
        double r18173 = r18171 - r18172;
        double r18174 = exp(r18169);
        double r18175 = r18174 - r18172;
        double r18176 = r18173 / r18175;
        double r18177 = sqrt(r18176);
        return r18177;
}
↓
double f(double x) {
        double r18178 = x;
        double r18179 = -2.5679454952614643e-15;
        bool r18180 = r18178 <= r18179;
        double r18181 = 2.0;
        double r18182 = r18181 * r18178;
        double r18183 = exp(r18182);
        double r18184 = 1.0;
        double r18185 = r18183 - r18184;
        double r18186 = 3.0;
        double r18187 = pow(r18185, r18186);
        double r18188 = cbrt(r18187);
        double r18189 = r18178 + r18178;
        double r18190 = exp(r18189);
        double r18191 = r18184 * r18184;
        double r18192 = r18190 - r18191;
        double r18193 = r18188 / r18192;
        double r18194 = sqrt(r18193);
        double r18195 = exp(r18178);
        double r18196 = r18195 + r18184;
        double r18197 = sqrt(r18196);
        double r18198 = r18194 * r18197;
        double r18199 = sqrt(r18181);
        double r18200 = 0.5;
        double r18201 = r18178 / r18199;
        double r18202 = r18200 * r18201;
        double r18203 = r18199 + r18202;
        double r18204 = 2.0;
        double r18205 = pow(r18178, r18204);
        double r18206 = r18205 / r18199;
        double r18207 = 0.25;
        double r18208 = 0.125;
        double r18209 = r18208 / r18181;
        double r18210 = r18207 - r18209;
        double r18211 = r18206 * r18210;
        double r18212 = r18203 + r18211;
        double r18213 = r18180 ? r18198 : r18212;
        return r18213;
}

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 < -2.5679454952614643e-15

    1. Initial program 0.7

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

    3. Applied flip--0.4

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

    4. Applied associate-/r/0.4

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

    5. Applied sqrt-prod0.4

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

    6. Simplified0.0

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

    8. Applied add-cbrt-cube0.0

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

    9. Simplified0.0

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

    if -2.5679454952614643e-15 < x

    1. Initial program 38.5

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

    2. Taylor expanded around 0 7.4

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

    3. Simplified7.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.5679454952614643 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))