Average Error: 4.3 → 0.9
Time: 7.1s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.403797957856243383586276017582772368542 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.403797957856243383586276017582772368542 \cdot 10^{-7}:\\
\;\;\;\;e^{\log \left(\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\right)}\\

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

\end{array}
double f(double x) {
        double r23175 = 2.0;
        double r23176 = x;
        double r23177 = r23175 * r23176;
        double r23178 = exp(r23177);
        double r23179 = 1.0;
        double r23180 = r23178 - r23179;
        double r23181 = exp(r23176);
        double r23182 = r23181 - r23179;
        double r23183 = r23180 / r23182;
        double r23184 = sqrt(r23183);
        return r23184;
}
↓
double f(double x) {
        double r23185 = x;
        double r23186 = -8.403797957856243e-07;
        bool r23187 = r23185 <= r23186;
        double r23188 = 2.0;
        double r23189 = r23188 * r23185;
        double r23190 = exp(r23189);
        double r23191 = sqrt(r23190);
        double r23192 = 1.0;
        double r23193 = sqrt(r23192);
        double r23194 = r23191 + r23193;
        double r23195 = r23191 - r23193;
        double r23196 = exp(r23185);
        double r23197 = r23196 - r23192;
        double r23198 = r23195 / r23197;
        double r23199 = r23194 * r23198;
        double r23200 = sqrt(r23199);
        double r23201 = log(r23200);
        double r23202 = exp(r23201);
        double r23203 = 0.5;
        double r23204 = sqrt(r23188);
        double r23205 = r23185 / r23204;
        double r23206 = r23203 * r23205;
        double r23207 = 2.0;
        double r23208 = pow(r23185, r23207);
        double r23209 = r23208 / r23204;
        double r23210 = 0.25;
        double r23211 = 0.125;
        double r23212 = r23211 / r23188;
        double r23213 = r23210 - r23212;
        double r23214 = r23209 * r23213;
        double r23215 = log(r23214);
        double r23216 = exp(r23215);
        double r23217 = r23204 + r23216;
        double r23218 = r23206 + r23217;
        double r23219 = r23187 ? r23202 : r23218;
        return r23219;
}

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 < -8.403797957856243e-07

    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-exp-log0.0

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

    if -8.403797957856243e-07 < x

    1. Initial program 35.2

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

    2. Taylor expanded around 0 6.9

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

    3. Simplified6.9

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

    5. Applied add-exp-log6.9

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

    6. Applied add-exp-log6.9

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

    7. Applied add-exp-log31.5

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

    8. Applied pow-exp31.5

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

    9. Applied div-exp31.5

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

    10. Applied prod-exp31.5

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

    11. Simplified6.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.403797957856243383586276017582772368542 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + e^{\log \left(\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))