Average Error: 4.7 → 0.8
Time: 5.5s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.5501934119102568 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.5501934119102568 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double x) {
        double r11153 = 2.0;
        double r11154 = x;
        double r11155 = r11153 * r11154;
        double r11156 = exp(r11155);
        double r11157 = 1.0;
        double r11158 = r11156 - r11157;
        double r11159 = exp(r11154);
        double r11160 = r11159 - r11157;
        double r11161 = r11158 / r11160;
        double r11162 = sqrt(r11161);
        return r11162;
}
↓
double f(double x) {
        double r11163 = x;
        double r11164 = -5.550193411910257e-17;
        bool r11165 = r11163 <= r11164;
        double r11166 = 2.0;
        double r11167 = r11166 * r11163;
        double r11168 = exp(r11167);
        double r11169 = sqrt(r11168);
        double r11170 = sqrt(r11169);
        double r11171 = r11170 * r11170;
        double r11172 = 1.0;
        double r11173 = sqrt(r11172);
        double r11174 = r11171 + r11173;
        double r11175 = 1.0;
        double r11176 = r11174 / r11175;
        double r11177 = sqrt(r11176);
        double r11178 = exp(r11166);
        double r11179 = 0.5;
        double r11180 = r11179 * r11163;
        double r11181 = pow(r11178, r11180);
        double r11182 = r11181 - r11173;
        double r11183 = exp(r11163);
        double r11184 = r11183 - r11172;
        double r11185 = r11182 / r11184;
        double r11186 = sqrt(r11185);
        double r11187 = r11177 * r11186;
        double r11188 = 0.5;
        double r11189 = sqrt(r11166);
        double r11190 = r11163 / r11189;
        double r11191 = r11188 * r11190;
        double r11192 = 2.0;
        double r11193 = pow(r11163, r11192);
        double r11194 = r11193 / r11189;
        double r11195 = 0.25;
        double r11196 = 0.125;
        double r11197 = r11196 / r11166;
        double r11198 = r11195 - r11197;
        double r11199 = r11194 * r11198;
        double r11200 = r11189 + r11199;
        double r11201 = r11191 + r11200;
        double r11202 = r11165 ? r11187 : r11201;
        return r11202;
}

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 < -5.550193411910257e-17

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied add-sqr-sqrt0.9

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

    5. Applied add-sqr-sqrt0.7

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

    6. Applied difference-of-squares0.3

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

    7. Applied times-frac0.3

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

    8. Applied sqrt-prod0.3

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

    10. Applied add-log-exp0.3

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

    11. Applied exp-to-pow0.3

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

    12. Applied sqrt-pow10.0

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

    13. Simplified0.0

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

    15. Applied add-sqr-sqrt0.0

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

    16. Applied sqrt-prod0.0

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

    if -5.550193411910257e-17 < x

    1. Initial program 38.3

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

    2. Taylor expanded around 0 8.1

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

    3. Simplified8.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.5501934119102568 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))