Average Error: 4.4 → 0.1
Time: 7.2s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.04863258415632608821798407282982479316 \cdot 10^{-7} \lor \neg \left(x \le 1.233819893157010264277302635793631679917 \cdot 10^{-6}\right):\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\left(\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}\right) \cdot \left(\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}\right)}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.04863258415632608821798407282982479316 \cdot 10^{-7} \lor \neg \left(x \le 1.233819893157010264277302635793631679917 \cdot 10^{-6}\right):\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\left(\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}\right) \cdot \left(\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}\right)}{\sqrt{e^{x}} - \sqrt{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 r23186 = 2.0;
        double r23187 = x;
        double r23188 = r23186 * r23187;
        double r23189 = exp(r23188);
        double r23190 = 1.0;
        double r23191 = r23189 - r23190;
        double r23192 = exp(r23187);
        double r23193 = r23192 - r23190;
        double r23194 = r23191 / r23193;
        double r23195 = sqrt(r23194);
        return r23195;
}
↓
double f(double x) {
        double r23196 = x;
        double r23197 = -1.0486325841563261e-07;
        bool r23198 = r23196 <= r23197;
        double r23199 = 1.2338198931570103e-06;
        bool r23200 = r23196 <= r23199;
        double r23201 = !r23200;
        bool r23202 = r23198 || r23201;
        double r23203 = 2.0;
        double r23204 = r23203 * r23196;
        double r23205 = exp(r23204);
        double r23206 = sqrt(r23205);
        double r23207 = 1.0;
        double r23208 = sqrt(r23207);
        double r23209 = r23206 + r23208;
        double r23210 = exp(r23196);
        double r23211 = sqrt(r23210);
        double r23212 = r23211 + r23208;
        double r23213 = r23209 / r23212;
        double r23214 = sqrt(r23206);
        double r23215 = sqrt(r23208);
        double r23216 = r23214 + r23215;
        double r23217 = r23214 - r23215;
        double r23218 = r23216 * r23217;
        double r23219 = r23211 - r23208;
        double r23220 = r23218 / r23219;
        double r23221 = r23213 * r23220;
        double r23222 = sqrt(r23221);
        double r23223 = 0.5;
        double r23224 = sqrt(r23203);
        double r23225 = r23196 / r23224;
        double r23226 = r23223 * r23225;
        double r23227 = 2.0;
        double r23228 = pow(r23196, r23227);
        double r23229 = r23228 / r23224;
        double r23230 = 0.25;
        double r23231 = 0.125;
        double r23232 = r23231 / r23203;
        double r23233 = r23230 - r23232;
        double r23234 = r23229 * r23233;
        double r23235 = r23224 + r23234;
        double r23236 = r23226 + r23235;
        double r23237 = r23202 ? r23222 : r23236;
        return r23237;
}

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 < -1.0486325841563261e-07 or 1.2338198931570103e-06 < x

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied add-sqr-sqrt0.5

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

    5. Applied difference-of-squares0.5

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

    6. Applied add-sqr-sqrt0.5

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

    7. Applied add-sqr-sqrt0.5

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

    8. Applied difference-of-squares0.5

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

    9. Applied times-frac0.5

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

    11. Applied add-sqr-sqrt0.5

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

    12. Applied sqrt-prod0.5

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

    13. Applied add-sqr-sqrt0.5

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

    14. Applied sqrt-prod0.4

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

    15. Applied difference-of-squares0.1

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

    if -1.0486325841563261e-07 < x < 1.2338198931570103e-06

    1. Initial program 42.4

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.04863258415632608821798407282982479316 \cdot 10^{-7} \lor \neg \left(x \le 1.233819893157010264277302635793631679917 \cdot 10^{-6}\right):\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\left(\sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{\sqrt{1}}\right) \cdot \left(\sqrt{\sqrt{e^{2 \cdot x}}} - \sqrt{\sqrt{1}}\right)}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))