Average Error: 4.4 → 0.7
Time: 5.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \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 r10960 = 2.0;
        double r10961 = x;
        double r10962 = r10960 * r10961;
        double r10963 = exp(r10962);
        double r10964 = 1.0;
        double r10965 = r10963 - r10964;
        double r10966 = exp(r10961);
        double r10967 = r10966 - r10964;
        double r10968 = r10965 / r10967;
        double r10969 = sqrt(r10968);
        return r10969;
}
↓
double f(double x) {
        double r10970 = x;
        double r10971 = -7.268542916058787e-16;
        bool r10972 = r10970 <= r10971;
        double r10973 = 2.0;
        double r10974 = exp(r10973);
        double r10975 = 0.5;
        double r10976 = r10975 * r10970;
        double r10977 = pow(r10974, r10976);
        double r10978 = 1.0;
        double r10979 = sqrt(r10978);
        double r10980 = r10977 + r10979;
        double r10981 = 1.0;
        double r10982 = r10980 / r10981;
        double r10983 = sqrt(r10982);
        double r10984 = r10977 - r10979;
        double r10985 = exp(r10970);
        double r10986 = r10985 - r10978;
        double r10987 = r10984 / r10986;
        double r10988 = sqrt(r10987);
        double r10989 = r10983 * r10988;
        double r10990 = 0.5;
        double r10991 = sqrt(r10973);
        double r10992 = r10970 / r10991;
        double r10993 = r10990 * r10992;
        double r10994 = 2.0;
        double r10995 = pow(r10970, r10994);
        double r10996 = r10995 / r10991;
        double r10997 = 0.25;
        double r10998 = 0.125;
        double r10999 = r10998 / r10973;
        double r11000 = r10997 - r10999;
        double r11001 = r10996 * r11000;
        double r11002 = r10991 + r11001;
        double r11003 = r10993 + r11002;
        double r11004 = r10972 ? r10989 : r11003;
        return r11004;
}

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 < -7.268542916058787e-16

    1. Initial program 0.7

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

    3. Applied *-un-lft-identity0.7

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

    4. Applied add-sqr-sqrt0.7

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

    5. Applied add-sqr-sqrt0.6

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

    6. Applied difference-of-squares0.2

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

    7. Applied times-frac0.2

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

    8. Applied sqrt-prod0.2

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

    10. Applied add-log-exp0.2

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

    11. Applied exp-to-pow0.2

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

    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-log-exp0.0

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

    16. Applied exp-to-pow0.0

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

    17. Applied sqrt-pow10.0

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

    18. Simplified0.0

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

    if -7.268542916058787e-16 < x

    1. Initial program 37.3

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

    2. Taylor expanded around 0 7.3

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

    3. Simplified7.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))