Average Error: 4.6 → 0.8
Time: 17.3s
Precision: 64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.336679594201055357367315112870898019537 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{{\left(\sqrt{1}\right)}^{3} + {\left({\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}\right)}^{3}}{\left({\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}\right) \cdot {\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} + 1} \cdot \frac{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 + x\right) + \frac{1}{2} \cdot \left(x \cdot x\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.336679594201055357367315112870898019537 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{{\left(\sqrt{1}\right)}^{3} + {\left({\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}\right)}^{3}}{\left({\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}\right) \cdot {\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} + 1} \cdot \frac{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(2 + x\right) + \frac{1}{2} \cdot \left(x \cdot x\right)}\\

\end{array}
double f(double x) {
        double r23918 = 2.0;
        double r23919 = x;
        double r23920 = r23918 * r23919;
        double r23921 = exp(r23920);
        double r23922 = 1.0;
        double r23923 = r23921 - r23922;
        double r23924 = exp(r23919);
        double r23925 = r23924 - r23922;
        double r23926 = r23923 / r23925;
        double r23927 = sqrt(r23926);
        return r23927;
}
↓
double f(double x) {
        double r23928 = x;
        double r23929 = -1.3366795942010554e-05;
        bool r23930 = r23928 <= r23929;
        double r23931 = 1.0;
        double r23932 = sqrt(r23931);
        double r23933 = 3.0;
        double r23934 = pow(r23932, r23933);
        double r23935 = exp(r23928);
        double r23936 = 2.0;
        double r23937 = 2.0;
        double r23938 = r23936 / r23937;
        double r23939 = pow(r23935, r23938);
        double r23940 = pow(r23939, r23933);
        double r23941 = r23934 + r23940;
        double r23942 = r23939 - r23932;
        double r23943 = r23942 * r23939;
        double r23944 = r23943 + r23931;
        double r23945 = r23941 / r23944;
        double r23946 = r23935 - r23931;
        double r23947 = r23942 / r23946;
        double r23948 = r23945 * r23947;
        double r23949 = sqrt(r23948);
        double r23950 = r23937 + r23928;
        double r23951 = 0.5;
        double r23952 = r23928 * r23928;
        double r23953 = r23951 * r23952;
        double r23954 = r23950 + r23953;
        double r23955 = sqrt(r23954);
        double r23956 = r23930 ? r23949 : r23955;
        return r23956;
}

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 < -1.3366795942010554e-05

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied *-un-lft-identity0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied sqr-pow0.1

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

    7. Applied difference-of-squares0.0

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

    8. Applied times-frac0.0

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

    9. Simplified0.0

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

    11. Applied flip3-+0.0

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

    12. Simplified0.0

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

    13. Simplified0.0

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

    if -1.3366795942010554e-05 < x

    1. Initial program 34.4

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

    2. Simplified29.6

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

    3. Taylor expanded around 0 6.2

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

    4. Simplified6.2

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{2} + \left(2 + x\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.336679594201055357367315112870898019537 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{{\left(\sqrt{1}\right)}^{3} + {\left({\left(e^{x}\right)}^{\left(\frac{2}{2}\right)}\right)}^{3}}{\left({\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}\right) \cdot {\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} + 1} \cdot \frac{{\left(e^{x}\right)}^{\left(\frac{2}{2}\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 + x\right) + \frac{1}{2} \cdot \left(x \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))