Average Error: 41.9 → 9.2
Time: 47.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.5141814096893173 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{\frac{t}{\sqrt{2}}}{2} - \frac{t}{\sqrt{2}}\right) \cdot \frac{2}{x \cdot x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 2.46097945070799 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\sqrt{\left(t \cdot \frac{t}{x}\right) \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}} \cdot \sqrt{\sqrt{\left(t \cdot \frac{t}{x}\right) \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) \cdot \frac{2}{x \cdot x} + \left(\sqrt{2} \cdot t + \frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.5141814096893173 \cdot 10^{+140}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{\frac{t}{\sqrt{2}}}{2} - \frac{t}{\sqrt{2}}\right) \cdot \frac{2}{x \cdot x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 2.46097945070799 \cdot 10^{+64}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\sqrt{\left(t \cdot \frac{t}{x}\right) \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}} \cdot \sqrt{\sqrt{\left(t \cdot \frac{t}{x}\right) \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r2179839 = 2.0;
        double r2179840 = sqrt(r2179839);
        double r2179841 = t;
        double r2179842 = r2179840 * r2179841;
        double r2179843 = x;
        double r2179844 = 1.0;
        double r2179845 = r2179843 + r2179844;
        double r2179846 = r2179843 - r2179844;
        double r2179847 = r2179845 / r2179846;
        double r2179848 = l;
        double r2179849 = r2179848 * r2179848;
        double r2179850 = r2179841 * r2179841;
        double r2179851 = r2179839 * r2179850;
        double r2179852 = r2179849 + r2179851;
        double r2179853 = r2179847 * r2179852;
        double r2179854 = r2179853 - r2179849;
        double r2179855 = sqrt(r2179854);
        double r2179856 = r2179842 / r2179855;
        return r2179856;
}
↓
double f(double x, double l, double t) {
        double r2179857 = t;
        double r2179858 = -3.5141814096893173e+140;
        bool r2179859 = r2179857 <= r2179858;
        double r2179860 = 2.0;
        double r2179861 = sqrt(r2179860);
        double r2179862 = r2179861 * r2179857;
        double r2179863 = r2179857 / r2179861;
        double r2179864 = r2179863 / r2179860;
        double r2179865 = r2179864 - r2179863;
        double r2179866 = x;
        double r2179867 = r2179866 * r2179866;
        double r2179868 = r2179860 / r2179867;
        double r2179869 = r2179865 * r2179868;
        double r2179870 = r2179869 - r2179862;
        double r2179871 = r2179860 / r2179866;
        double r2179872 = r2179863 * r2179871;
        double r2179873 = r2179870 - r2179872;
        double r2179874 = r2179862 / r2179873;
        double r2179875 = 2.46097945070799e+64;
        bool r2179876 = r2179857 <= r2179875;
        double r2179877 = cbrt(r2179861);
        double r2179878 = r2179877 * r2179857;
        double r2179879 = r2179877 * r2179877;
        double r2179880 = r2179878 * r2179879;
        double r2179881 = r2179857 / r2179866;
        double r2179882 = r2179857 * r2179881;
        double r2179883 = 4.0;
        double r2179884 = r2179882 * r2179883;
        double r2179885 = l;
        double r2179886 = r2179885 / r2179866;
        double r2179887 = r2179885 * r2179886;
        double r2179888 = r2179857 * r2179857;
        double r2179889 = r2179887 + r2179888;
        double r2179890 = r2179860 * r2179889;
        double r2179891 = r2179884 + r2179890;
        double r2179892 = sqrt(r2179891);
        double r2179893 = sqrt(r2179892);
        double r2179894 = r2179893 * r2179893;
        double r2179895 = r2179880 / r2179894;
        double r2179896 = r2179863 - r2179864;
        double r2179897 = r2179896 * r2179868;
        double r2179898 = r2179862 + r2179872;
        double r2179899 = r2179897 + r2179898;
        double r2179900 = r2179862 / r2179899;
        double r2179901 = r2179876 ? r2179895 : r2179900;
        double r2179902 = r2179859 ? r2179874 : r2179901;
        return r2179902;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -3.5141814096893173e+140

    1. Initial program 58.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 2.4

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

    3. Simplified2.4

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

    if -3.5141814096893173e+140 < t < 2.46097945070799e+64

    1. Initial program 35.9

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 17.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified17.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity17.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{\color{blue}{1 \cdot x}}\right)}}\]

    6. Applied times-frac13.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \color{blue}{\frac{\ell}{1} \cdot \frac{\ell}{x}}\right)}}\]

    7. Simplified13.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \color{blue}{\ell} \cdot \frac{\ell}{x}\right)}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt13.1

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot t}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\]

    10. Applied associate-*l*13.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt13.0

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\color{blue}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}}}\]

    13. Applied sqrt-prod13.2

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\color{blue}{\sqrt{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}} \cdot \sqrt{\sqrt{4 \cdot \left(\frac{t}{x} \cdot t\right) + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}}}\]

    if 2.46097945070799e+64 < t

    1. Initial program 45.6

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 3.7

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

    3. Simplified3.7

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.5141814096893173 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{\frac{t}{\sqrt{2}}}{2} - \frac{t}{\sqrt{2}}\right) \cdot \frac{2}{x \cdot x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 2.46097945070799 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\sqrt{\left(t \cdot \frac{t}{x}\right) \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}} \cdot \sqrt{\sqrt{\left(t \cdot \frac{t}{x}\right) \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) \cdot \frac{2}{x \cdot x} + \left(\sqrt{2} \cdot t + \frac{t}{\sqrt{2}} \cdot \frac{2}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))