Average Error: 42.8 → 10.2
Time: 8.6s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -30300791297250332:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -5.511170609100405 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}} \cdot \sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}}\\ \mathbf{elif}\;t \le -1.21751320251488658 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 3.27940796217429109 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{elif}\;t \le 2.2209729288484951 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \mathbf{elif}\;t \le 1.73405392707011745 \cdot 10^{57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]
\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 -30300791297250332:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le -5.511170609100405 \cdot 10^{-151}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}} \cdot \sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}}\\

\mathbf{elif}\;t \le -1.21751320251488658 \cdot 10^{-295}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 3.27940796217429109 \cdot 10^{-264}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\

\mathbf{elif}\;t \le 2.2209729288484951 \cdot 10^{-162}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\

\mathbf{elif}\;t \le 1.73405392707011745 \cdot 10^{57}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\

\end{array}
double f(double x, double l, double t) {
        double r35851 = 2.0;
        double r35852 = sqrt(r35851);
        double r35853 = t;
        double r35854 = r35852 * r35853;
        double r35855 = x;
        double r35856 = 1.0;
        double r35857 = r35855 + r35856;
        double r35858 = r35855 - r35856;
        double r35859 = r35857 / r35858;
        double r35860 = l;
        double r35861 = r35860 * r35860;
        double r35862 = r35853 * r35853;
        double r35863 = r35851 * r35862;
        double r35864 = r35861 + r35863;
        double r35865 = r35859 * r35864;
        double r35866 = r35865 - r35861;
        double r35867 = sqrt(r35866);
        double r35868 = r35854 / r35867;
        return r35868;
}
↓
double f(double x, double l, double t) {
        double r35869 = t;
        double r35870 = -30300791297250332.0;
        bool r35871 = r35869 <= r35870;
        double r35872 = 2.0;
        double r35873 = sqrt(r35872);
        double r35874 = r35873 * r35869;
        double r35875 = 3.0;
        double r35876 = pow(r35873, r35875);
        double r35877 = x;
        double r35878 = 2.0;
        double r35879 = pow(r35877, r35878);
        double r35880 = r35876 * r35879;
        double r35881 = r35869 / r35880;
        double r35882 = r35873 * r35879;
        double r35883 = r35869 / r35882;
        double r35884 = r35881 - r35883;
        double r35885 = r35872 * r35884;
        double r35886 = r35885 - r35874;
        double r35887 = r35873 * r35877;
        double r35888 = r35869 / r35887;
        double r35889 = r35872 * r35888;
        double r35890 = r35886 - r35889;
        double r35891 = r35874 / r35890;
        double r35892 = -5.511170609100405e-151;
        bool r35893 = r35869 <= r35892;
        double r35894 = 4.0;
        double r35895 = pow(r35869, r35878);
        double r35896 = r35895 / r35877;
        double r35897 = r35894 * r35896;
        double r35898 = l;
        double r35899 = fabs(r35898);
        double r35900 = r35899 / r35877;
        double r35901 = r35899 * r35900;
        double r35902 = r35895 + r35901;
        double r35903 = r35872 * r35902;
        double r35904 = r35897 + r35903;
        double r35905 = sqrt(r35904);
        double r35906 = sqrt(r35905);
        double r35907 = r35906 * r35906;
        double r35908 = r35874 / r35907;
        double r35909 = -1.2175132025148866e-295;
        bool r35910 = r35869 <= r35909;
        double r35911 = 3.279407962174291e-264;
        bool r35912 = r35869 <= r35911;
        double r35913 = r35874 / r35905;
        double r35914 = 2.220972928848495e-162;
        bool r35915 = r35869 <= r35914;
        double r35916 = r35869 * r35873;
        double r35917 = r35889 + r35916;
        double r35918 = r35872 * r35881;
        double r35919 = r35917 - r35918;
        double r35920 = r35874 / r35919;
        double r35921 = 1.7340539270701175e+57;
        bool r35922 = r35869 <= r35921;
        double r35923 = r35922 ? r35913 : r35920;
        double r35924 = r35915 ? r35920 : r35923;
        double r35925 = r35912 ? r35913 : r35924;
        double r35926 = r35910 ? r35891 : r35925;
        double r35927 = r35893 ? r35908 : r35926;
        double r35928 = r35871 ? r35891 : r35927;
        return r35928;
}

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 4 regimes

  2. if t < -30300791297250332.0 or -5.511170609100405e-151 < t < -1.2175132025148866e-295

    1. Initial program 46.1

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

    2. Taylor expanded around -inf 12.5

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

    3. Simplified12.5

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

    if -30300791297250332.0 < t < -5.511170609100405e-151

    1. Initial program 28.6

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

    2. Taylor expanded around inf 9.0

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

    3. Simplified9.0

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

    5. Applied *-un-lft-identity9.0

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

    6. Applied add-sqr-sqrt9.0

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

    7. Applied times-frac9.0

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

    8. Simplified9.0

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

    9. Simplified5.0

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

    11. Applied add-sqr-sqrt5.0

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

    12. Applied sqrt-prod5.2

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

    if -1.2175132025148866e-295 < t < 3.279407962174291e-264 or 2.220972928848495e-162 < t < 1.7340539270701175e+57

    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 13.4

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

    3. Simplified13.4

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

    5. Applied *-un-lft-identity13.4

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

    6. Applied add-sqr-sqrt13.4

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

    7. Applied times-frac13.4

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

    8. Simplified13.4

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

    9. Simplified9.1

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

    if 3.279407962174291e-264 < t < 2.220972928848495e-162 or 1.7340539270701175e+57 < t

    1. Initial program 49.3

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

    2. Taylor expanded around inf 42.8

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

    3. Simplified42.8

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

    4. Taylor expanded around inf 10.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -30300791297250332:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -5.511170609100405 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}} \cdot \sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}}\\ \mathbf{elif}\;t \le -1.21751320251488658 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 3.27940796217429109 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{elif}\;t \le 2.2209729288484951 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \mathbf{elif}\;t \le 1.73405392707011745 \cdot 10^{57}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))