Average Error: 42.6 → 8.9
Time: 1.7m
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.3583535173842932 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.8956945634134774 \cdot 10^{-265}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\frac{4}{x} + 2\right) \cdot t\right) \cdot t + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{elif}\;t \le 2.6753623478297 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}} + \left(\sqrt{2} \cdot t - \frac{t \cdot \frac{1}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{elif}\;t \le 2.0371843034385597 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\frac{4}{x} + 2\right) \cdot t\right) \cdot t + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}} + \left(\sqrt{2} \cdot t - \frac{t \cdot \frac{1}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]
\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 -3.3583535173842932 \cdot 10^{+75}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 1.8956945634134774 \cdot 10^{-265}:\\
\;\;\;\;\frac{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\frac{4}{x} + 2\right) \cdot t\right) \cdot t + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\

\mathbf{elif}\;t \le 2.6753623478297 \cdot 10^{-158}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}} + \left(\sqrt{2} \cdot t - \frac{t \cdot \frac{1}{\sqrt{2}}}{x \cdot x}\right)}\\

\mathbf{elif}\;t \le 2.0371843034385597 \cdot 10^{+68}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\frac{4}{x} + 2\right) \cdot t\right) \cdot t + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r6058821 = 2.0;
        double r6058822 = sqrt(r6058821);
        double r6058823 = t;
        double r6058824 = r6058822 * r6058823;
        double r6058825 = x;
        double r6058826 = 1.0;
        double r6058827 = r6058825 + r6058826;
        double r6058828 = r6058825 - r6058826;
        double r6058829 = r6058827 / r6058828;
        double r6058830 = l;
        double r6058831 = r6058830 * r6058830;
        double r6058832 = r6058823 * r6058823;
        double r6058833 = r6058821 * r6058832;
        double r6058834 = r6058831 + r6058833;
        double r6058835 = r6058829 * r6058834;
        double r6058836 = r6058835 - r6058831;
        double r6058837 = sqrt(r6058836);
        double r6058838 = r6058824 / r6058837;
        return r6058838;
}
↓
double f(double x, double l, double t) {
        double r6058839 = t;
        double r6058840 = -3.3583535173842932e+75;
        bool r6058841 = r6058839 <= r6058840;
        double r6058842 = 2.0;
        double r6058843 = sqrt(r6058842);
        double r6058844 = r6058843 * r6058839;
        double r6058845 = 1.0;
        double r6058846 = r6058845 / r6058843;
        double r6058847 = r6058842 / r6058843;
        double r6058848 = r6058846 - r6058847;
        double r6058849 = x;
        double r6058850 = r6058849 * r6058849;
        double r6058851 = r6058839 / r6058850;
        double r6058852 = r6058848 * r6058851;
        double r6058853 = r6058849 * r6058843;
        double r6058854 = r6058839 / r6058853;
        double r6058855 = r6058854 * r6058842;
        double r6058856 = r6058855 + r6058844;
        double r6058857 = r6058852 - r6058856;
        double r6058858 = r6058844 / r6058857;
        double r6058859 = 1.8956945634134774e-265;
        bool r6058860 = r6058839 <= r6058859;
        double r6058861 = cbrt(r6058843);
        double r6058862 = cbrt(r6058861);
        double r6058863 = r6058862 * r6058862;
        double r6058864 = r6058839 * r6058862;
        double r6058865 = r6058863 * r6058864;
        double r6058866 = r6058861 * r6058861;
        double r6058867 = r6058865 * r6058866;
        double r6058868 = 4.0;
        double r6058869 = r6058868 / r6058849;
        double r6058870 = r6058869 + r6058842;
        double r6058871 = r6058870 * r6058839;
        double r6058872 = r6058871 * r6058839;
        double r6058873 = l;
        double r6058874 = r6058873 * r6058842;
        double r6058875 = r6058873 / r6058849;
        double r6058876 = r6058874 * r6058875;
        double r6058877 = r6058872 + r6058876;
        double r6058878 = sqrt(r6058877);
        double r6058879 = r6058867 / r6058878;
        double r6058880 = 2.6753623478297e-158;
        bool r6058881 = r6058839 <= r6058880;
        double r6058882 = r6058842 / r6058849;
        double r6058883 = r6058882 / r6058849;
        double r6058884 = r6058883 + r6058882;
        double r6058885 = r6058839 / r6058843;
        double r6058886 = r6058884 * r6058885;
        double r6058887 = r6058839 * r6058846;
        double r6058888 = r6058887 / r6058850;
        double r6058889 = r6058844 - r6058888;
        double r6058890 = r6058886 + r6058889;
        double r6058891 = r6058844 / r6058890;
        double r6058892 = 2.0371843034385597e+68;
        bool r6058893 = r6058839 <= r6058892;
        double r6058894 = r6058839 * r6058861;
        double r6058895 = r6058866 * r6058894;
        double r6058896 = r6058895 / r6058878;
        double r6058897 = r6058893 ? r6058896 : r6058891;
        double r6058898 = r6058881 ? r6058891 : r6058897;
        double r6058899 = r6058860 ? r6058879 : r6058898;
        double r6058900 = r6058841 ? r6058858 : r6058899;
        return r6058900;
}

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 < -3.3583535173842932e+75

    1. Initial program 47.9

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

    2. Taylor expanded around -inf 3.4

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

    3. Simplified3.4

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

    if -3.3583535173842932e+75 < t < 1.8956945634134774e-265

    1. Initial program 40.7

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

    2. Taylor expanded around -inf 18.7

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

    3. Simplified14.9

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

    5. Applied add-cube-cbrt14.9

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

    6. Applied associate-*l*14.9

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

    8. Applied add-cube-cbrt14.9

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

    9. Applied associate-*l*15.0

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

    if 1.8956945634134774e-265 < t < 2.6753623478297e-158 or 2.0371843034385597e+68 < t

    1. Initial program 49.5

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

    2. Taylor expanded around inf 9.9

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

    3. Simplified9.9

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

    if 2.6753623478297e-158 < t < 2.0371843034385597e+68

    1. Initial program 27.0

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

    2. Taylor expanded around -inf 9.7

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

    3. Simplified4.0

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

    5. Applied add-cube-cbrt4.0

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

    6. Applied associate-*l*4.0

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.3583535173842932 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} \cdot 2 + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.8956945634134774 \cdot 10^{-265}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\frac{4}{x} + 2\right) \cdot t\right) \cdot t + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{elif}\;t \le 2.6753623478297 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}} + \left(\sqrt{2} \cdot t - \frac{t \cdot \frac{1}{\sqrt{2}}}{x \cdot x}\right)}\\ \mathbf{elif}\;t \le 2.0371843034385597 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(t \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\left(\frac{4}{x} + 2\right) \cdot t\right) \cdot t + \left(\ell \cdot 2\right) \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}} + \left(\sqrt{2} \cdot t - \frac{t \cdot \frac{1}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))))