Average Error: 42.1 → 8.5
Time: 52.0s
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 -3.3972712623864903 \cdot 10^{+34}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le -5.1631397836325573 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -8.117479547197517 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 8.566610103260225 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{\sqrt{\left(\left(4 - \left(2 - \frac{4}{x}\right) \cdot \frac{4}{x}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}{\sqrt{x \cdot \left(\left(4 - 2 \cdot \frac{4}{x}\right) + \frac{4}{x} \cdot \frac{4}{x}\right)}}}\\ \mathbf{elif}\;t \le 1.6397433314236832 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 1.4454171454576948 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{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 -3.3972712623864903 \cdot 10^{+34}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le -5.1631397836325573 \cdot 10^{-172}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\

\mathbf{elif}\;t \le -8.117479547197517 \cdot 10^{-196}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 8.566610103260225 \cdot 10^{-208}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{\sqrt{\left(\left(4 - \left(2 - \frac{4}{x}\right) \cdot \frac{4}{x}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}{\sqrt{x \cdot \left(\left(4 - 2 \cdot \frac{4}{x}\right) + \frac{4}{x} \cdot \frac{4}{x}\right)}}}\\

\mathbf{elif}\;t \le 1.6397433314236832 \cdot 10^{-160}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\

\mathbf{elif}\;t \le 1.4454171454576948 \cdot 10^{+136}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r2410856 = 2.0;
        double r2410857 = sqrt(r2410856);
        double r2410858 = t;
        double r2410859 = r2410857 * r2410858;
        double r2410860 = x;
        double r2410861 = 1.0;
        double r2410862 = r2410860 + r2410861;
        double r2410863 = r2410860 - r2410861;
        double r2410864 = r2410862 / r2410863;
        double r2410865 = l;
        double r2410866 = r2410865 * r2410865;
        double r2410867 = r2410858 * r2410858;
        double r2410868 = r2410856 * r2410867;
        double r2410869 = r2410866 + r2410868;
        double r2410870 = r2410864 * r2410869;
        double r2410871 = r2410870 - r2410866;
        double r2410872 = sqrt(r2410871);
        double r2410873 = r2410859 / r2410872;
        return r2410873;
}
↓
double f(double x, double l, double t) {
        double r2410874 = t;
        double r2410875 = -3.3972712623864903e+34;
        bool r2410876 = r2410874 <= r2410875;
        double r2410877 = 2.0;
        double r2410878 = sqrt(r2410877);
        double r2410879 = r2410878 * r2410874;
        double r2410880 = x;
        double r2410881 = r2410880 * r2410880;
        double r2410882 = r2410874 / r2410881;
        double r2410883 = 1.0;
        double r2410884 = r2410883 / r2410878;
        double r2410885 = r2410882 * r2410884;
        double r2410886 = r2410877 / r2410878;
        double r2410887 = r2410874 / r2410880;
        double r2410888 = r2410887 + r2410882;
        double r2410889 = r2410886 * r2410888;
        double r2410890 = r2410889 + r2410879;
        double r2410891 = r2410885 - r2410890;
        double r2410892 = r2410879 / r2410891;
        double r2410893 = -5.1631397836325573e-172;
        bool r2410894 = r2410874 <= r2410893;
        double r2410895 = cbrt(r2410878);
        double r2410896 = r2410895 * r2410874;
        double r2410897 = r2410895 * r2410895;
        double r2410898 = r2410896 * r2410897;
        double r2410899 = l;
        double r2410900 = r2410899 * r2410877;
        double r2410901 = r2410899 / r2410880;
        double r2410902 = r2410900 * r2410901;
        double r2410903 = 4.0;
        double r2410904 = r2410903 / r2410880;
        double r2410905 = r2410904 + r2410877;
        double r2410906 = r2410874 * r2410874;
        double r2410907 = r2410905 * r2410906;
        double r2410908 = r2410902 + r2410907;
        double r2410909 = sqrt(r2410908);
        double r2410910 = r2410898 / r2410909;
        double r2410911 = -8.117479547197517e-196;
        bool r2410912 = r2410874 <= r2410911;
        double r2410913 = 8.566610103260225e-208;
        bool r2410914 = r2410874 <= r2410913;
        double r2410915 = r2410877 - r2410904;
        double r2410916 = r2410915 * r2410904;
        double r2410917 = r2410903 - r2410916;
        double r2410918 = r2410899 * r2410899;
        double r2410919 = r2410917 * r2410918;
        double r2410920 = r2410919 * r2410877;
        double r2410921 = r2410880 * r2410906;
        double r2410922 = 64.0;
        double r2410923 = r2410922 / r2410881;
        double r2410924 = r2410923 / r2410880;
        double r2410925 = 8.0;
        double r2410926 = r2410924 + r2410925;
        double r2410927 = r2410921 * r2410926;
        double r2410928 = r2410920 + r2410927;
        double r2410929 = sqrt(r2410928);
        double r2410930 = r2410877 * r2410904;
        double r2410931 = r2410903 - r2410930;
        double r2410932 = r2410904 * r2410904;
        double r2410933 = r2410931 + r2410932;
        double r2410934 = r2410880 * r2410933;
        double r2410935 = sqrt(r2410934);
        double r2410936 = r2410929 / r2410935;
        double r2410937 = r2410898 / r2410936;
        double r2410938 = 1.6397433314236832e-160;
        bool r2410939 = r2410874 <= r2410938;
        double r2410940 = r2410879 - r2410885;
        double r2410941 = r2410877 / r2410880;
        double r2410942 = r2410941 / r2410880;
        double r2410943 = r2410942 + r2410941;
        double r2410944 = r2410874 / r2410878;
        double r2410945 = r2410943 * r2410944;
        double r2410946 = r2410940 + r2410945;
        double r2410947 = r2410879 / r2410946;
        double r2410948 = 1.4454171454576948e+136;
        bool r2410949 = r2410874 <= r2410948;
        double r2410950 = r2410949 ? r2410910 : r2410947;
        double r2410951 = r2410939 ? r2410947 : r2410950;
        double r2410952 = r2410914 ? r2410937 : r2410951;
        double r2410953 = r2410912 ? r2410892 : r2410952;
        double r2410954 = r2410894 ? r2410910 : r2410953;
        double r2410955 = r2410876 ? r2410892 : r2410954;
        return r2410955;
}

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 < -3.3972712623864903e+34 or -5.1631397836325573e-172 < t < -8.117479547197517e-196

    1. Initial program 44.7

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

    2. Taylor expanded around -inf 6.1

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

    3. Simplified6.1

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

    if -3.3972712623864903e+34 < t < -5.1631397836325573e-172 or 1.6397433314236832e-160 < t < 1.4454171454576948e+136

    1. Initial program 26.3

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

    2. Taylor expanded around inf 10.6

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

    3. Simplified5.4

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

    5. Applied add-cube-cbrt5.4

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

    6. Applied associate-*l*5.3

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

    if -8.117479547197517e-196 < t < 8.566610103260225e-208

    1. Initial program 61.7

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

    2. Taylor expanded around inf 30.7

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

    3. Simplified29.1

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

    5. Applied add-cube-cbrt29.1

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

    6. Applied associate-*l*29.1

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

    8. Applied associate-*r/30.7

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

    9. Applied flip3-+30.7

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

    10. Applied associate-*r/30.7

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

    11. Applied frac-add31.1

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

    12. Applied sqrt-div25.5

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

    13. Simplified25.6

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\frac{\color{blue}{\sqrt{\left(\left(4 - \frac{4}{x} \cdot \left(2 - \frac{4}{x}\right)\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}}{\sqrt{\left(\frac{4}{x} \cdot \frac{4}{x} + \left(2 \cdot 2 - \frac{4}{x} \cdot 2\right)\right) \cdot x}}}\]

    if 8.566610103260225e-208 < t < 1.6397433314236832e-160 or 1.4454171454576948e+136 < t

    1. Initial program 58.6

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

    2. Taylor expanded around inf 7.6

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

    3. Simplified7.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right) + \left(\sqrt{2} \cdot t - \frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x}\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.3972712623864903 \cdot 10^{+34}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le -5.1631397836325573 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -8.117479547197517 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 8.566610103260225 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{\sqrt{\left(\left(4 - \left(2 - \frac{4}{x}\right) \cdot \frac{4}{x}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 2 + \left(x \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\frac{64}{x \cdot x}}{x} + 8\right)}}{\sqrt{x \cdot \left(\left(4 - 2 \cdot \frac{4}{x}\right) + \frac{4}{x} \cdot \frac{4}{x}\right)}}}\\ \mathbf{elif}\;t \le 1.6397433314236832 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \mathbf{elif}\;t \le 1.4454171454576948 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot t\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{\left(\ell \cdot 2\right) \cdot \frac{\ell}{x} + \left(\frac{4}{x} + 2\right) \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t - \frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}}\right) + \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right) \cdot \frac{t}{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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)))))