Average Error: 43.3 → 9.2
Time: 27.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.615673507435036296435750238402039962931 \cdot 10^{50}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le -6.111047820265303935317690964852722629158 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{elif}\;t \le -1.323606909102099087479133351528995697428 \cdot 10^{-261}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 1.277987426228136459634363360568383475924 \cdot 10^{119}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(t \cdot \sqrt{2} - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -2.615673507435036296435750238402039962931 \cdot 10^{50}:\\
\;\;\;\;\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) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\

\mathbf{elif}\;t \le -6.111047820265303935317690964852722629158 \cdot 10^{-179}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\

\mathbf{elif}\;t \le -1.323606909102099087479133351528995697428 \cdot 10^{-261}:\\
\;\;\;\;\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) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\

\mathbf{elif}\;t \le 1.277987426228136459634363360568383475924 \cdot 10^{119}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r48805 = 2.0;
        double r48806 = sqrt(r48805);
        double r48807 = t;
        double r48808 = r48806 * r48807;
        double r48809 = x;
        double r48810 = 1.0;
        double r48811 = r48809 + r48810;
        double r48812 = r48809 - r48810;
        double r48813 = r48811 / r48812;
        double r48814 = l;
        double r48815 = r48814 * r48814;
        double r48816 = r48807 * r48807;
        double r48817 = r48805 * r48816;
        double r48818 = r48815 + r48817;
        double r48819 = r48813 * r48818;
        double r48820 = r48819 - r48815;
        double r48821 = sqrt(r48820);
        double r48822 = r48808 / r48821;
        return r48822;
}
↓
double f(double x, double l, double t) {
        double r48823 = t;
        double r48824 = -2.6156735074350363e+50;
        bool r48825 = r48823 <= r48824;
        double r48826 = 2.0;
        double r48827 = sqrt(r48826);
        double r48828 = r48827 * r48823;
        double r48829 = 3.0;
        double r48830 = pow(r48827, r48829);
        double r48831 = x;
        double r48832 = 2.0;
        double r48833 = pow(r48831, r48832);
        double r48834 = r48830 * r48833;
        double r48835 = r48823 / r48834;
        double r48836 = r48827 * r48833;
        double r48837 = r48823 / r48836;
        double r48838 = r48835 - r48837;
        double r48839 = r48826 * r48838;
        double r48840 = r48827 * r48831;
        double r48841 = r48823 / r48840;
        double r48842 = r48826 * r48841;
        double r48843 = r48839 - r48842;
        double r48844 = r48823 * r48827;
        double r48845 = r48843 - r48844;
        double r48846 = r48828 / r48845;
        double r48847 = -6.111047820265304e-179;
        bool r48848 = r48823 <= r48847;
        double r48849 = r48823 * r48823;
        double r48850 = l;
        double r48851 = fabs(r48850);
        double r48852 = r48851 / r48831;
        double r48853 = r48851 * r48852;
        double r48854 = r48849 + r48853;
        double r48855 = r48826 * r48854;
        double r48856 = 4.0;
        double r48857 = pow(r48823, r48832);
        double r48858 = r48857 / r48831;
        double r48859 = r48856 * r48858;
        double r48860 = r48855 + r48859;
        double r48861 = sqrt(r48860);
        double r48862 = r48828 / r48861;
        double r48863 = -1.323606909102099e-261;
        bool r48864 = r48823 <= r48863;
        double r48865 = 1.2779874262281365e+119;
        bool r48866 = r48823 <= r48865;
        double r48867 = r48837 + r48841;
        double r48868 = r48826 * r48867;
        double r48869 = r48826 * r48835;
        double r48870 = r48844 - r48869;
        double r48871 = r48868 + r48870;
        double r48872 = r48828 / r48871;
        double r48873 = r48866 ? r48862 : r48872;
        double r48874 = r48864 ? r48846 : r48873;
        double r48875 = r48848 ? r48862 : r48874;
        double r48876 = r48825 ? r48846 : r48875;
        return r48876;
}

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 < -2.6156735074350363e+50 or -6.111047820265304e-179 < t < -1.323606909102099e-261

    1. Initial program 48.5

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified9.3

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

    if -2.6156735074350363e+50 < t < -6.111047820265304e-179 or -1.323606909102099e-261 < t < 1.2779874262281365e+119

    1. Initial program 36.4

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

    2. Taylor expanded around inf 16.1

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

    3. Simplified16.1

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

    5. Applied *-un-lft-identity16.1

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

    6. Applied add-sqr-sqrt16.1

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

    7. Applied times-frac16.1

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

    8. Simplified16.1

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

    9. Simplified11.7

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

    if 1.2779874262281365e+119 < t

    1. Initial program 54.6

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

    2. Taylor expanded around inf 2.1

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

    3. Simplified2.1

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.615673507435036296435750238402039962931 \cdot 10^{50}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le -6.111047820265303935317690964852722629158 \cdot 10^{-179}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{elif}\;t \le -1.323606909102099087479133351528995697428 \cdot 10^{-261}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 1.277987426228136459634363360568383475924 \cdot 10^{119}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(t \cdot \sqrt{2} - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))))