Average Error: 42.5 → 9.0
Time: 46.8s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.429472263354247 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le -1.8034323523291548 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -3.737726633798208 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 5.047696061686955 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le 1.351904254863679 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} + \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.1429092917229189 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} + \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right)}\\ \end{array}\]
\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 -1.429472263354247 \cdot 10^{+67}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le -1.8034323523291548 \cdot 10^{-167}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\

\mathbf{elif}\;t \le -3.737726633798208 \cdot 10^{-276}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 5.047696061686955 \cdot 10^{-240}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\

\mathbf{elif}\;t \le 1.351904254863679 \cdot 10^{-161}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} + \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 1.1429092917229189 \cdot 10^{+136}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1810787 = 2.0;
        double r1810788 = sqrt(r1810787);
        double r1810789 = t;
        double r1810790 = r1810788 * r1810789;
        double r1810791 = x;
        double r1810792 = 1.0;
        double r1810793 = r1810791 + r1810792;
        double r1810794 = r1810791 - r1810792;
        double r1810795 = r1810793 / r1810794;
        double r1810796 = l;
        double r1810797 = r1810796 * r1810796;
        double r1810798 = r1810789 * r1810789;
        double r1810799 = r1810787 * r1810798;
        double r1810800 = r1810797 + r1810799;
        double r1810801 = r1810795 * r1810800;
        double r1810802 = r1810801 - r1810797;
        double r1810803 = sqrt(r1810802);
        double r1810804 = r1810790 / r1810803;
        return r1810804;
}
↓
double f(double x, double l, double t) {
        double r1810805 = t;
        double r1810806 = -1.429472263354247e+67;
        bool r1810807 = r1810805 <= r1810806;
        double r1810808 = 2.0;
        double r1810809 = sqrt(r1810808);
        double r1810810 = r1810809 * r1810805;
        double r1810811 = r1810808 * r1810809;
        double r1810812 = r1810805 / r1810811;
        double r1810813 = r1810805 / r1810809;
        double r1810814 = r1810812 - r1810813;
        double r1810815 = x;
        double r1810816 = r1810808 / r1810815;
        double r1810817 = r1810816 / r1810815;
        double r1810818 = r1810814 * r1810817;
        double r1810819 = r1810818 - r1810810;
        double r1810820 = r1810813 * r1810816;
        double r1810821 = r1810819 - r1810820;
        double r1810822 = r1810810 / r1810821;
        double r1810823 = -1.8034323523291548e-167;
        bool r1810824 = r1810805 <= r1810823;
        double r1810825 = r1810805 * r1810805;
        double r1810826 = l;
        double r1810827 = r1810815 / r1810826;
        double r1810828 = r1810826 / r1810827;
        double r1810829 = r1810825 + r1810828;
        double r1810830 = r1810808 * r1810829;
        double r1810831 = 4.0;
        double r1810832 = r1810831 / r1810815;
        double r1810833 = r1810832 * r1810825;
        double r1810834 = r1810830 + r1810833;
        double r1810835 = sqrt(r1810834);
        double r1810836 = r1810810 / r1810835;
        double r1810837 = -3.737726633798208e-276;
        bool r1810838 = r1810805 <= r1810837;
        double r1810839 = 5.047696061686955e-240;
        bool r1810840 = r1810805 <= r1810839;
        double r1810841 = 1.351904254863679e-161;
        bool r1810842 = r1810805 <= r1810841;
        double r1810843 = r1810813 - r1810812;
        double r1810844 = r1810843 * r1810817;
        double r1810845 = r1810820 + r1810810;
        double r1810846 = r1810844 + r1810845;
        double r1810847 = r1810810 / r1810846;
        double r1810848 = 1.1429092917229189e+136;
        bool r1810849 = r1810805 <= r1810848;
        double r1810850 = r1810849 ? r1810836 : r1810847;
        double r1810851 = r1810842 ? r1810847 : r1810850;
        double r1810852 = r1810840 ? r1810836 : r1810851;
        double r1810853 = r1810838 ? r1810822 : r1810852;
        double r1810854 = r1810824 ? r1810836 : r1810853;
        double r1810855 = r1810807 ? r1810822 : r1810854;
        return r1810855;
}

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 < -1.429472263354247e+67 or -1.8034323523291548e-167 < t < -3.737726633798208e-276

    1. Initial program 49.2

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

    2. Taylor expanded around -inf 10.7

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

    3. Simplified10.7

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

    if -1.429472263354247e+67 < t < -1.8034323523291548e-167 or -3.737726633798208e-276 < t < 5.047696061686955e-240 or 1.351904254863679e-161 < t < 1.1429092917229189e+136

    1. Initial program 30.6

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

    2. Taylor expanded around inf 12.6

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

    3. Simplified8.1

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

    if 5.047696061686955e-240 < t < 1.351904254863679e-161 or 1.1429092917229189e+136 < t

    1. Initial program 58.2

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

    2. Taylor expanded around inf 8.8

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

    3. Simplified8.8

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.429472263354247 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le -1.8034323523291548 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le -3.737726633798208 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 5.047696061686955 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le 1.351904254863679 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} + \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.1429092917229189 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right) + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} + \left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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)))))