Average Error: 43.8 → 9.8
Time: 14.6s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.1715607139623251 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]
\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 -8.1715607139623251 \cdot 10^{60}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

\mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r38800 = 2.0;
        double r38801 = sqrt(r38800);
        double r38802 = t;
        double r38803 = r38801 * r38802;
        double r38804 = x;
        double r38805 = 1.0;
        double r38806 = r38804 + r38805;
        double r38807 = r38804 - r38805;
        double r38808 = r38806 / r38807;
        double r38809 = l;
        double r38810 = r38809 * r38809;
        double r38811 = r38802 * r38802;
        double r38812 = r38800 * r38811;
        double r38813 = r38810 + r38812;
        double r38814 = r38808 * r38813;
        double r38815 = r38814 - r38810;
        double r38816 = sqrt(r38815);
        double r38817 = r38803 / r38816;
        return r38817;
}
↓
double f(double x, double l, double t) {
        double r38818 = t;
        double r38819 = -8.171560713962325e+60;
        bool r38820 = r38818 <= r38819;
        double r38821 = 2.0;
        double r38822 = sqrt(r38821);
        double r38823 = r38822 * r38818;
        double r38824 = 3.0;
        double r38825 = pow(r38822, r38824);
        double r38826 = x;
        double r38827 = 2.0;
        double r38828 = pow(r38826, r38827);
        double r38829 = r38825 * r38828;
        double r38830 = r38818 / r38829;
        double r38831 = r38822 * r38828;
        double r38832 = r38818 / r38831;
        double r38833 = r38830 - r38832;
        double r38834 = r38821 * r38833;
        double r38835 = r38822 * r38826;
        double r38836 = r38818 / r38835;
        double r38837 = r38818 * r38822;
        double r38838 = fma(r38821, r38836, r38837);
        double r38839 = r38834 - r38838;
        double r38840 = r38823 / r38839;
        double r38841 = -1.1776830540197237e-181;
        bool r38842 = r38818 <= r38841;
        double r38843 = r38818 * r38818;
        double r38844 = l;
        double r38845 = fabs(r38844);
        double r38846 = r38845 / r38826;
        double r38847 = r38845 * r38846;
        double r38848 = 4.0;
        double r38849 = pow(r38818, r38827);
        double r38850 = r38849 / r38826;
        double r38851 = r38848 * r38850;
        double r38852 = fma(r38821, r38847, r38851);
        double r38853 = fma(r38821, r38843, r38852);
        double r38854 = sqrt(r38853);
        double r38855 = r38823 / r38854;
        double r38856 = -5.07338813022938e-287;
        bool r38857 = r38818 <= r38856;
        double r38858 = 6.769189022626266e+24;
        bool r38859 = r38818 <= r38858;
        double r38860 = r38821 * r38836;
        double r38861 = fma(r38818, r38822, r38860);
        double r38862 = r38823 / r38861;
        double r38863 = r38859 ? r38855 : r38862;
        double r38864 = r38857 ? r38840 : r38863;
        double r38865 = r38842 ? r38855 : r38864;
        double r38866 = r38820 ? r38840 : r38865;
        return r38866;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -8.171560713962325e+60 or -1.1776830540197237e-181 < t < -5.07338813022938e-287

    1. Initial program 51.2

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

    2. Taylor expanded around -inf 11.5

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

    3. Simplified11.5

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

    if -8.171560713962325e+60 < t < -1.1776830540197237e-181 or -5.07338813022938e-287 < t < 6.769189022626266e+24

    1. Initial program 39.5

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

    2. Taylor expanded around inf 16.3

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

    3. Simplified16.3

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

    5. Applied *-un-lft-identity16.3

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

    6. Applied add-sqr-sqrt16.3

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

    7. Applied times-frac16.3

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

    8. Simplified16.3

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

    9. Simplified12.0

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

    if 6.769189022626266e+24 < t

    1. Initial program 41.9

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

    2. Taylor expanded around inf 39.8

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

    3. Simplified39.8

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

    4. Taylor expanded around inf 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    5. Simplified4.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.1715607139623251 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))))