Average Error: 42.6 → 8.9
Time: 1.3m
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.861930782743277 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{2}{x}}{x}\right), \left(\frac{t}{2 \cdot \sqrt{2}}\right), \left(-\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.2059628825355719 \cdot 10^{-265}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 1.0535291832943578 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\sqrt{2}\right), t, \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\ \mathbf{elif}\;t \le 2.9199902382812565 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\sqrt{2}\right), t, \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\ \end{array}\]
\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 -1.861930782743277 \cdot 10^{+82}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{2}{x}}{x}\right), \left(\frac{t}{2 \cdot \sqrt{2}}\right), \left(-\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right)\right)\right)}\\

\mathbf{elif}\;t \le 1.2059628825355719 \cdot 10^{-265}:\\
\;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}}\\

\mathbf{elif}\;t \le 1.0535291832943578 \cdot 10^{-158}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\sqrt{2}\right), t, \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\

\mathbf{elif}\;t \le 2.9199902382812565 \cdot 10^{+64}:\\
\;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r4179813 = 2.0;
        double r4179814 = sqrt(r4179813);
        double r4179815 = t;
        double r4179816 = r4179814 * r4179815;
        double r4179817 = x;
        double r4179818 = 1.0;
        double r4179819 = r4179817 + r4179818;
        double r4179820 = r4179817 - r4179818;
        double r4179821 = r4179819 / r4179820;
        double r4179822 = l;
        double r4179823 = r4179822 * r4179822;
        double r4179824 = r4179815 * r4179815;
        double r4179825 = r4179813 * r4179824;
        double r4179826 = r4179823 + r4179825;
        double r4179827 = r4179821 * r4179826;
        double r4179828 = r4179827 - r4179823;
        double r4179829 = sqrt(r4179828);
        double r4179830 = r4179816 / r4179829;
        return r4179830;
}
↓
double f(double x, double l, double t) {
        double r4179831 = t;
        double r4179832 = -1.861930782743277e+82;
        bool r4179833 = r4179831 <= r4179832;
        double r4179834 = 2.0;
        double r4179835 = sqrt(r4179834);
        double r4179836 = r4179835 * r4179831;
        double r4179837 = x;
        double r4179838 = r4179834 / r4179837;
        double r4179839 = r4179838 / r4179837;
        double r4179840 = r4179834 * r4179835;
        double r4179841 = r4179831 / r4179840;
        double r4179842 = r4179831 / r4179835;
        double r4179843 = r4179838 + r4179839;
        double r4179844 = r4179842 * r4179843;
        double r4179845 = fma(r4179831, r4179835, r4179844);
        double r4179846 = -r4179845;
        double r4179847 = fma(r4179839, r4179841, r4179846);
        double r4179848 = r4179836 / r4179847;
        double r4179849 = 1.2059628825355719e-265;
        bool r4179850 = r4179831 <= r4179849;
        double r4179851 = sqrt(r4179835);
        double r4179852 = sqrt(r4179851);
        double r4179853 = r4179831 * r4179851;
        double r4179854 = r4179852 * r4179853;
        double r4179855 = r4179854 * r4179852;
        double r4179856 = l;
        double r4179857 = r4179856 / r4179837;
        double r4179858 = r4179831 * r4179831;
        double r4179859 = fma(r4179857, r4179856, r4179858);
        double r4179860 = 4.0;
        double r4179861 = r4179860 * r4179858;
        double r4179862 = r4179861 / r4179837;
        double r4179863 = fma(r4179859, r4179834, r4179862);
        double r4179864 = sqrt(r4179863);
        double r4179865 = r4179855 / r4179864;
        double r4179866 = 1.0535291832943578e-158;
        bool r4179867 = r4179831 <= r4179866;
        double r4179868 = fma(r4179835, r4179831, r4179844);
        double r4179869 = r4179837 * r4179837;
        double r4179870 = r4179842 / r4179869;
        double r4179871 = r4179868 - r4179870;
        double r4179872 = r4179836 / r4179871;
        double r4179873 = 2.9199902382812565e+64;
        bool r4179874 = r4179831 <= r4179873;
        double r4179875 = r4179874 ? r4179865 : r4179872;
        double r4179876 = r4179867 ? r4179872 : r4179875;
        double r4179877 = r4179850 ? r4179865 : r4179876;
        double r4179878 = r4179833 ? r4179848 : r4179877;
        return r4179878;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.861930782743277e+82

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.2

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

    3. Simplified3.2

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

    if -1.861930782743277e+82 < t < 1.2059628825355719e-265 or 1.0535291832943578e-158 < t < 2.9199902382812565e+64

    1. Initial program 35.7

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

    2. Taylor expanded around -inf 15.5

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

    3. Simplified11.1

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

    5. Applied add-sqr-sqrt11.2

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

    6. Applied associate-*l*11.1

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

    8. Applied add-sqr-sqrt11.1

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

    9. Applied sqrt-prod11.1

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

    10. Applied sqrt-prod11.1

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

    11. Applied associate-*l*11.0

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

    if 1.2059628825355719e-265 < t < 1.0535291832943578e-158 or 2.9199902382812565e+64 < t

    1. Initial program 49.1

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

    2. Taylor expanded around inf 9.8

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

    3. Simplified9.8

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.861930782743277 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\frac{\frac{2}{x}}{x}\right), \left(\frac{t}{2 \cdot \sqrt{2}}\right), \left(-\mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.2059628825355719 \cdot 10^{-265}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 1.0535291832943578 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\sqrt{2}\right), t, \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\ \mathbf{elif}\;t \le 2.9199902382812565 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\sqrt{2}}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)\right) \cdot \sqrt{\sqrt{\sqrt{2}}}}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\left(\sqrt{2}\right), t, \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))))