Average Error: 42.7 → 9.3
Time: 42.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.2693742662560688 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le -4.157152618984683 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le -2.3234922006583374 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.512500005971486 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le 1.672688434817364 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \mathbf{elif}\;t \le 1.0420326231034605 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \end{array}\]
\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 -2.2693742662560688 \cdot 10^{+103}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le -4.157152618984683 \cdot 10^{-167}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\

\mathbf{elif}\;t \le -2.3234922006583374 \cdot 10^{-268}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 1.512500005971486 \cdot 10^{-194}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\

\mathbf{elif}\;t \le 1.672688434817364 \cdot 10^{-158}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\

\mathbf{elif}\;t \le 1.0420326231034605 \cdot 10^{+133}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1665854 = 2.0;
        double r1665855 = sqrt(r1665854);
        double r1665856 = t;
        double r1665857 = r1665855 * r1665856;
        double r1665858 = x;
        double r1665859 = 1.0;
        double r1665860 = r1665858 + r1665859;
        double r1665861 = r1665858 - r1665859;
        double r1665862 = r1665860 / r1665861;
        double r1665863 = l;
        double r1665864 = r1665863 * r1665863;
        double r1665865 = r1665856 * r1665856;
        double r1665866 = r1665854 * r1665865;
        double r1665867 = r1665864 + r1665866;
        double r1665868 = r1665862 * r1665867;
        double r1665869 = r1665868 - r1665864;
        double r1665870 = sqrt(r1665869);
        double r1665871 = r1665857 / r1665870;
        return r1665871;
}
↓
double f(double x, double l, double t) {
        double r1665872 = t;
        double r1665873 = -2.2693742662560688e+103;
        bool r1665874 = r1665872 <= r1665873;
        double r1665875 = 2.0;
        double r1665876 = sqrt(r1665875);
        double r1665877 = r1665876 * r1665872;
        double r1665878 = 1.0;
        double r1665879 = r1665878 / r1665876;
        double r1665880 = r1665875 / r1665876;
        double r1665881 = r1665879 - r1665880;
        double r1665882 = x;
        double r1665883 = r1665882 * r1665882;
        double r1665884 = r1665872 / r1665883;
        double r1665885 = r1665881 * r1665884;
        double r1665886 = r1665882 * r1665876;
        double r1665887 = r1665875 / r1665886;
        double r1665888 = r1665887 * r1665872;
        double r1665889 = r1665888 + r1665877;
        double r1665890 = r1665885 - r1665889;
        double r1665891 = r1665877 / r1665890;
        double r1665892 = -4.157152618984683e-167;
        bool r1665893 = r1665872 <= r1665892;
        double r1665894 = 4.0;
        double r1665895 = r1665882 / r1665872;
        double r1665896 = r1665895 / r1665872;
        double r1665897 = r1665894 / r1665896;
        double r1665898 = l;
        double r1665899 = r1665882 / r1665898;
        double r1665900 = r1665898 / r1665899;
        double r1665901 = r1665872 * r1665872;
        double r1665902 = r1665900 + r1665901;
        double r1665903 = r1665875 * r1665902;
        double r1665904 = r1665897 + r1665903;
        double r1665905 = sqrt(r1665904);
        double r1665906 = r1665877 / r1665905;
        double r1665907 = -2.3234922006583374e-268;
        bool r1665908 = r1665872 <= r1665907;
        double r1665909 = 1.512500005971486e-194;
        bool r1665910 = r1665872 <= r1665909;
        double r1665911 = 1.672688434817364e-158;
        bool r1665912 = r1665872 <= r1665911;
        double r1665913 = r1665889 - r1665885;
        double r1665914 = r1665877 / r1665913;
        double r1665915 = 1.0420326231034605e+133;
        bool r1665916 = r1665872 <= r1665915;
        double r1665917 = r1665916 ? r1665906 : r1665914;
        double r1665918 = r1665912 ? r1665914 : r1665917;
        double r1665919 = r1665910 ? r1665906 : r1665918;
        double r1665920 = r1665908 ? r1665891 : r1665919;
        double r1665921 = r1665893 ? r1665906 : r1665920;
        double r1665922 = r1665874 ? r1665891 : r1665921;
        return r1665922;
}

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 < -2.2693742662560688e+103 or -4.157152618984683e-167 < t < -2.3234922006583374e-268

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 11.0

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

    3. Simplified11.0

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

    if -2.2693742662560688e+103 < t < -4.157152618984683e-167 or -2.3234922006583374e-268 < t < 1.512500005971486e-194 or 1.672688434817364e-158 < t < 1.0420326231034605e+133

    1. Initial program 32.3

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

    2. Taylor expanded around inf 14.3

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

    3. Simplified14.3

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

    5. Applied *-un-lft-identity14.3

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

    6. Applied times-frac9.8

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

    7. Simplified9.8

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

    8. Taylor expanded around inf 14.3

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

    9. Simplified9.8

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

    if 1.512500005971486e-194 < t < 1.672688434817364e-158 or 1.0420326231034605e+133 < t

    1. Initial program 56.6

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

    2. Taylor expanded around inf 5.4

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

    3. Simplified5.4

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2693742662560688 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le -4.157152618984683 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le -2.3234922006583374 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.512500005971486 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le 1.672688434817364 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \mathbf{elif}\;t \le 1.0420326231034605 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{4}{\frac{\frac{x}{t}}{t}} + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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)))))