Average Error: 43.5 → 9.8
Time: 37.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.24254393201664138044439133386542443571 \cdot 10^{86}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{-\left(\sqrt{2} \cdot t + \frac{2 \cdot t}{x \cdot \sqrt{2}}\right)}{t}}\\ \mathbf{elif}\;t \le 3.424779697706979355832148987272206628453 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(2 \cdot \left(t \cdot t\right) + \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \left(\sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}}\right)\right)}}{t}}\\ \mathbf{elif}\;t \le 6.783164823431965385056094361766235146363 \cdot 10^{-165} \lor \neg \left(t \le 5.469601309486936787300845358655254676933 \cdot 10^{45}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot t + \frac{2 \cdot t}{x \cdot \sqrt{2}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\sqrt{\left(\frac{4}{\frac{\frac{x}{t}}{t}} + \frac{\ell \cdot 2}{\frac{x}{\ell}}\right) + t \cdot \left(2 \cdot t\right)}} \cdot \sqrt{\sqrt{\left(\frac{4}{\frac{\frac{x}{t}}{t}} + \frac{\ell \cdot 2}{\frac{x}{\ell}}\right) + t \cdot \left(2 \cdot t\right)}}}{t}}\\ \end{array}\]
\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 -2.24254393201664138044439133386542443571 \cdot 10^{86}:\\
\;\;\;\;\frac{\sqrt{2}}{\frac{-\left(\sqrt{2} \cdot t + \frac{2 \cdot t}{x \cdot \sqrt{2}}\right)}{t}}\\

\mathbf{elif}\;t \le 3.424779697706979355832148987272206628453 \cdot 10^{-203}:\\
\;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(2 \cdot \left(t \cdot t\right) + \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \left(\sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}}\right)\right)}}{t}}\\

\mathbf{elif}\;t \le 6.783164823431965385056094361766235146363 \cdot 10^{-165} \lor \neg \left(t \le 5.469601309486936787300845358655254676933 \cdot 10^{45}\right):\\
\;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot t + \frac{2 \cdot t}{x \cdot \sqrt{2}}}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\sqrt{\left(\frac{4}{\frac{\frac{x}{t}}{t}} + \frac{\ell \cdot 2}{\frac{x}{\ell}}\right) + t \cdot \left(2 \cdot t\right)}} \cdot \sqrt{\sqrt{\left(\frac{4}{\frac{\frac{x}{t}}{t}} + \frac{\ell \cdot 2}{\frac{x}{\ell}}\right) + t \cdot \left(2 \cdot t\right)}}}{t}}\\

\end{array}
double f(double x, double l, double t) {
        double r45813 = 2.0;
        double r45814 = sqrt(r45813);
        double r45815 = t;
        double r45816 = r45814 * r45815;
        double r45817 = x;
        double r45818 = 1.0;
        double r45819 = r45817 + r45818;
        double r45820 = r45817 - r45818;
        double r45821 = r45819 / r45820;
        double r45822 = l;
        double r45823 = r45822 * r45822;
        double r45824 = r45815 * r45815;
        double r45825 = r45813 * r45824;
        double r45826 = r45823 + r45825;
        double r45827 = r45821 * r45826;
        double r45828 = r45827 - r45823;
        double r45829 = sqrt(r45828);
        double r45830 = r45816 / r45829;
        return r45830;
}
↓
double f(double x, double l, double t) {
        double r45831 = t;
        double r45832 = -2.2425439320166414e+86;
        bool r45833 = r45831 <= r45832;
        double r45834 = 2.0;
        double r45835 = sqrt(r45834);
        double r45836 = r45835 * r45831;
        double r45837 = r45834 * r45831;
        double r45838 = x;
        double r45839 = r45838 * r45835;
        double r45840 = r45837 / r45839;
        double r45841 = r45836 + r45840;
        double r45842 = -r45841;
        double r45843 = r45842 / r45831;
        double r45844 = r45835 / r45843;
        double r45845 = 3.4247796977069794e-203;
        bool r45846 = r45831 <= r45845;
        double r45847 = 4.0;
        double r45848 = r45831 * r45831;
        double r45849 = r45838 / r45848;
        double r45850 = r45847 / r45849;
        double r45851 = r45834 * r45848;
        double r45852 = l;
        double r45853 = r45852 * r45834;
        double r45854 = r45838 / r45852;
        double r45855 = r45853 / r45854;
        double r45856 = cbrt(r45855);
        double r45857 = r45856 * r45856;
        double r45858 = r45856 * r45857;
        double r45859 = r45851 + r45858;
        double r45860 = r45850 + r45859;
        double r45861 = sqrt(r45860);
        double r45862 = r45861 / r45831;
        double r45863 = r45835 / r45862;
        double r45864 = 6.783164823431965e-165;
        bool r45865 = r45831 <= r45864;
        double r45866 = 5.469601309486937e+45;
        bool r45867 = r45831 <= r45866;
        double r45868 = !r45867;
        bool r45869 = r45865 || r45868;
        double r45870 = r45841 / r45831;
        double r45871 = r45835 / r45870;
        double r45872 = r45838 / r45831;
        double r45873 = r45872 / r45831;
        double r45874 = r45847 / r45873;
        double r45875 = r45874 + r45855;
        double r45876 = r45831 * r45837;
        double r45877 = r45875 + r45876;
        double r45878 = sqrt(r45877);
        double r45879 = sqrt(r45878);
        double r45880 = r45879 * r45879;
        double r45881 = r45880 / r45831;
        double r45882 = r45835 / r45881;
        double r45883 = r45869 ? r45871 : r45882;
        double r45884 = r45846 ? r45863 : r45883;
        double r45885 = r45833 ? r45844 : r45884;
        return r45885;
}

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 4 regimes

  2. if t < -2.2425439320166414e+86

    1. Initial program 49.0

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

    2. Simplified49.0

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

    3. Taylor expanded around -inf 3.7

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

    4. Simplified3.7

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

    if -2.2425439320166414e+86 < t < 3.4247796977069794e-203

    1. Initial program 43.2

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

    2. Simplified43.2

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

    3. Taylor expanded around inf 20.9

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

    4. Simplified20.9

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

    6. Applied add-cube-cbrt21.0

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(\color{blue}{\left(\sqrt[3]{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}} \cdot \sqrt[3]{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}\right) \cdot \sqrt[3]{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}} + \left(t \cdot t\right) \cdot 2\right)}}{t}}\]

    7. Simplified21.0

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(\color{blue}{\left(\sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}}\right)} \cdot \sqrt[3]{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}} + \left(t \cdot t\right) \cdot 2\right)}}{t}}\]

    8. Simplified17.5

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(\left(\sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}}} + \left(t \cdot t\right) \cdot 2\right)}}{t}}\]

    if 3.4247796977069794e-203 < t < 6.783164823431965e-165 or 5.469601309486937e+45 < t

    1. Initial program 47.4

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

    2. Simplified47.4

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

    3. Taylor expanded around inf 6.8

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

    4. Simplified6.8

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

    if 6.783164823431965e-165 < t < 5.469601309486937e+45

    1. Initial program 30.5

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

    2. Simplified30.5

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

    3. Taylor expanded around inf 10.5

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

    4. Simplified10.5

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

    6. Applied add-sqr-sqrt10.5

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

    7. Applied sqrt-prod10.6

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

    8. Simplified10.6

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

    9. Simplified5.7

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + \frac{4}{\frac{\frac{x}{t}}{t}}\right) + t \cdot \left(t \cdot 2\right)}} \cdot \color{blue}{\sqrt{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + \frac{4}{\frac{\frac{x}{t}}{t}}\right) + t \cdot \left(t \cdot 2\right)}}}}{t}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.24254393201664138044439133386542443571 \cdot 10^{86}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{-\left(\sqrt{2} \cdot t + \frac{2 \cdot t}{x \cdot \sqrt{2}}\right)}{t}}\\ \mathbf{elif}\;t \le 3.424779697706979355832148987272206628453 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\frac{4}{\frac{x}{t \cdot t}} + \left(2 \cdot \left(t \cdot t\right) + \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \left(\sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}} \cdot \sqrt[3]{\frac{\ell \cdot 2}{\frac{x}{\ell}}}\right)\right)}}{t}}\\ \mathbf{elif}\;t \le 6.783164823431965385056094361766235146363 \cdot 10^{-165} \lor \neg \left(t \le 5.469601309486936787300845358655254676933 \cdot 10^{45}\right):\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{2} \cdot t + \frac{2 \cdot t}{x \cdot \sqrt{2}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\sqrt{\left(\frac{4}{\frac{\frac{x}{t}}{t}} + \frac{\ell \cdot 2}{\frac{x}{\ell}}\right) + t \cdot \left(2 \cdot t\right)}} \cdot \sqrt{\sqrt{\left(\frac{4}{\frac{\frac{x}{t}}{t}} + \frac{\ell \cdot 2}{\frac{x}{\ell}}\right) + t \cdot \left(2 \cdot t\right)}}}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))