Average Error: 42.6 → 10.0
Time: 32.3s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.715046482673698729450242436769280905141 \cdot 10^{110}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le -1.993349418678108800529215067343269974318 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{elif}\;t \le -9.405973531316698506909205844553614944599 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 8.799119845230040103380880957879300899262 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{elif}\;t \le 3.517775087045906516144188620937021386678 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\ \mathbf{elif}\;t \le 4.39891585056256545899226457368516974501 \cdot 10^{111}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\ \end{array}\]
\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 -3.715046482673698729450242436769280905141 \cdot 10^{110}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le -1.993349418678108800529215067343269974318 \cdot 10^{-159}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\

\mathbf{elif}\;t \le -9.405973531316698506909205844553614944599 \cdot 10^{-275}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 8.799119845230040103380880957879300899262 \cdot 10^{-305}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\

\mathbf{elif}\;t \le 3.517775087045906516144188620937021386678 \cdot 10^{-239}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\

\mathbf{elif}\;t \le 4.39891585056256545899226457368516974501 \cdot 10^{111}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1756883 = 2.0;
        double r1756884 = sqrt(r1756883);
        double r1756885 = t;
        double r1756886 = r1756884 * r1756885;
        double r1756887 = x;
        double r1756888 = 1.0;
        double r1756889 = r1756887 + r1756888;
        double r1756890 = r1756887 - r1756888;
        double r1756891 = r1756889 / r1756890;
        double r1756892 = l;
        double r1756893 = r1756892 * r1756892;
        double r1756894 = r1756885 * r1756885;
        double r1756895 = r1756883 * r1756894;
        double r1756896 = r1756893 + r1756895;
        double r1756897 = r1756891 * r1756896;
        double r1756898 = r1756897 - r1756893;
        double r1756899 = sqrt(r1756898);
        double r1756900 = r1756886 / r1756899;
        return r1756900;
}
↓
double f(double x, double l, double t) {
        double r1756901 = t;
        double r1756902 = -3.715046482673699e+110;
        bool r1756903 = r1756901 <= r1756902;
        double r1756904 = 2.0;
        double r1756905 = sqrt(r1756904);
        double r1756906 = r1756905 * r1756901;
        double r1756907 = x;
        double r1756908 = r1756904 * r1756905;
        double r1756909 = r1756907 * r1756908;
        double r1756910 = r1756904 / r1756909;
        double r1756911 = r1756901 / r1756907;
        double r1756912 = r1756910 * r1756911;
        double r1756913 = r1756904 / r1756907;
        double r1756914 = r1756901 / r1756905;
        double r1756915 = r1756913 * r1756914;
        double r1756916 = r1756906 + r1756915;
        double r1756917 = r1756912 - r1756916;
        double r1756918 = r1756914 / r1756907;
        double r1756919 = r1756918 * r1756913;
        double r1756920 = r1756917 - r1756919;
        double r1756921 = r1756906 / r1756920;
        double r1756922 = -1.9933494186781088e-159;
        bool r1756923 = r1756901 <= r1756922;
        double r1756924 = l;
        double r1756925 = cbrt(r1756907);
        double r1756926 = r1756925 * r1756925;
        double r1756927 = r1756924 / r1756926;
        double r1756928 = r1756924 / r1756925;
        double r1756929 = r1756927 * r1756928;
        double r1756930 = r1756901 * r1756901;
        double r1756931 = r1756929 + r1756930;
        double r1756932 = r1756904 * r1756931;
        double r1756933 = r1756930 / r1756907;
        double r1756934 = 4.0;
        double r1756935 = r1756933 * r1756934;
        double r1756936 = r1756932 + r1756935;
        double r1756937 = sqrt(r1756936);
        double r1756938 = r1756906 / r1756937;
        double r1756939 = -9.405973531316699e-275;
        bool r1756940 = r1756901 <= r1756939;
        double r1756941 = 8.79911984523004e-305;
        bool r1756942 = r1756901 <= r1756941;
        double r1756943 = r1756924 * r1756924;
        double r1756944 = r1756943 / r1756907;
        double r1756945 = r1756944 + r1756930;
        double r1756946 = r1756904 * r1756945;
        double r1756947 = r1756946 + r1756935;
        double r1756948 = sqrt(r1756947);
        double r1756949 = r1756906 / r1756948;
        double r1756950 = 3.5177750870459065e-239;
        bool r1756951 = r1756901 <= r1756950;
        double r1756952 = r1756919 - r1756912;
        double r1756953 = r1756916 + r1756952;
        double r1756954 = r1756906 / r1756953;
        double r1756955 = 4.3989158505625655e+111;
        bool r1756956 = r1756901 <= r1756955;
        double r1756957 = r1756956 ? r1756938 : r1756954;
        double r1756958 = r1756951 ? r1756954 : r1756957;
        double r1756959 = r1756942 ? r1756949 : r1756958;
        double r1756960 = r1756940 ? r1756921 : r1756959;
        double r1756961 = r1756923 ? r1756938 : r1756960;
        double r1756962 = r1756903 ? r1756921 : r1756961;
        return r1756962;
}

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 < -3.715046482673699e+110 or -1.9933494186781088e-159 < t < -9.405973531316699e-275

    1. Initial program 55.3

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

    2. Taylor expanded around -inf 11.2

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

    3. Simplified11.2

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

    if -3.715046482673699e+110 < t < -1.9933494186781088e-159 or 3.5177750870459065e-239 < t < 4.3989158505625655e+111

    1. Initial program 29.2

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

    2. Taylor expanded around inf 13.0

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

    3. Simplified13.0

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

    5. Applied add-cube-cbrt13.0

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

    6. Applied times-frac8.5

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

    if -9.405973531316699e-275 < t < 8.79911984523004e-305

    1. Initial program 62.1

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

    2. Taylor expanded around inf 27.6

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

    3. Simplified27.6

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

    if 8.79911984523004e-305 < t < 3.5177750870459065e-239 or 4.3989158505625655e+111 < t

    1. Initial program 54.1

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

    2. Taylor expanded around inf 10.0

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

    3. Simplified10.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.715046482673698729450242436769280905141 \cdot 10^{110}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le -1.993349418678108800529215067343269974318 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{elif}\;t \le -9.405973531316698506909205844553614944599 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x} - \left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right)\right) - \frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 8.799119845230040103380880957879300899262 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{elif}\;t \le 3.517775087045906516144188620937021386678 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\ \mathbf{elif}\;t \le 4.39891585056256545899226457368516974501 \cdot 10^{111}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}} + t \cdot t\right) + \frac{t \cdot t}{x} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t + \frac{2}{x} \cdot \frac{t}{\sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x} \cdot \frac{2}{x} - \frac{2}{x \cdot \left(2 \cdot \sqrt{2}\right)} \cdot \frac{t}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))))