Average Error: 42.7 → 9.0
Time: 16.2s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.606623675187166320307147356355623739061 \cdot 10^{75}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 4.894745026611551231238356257343815376774 \cdot 10^{-226} \lor \neg \left(t \le 5.832383717323896654554068457322403291209 \cdot 10^{-160}\right) \land t \le 4869868199152865435985673923804373450752:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} - 2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]
\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 -1.606623675187166320307147356355623739061 \cdot 10^{75}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 4.894745026611551231238356257343815376774 \cdot 10^{-226} \lor \neg \left(t \le 5.832383717323896654554068457322403291209 \cdot 10^{-160}\right) \land t \le 4869868199152865435985673923804373450752:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} - 2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r37904 = 2.0;
        double r37905 = sqrt(r37904);
        double r37906 = t;
        double r37907 = r37905 * r37906;
        double r37908 = x;
        double r37909 = 1.0;
        double r37910 = r37908 + r37909;
        double r37911 = r37908 - r37909;
        double r37912 = r37910 / r37911;
        double r37913 = l;
        double r37914 = r37913 * r37913;
        double r37915 = r37906 * r37906;
        double r37916 = r37904 * r37915;
        double r37917 = r37914 + r37916;
        double r37918 = r37912 * r37917;
        double r37919 = r37918 - r37914;
        double r37920 = sqrt(r37919);
        double r37921 = r37907 / r37920;
        return r37921;
}
↓
double f(double x, double l, double t) {
        double r37922 = t;
        double r37923 = -1.6066236751871663e+75;
        bool r37924 = r37922 <= r37923;
        double r37925 = 2.0;
        double r37926 = sqrt(r37925);
        double r37927 = r37926 * r37922;
        double r37928 = x;
        double r37929 = 2.0;
        double r37930 = pow(r37928, r37929);
        double r37931 = r37922 / r37930;
        double r37932 = r37926 * r37925;
        double r37933 = r37925 / r37932;
        double r37934 = r37925 / r37926;
        double r37935 = r37933 - r37934;
        double r37936 = r37931 * r37935;
        double r37937 = r37936 - r37927;
        double r37938 = r37926 * r37928;
        double r37939 = r37922 / r37938;
        double r37940 = r37925 * r37939;
        double r37941 = r37937 - r37940;
        double r37942 = r37927 / r37941;
        double r37943 = 4.894745026611551e-226;
        bool r37944 = r37922 <= r37943;
        double r37945 = 5.832383717323897e-160;
        bool r37946 = r37922 <= r37945;
        double r37947 = !r37946;
        double r37948 = 4.8698681991528654e+39;
        bool r37949 = r37922 <= r37948;
        bool r37950 = r37947 && r37949;
        bool r37951 = r37944 || r37950;
        double r37952 = cbrt(r37926);
        double r37953 = r37952 * r37952;
        double r37954 = r37952 * r37922;
        double r37955 = r37953 * r37954;
        double r37956 = 4.0;
        double r37957 = pow(r37922, r37929);
        double r37958 = r37957 / r37928;
        double r37959 = r37956 * r37958;
        double r37960 = r37922 * r37922;
        double r37961 = l;
        double r37962 = r37929 / r37929;
        double r37963 = pow(r37961, r37962);
        double r37964 = r37928 / r37961;
        double r37965 = r37963 / r37964;
        double r37966 = r37960 + r37965;
        double r37967 = r37925 * r37966;
        double r37968 = r37959 + r37967;
        double r37969 = sqrt(r37968);
        double r37970 = r37955 / r37969;
        double r37971 = r37922 * r37926;
        double r37972 = 3.0;
        double r37973 = pow(r37926, r37972);
        double r37974 = r37973 * r37930;
        double r37975 = r37922 / r37974;
        double r37976 = r37975 - r37939;
        double r37977 = r37925 * r37976;
        double r37978 = r37971 - r37977;
        double r37979 = r37927 / r37978;
        double r37980 = r37951 ? r37970 : r37979;
        double r37981 = r37924 ? r37942 : r37980;
        return r37981;
}

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 < -1.6066236751871663e+75

    1. Initial program 45.7

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -1.6066236751871663e+75 < t < 4.894745026611551e-226 or 5.832383717323897e-160 < t < 4.8698681991528654e+39

    1. Initial program 38.5

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

    2. Taylor expanded around inf 16.4

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

    3. Simplified16.4

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

    5. Applied sqr-pow16.4

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

    6. Applied associate-/l*12.5

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

    7. Simplified12.5

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

    9. Applied add-cube-cbrt12.5

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

    10. Applied associate-*l*12.5

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

    if 4.894745026611551e-226 < t < 5.832383717323897e-160 or 4.8698681991528654e+39 < t

    1. Initial program 47.3

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

    2. Taylor expanded around inf 42.4

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

    3. Simplified42.4

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

    5. Applied sqr-pow42.4

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

    6. Applied associate-/l*40.0

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

    7. Simplified40.0

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

    8. Taylor expanded around inf 8.0

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

    9. Simplified8.0

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.606623675187166320307147356355623739061 \cdot 10^{75}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 4.894745026611551231238356257343815376774 \cdot 10^{-226} \lor \neg \left(t \le 5.832383717323896654554068457322403291209 \cdot 10^{-160}\right) \land t \le 4869868199152865435985673923804373450752:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(t \cdot t + \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} - 2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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)))))