Average Error: 42.2 → 9.0
Time: 30.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.491422032829388 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -2.2098329558361532 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le -5.237639958129932 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 3.5070992056927235 \cdot 10^{+116}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]
\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 -5.491422032829388 \cdot 10^{+54}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le -2.2098329558361532 \cdot 10^{-133}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\

\mathbf{elif}\;t \le -5.237639958129932 \cdot 10^{-222}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 3.5070992056927235 \cdot 10^{+116}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1381935 = 2.0;
        double r1381936 = sqrt(r1381935);
        double r1381937 = t;
        double r1381938 = r1381936 * r1381937;
        double r1381939 = x;
        double r1381940 = 1.0;
        double r1381941 = r1381939 + r1381940;
        double r1381942 = r1381939 - r1381940;
        double r1381943 = r1381941 / r1381942;
        double r1381944 = l;
        double r1381945 = r1381944 * r1381944;
        double r1381946 = r1381937 * r1381937;
        double r1381947 = r1381935 * r1381946;
        double r1381948 = r1381945 + r1381947;
        double r1381949 = r1381943 * r1381948;
        double r1381950 = r1381949 - r1381945;
        double r1381951 = sqrt(r1381950);
        double r1381952 = r1381938 / r1381951;
        return r1381952;
}
↓
double f(double x, double l, double t) {
        double r1381953 = t;
        double r1381954 = -5.491422032829388e+54;
        bool r1381955 = r1381953 <= r1381954;
        double r1381956 = 2.0;
        double r1381957 = sqrt(r1381956);
        double r1381958 = r1381957 * r1381953;
        double r1381959 = r1381953 / r1381957;
        double r1381960 = x;
        double r1381961 = r1381960 * r1381960;
        double r1381962 = r1381959 / r1381961;
        double r1381963 = r1381960 * r1381957;
        double r1381964 = r1381953 / r1381963;
        double r1381965 = r1381964 + r1381962;
        double r1381966 = r1381965 * r1381956;
        double r1381967 = r1381962 - r1381966;
        double r1381968 = r1381967 - r1381958;
        double r1381969 = r1381958 / r1381968;
        double r1381970 = -2.2098329558361532e-133;
        bool r1381971 = r1381953 <= r1381970;
        double r1381972 = r1381953 * r1381953;
        double r1381973 = r1381972 / r1381960;
        double r1381974 = 4.0;
        double r1381975 = r1381973 * r1381974;
        double r1381976 = l;
        double r1381977 = r1381976 / r1381960;
        double r1381978 = r1381976 * r1381977;
        double r1381979 = r1381978 + r1381972;
        double r1381980 = r1381956 * r1381979;
        double r1381981 = r1381975 + r1381980;
        double r1381982 = sqrt(r1381981);
        double r1381983 = r1381958 / r1381982;
        double r1381984 = -5.237639958129932e-222;
        bool r1381985 = r1381953 <= r1381984;
        double r1381986 = 3.5070992056927235e+116;
        bool r1381987 = r1381953 <= r1381986;
        double r1381988 = r1381958 - r1381962;
        double r1381989 = r1381966 + r1381988;
        double r1381990 = r1381958 / r1381989;
        double r1381991 = r1381987 ? r1381983 : r1381990;
        double r1381992 = r1381985 ? r1381969 : r1381991;
        double r1381993 = r1381971 ? r1381983 : r1381992;
        double r1381994 = r1381955 ? r1381969 : r1381993;
        return r1381994;
}

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 < -5.491422032829388e+54 or -2.2098329558361532e-133 < t < -5.237639958129932e-222

    1. Initial program 46.2

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -5.491422032829388e+54 < t < -2.2098329558361532e-133 or -5.237639958129932e-222 < t < 3.5070992056927235e+116

    1. Initial program 35.8

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

    2. Taylor expanded around inf 15.7

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

    3. Simplified15.7

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

    5. Applied associate-/l*11.6

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

    6. Taylor expanded around inf 15.7

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

    7. Simplified11.6

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

    if 3.5070992056927235e+116 < t

    1. Initial program 53.3

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

    2. Taylor expanded around inf 2.5

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

    3. Simplified2.5

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.491422032829388 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -2.2098329558361532 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{elif}\;t \le -5.237639958129932 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} - \left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 3.5070992056927235 \cdot 10^{+116}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{t \cdot t}{x} \cdot 4 + 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{x \cdot \sqrt{2}} + \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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)))))