Average Error: 48.1 → 10.2
Time: 1.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 4.421956979468318490074797826886651301589 \cdot 10^{-50} \lor \neg \left(t \le 4.542715892992587199784907751836612646733 \cdot 10^{84}\right):\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{{\left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right)}^{-1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right)\right) \cdot \left(\frac{1}{\tan k} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\ell \cdot \frac{2}{\sin k}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\tan k}\right)\\ \end{array}\]
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;t \le 4.421956979468318490074797826886651301589 \cdot 10^{-50} \lor \neg \left(t \le 4.542715892992587199784907751836612646733 \cdot 10^{84}\right):\\
\;\;\;\;\left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{{\left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right)}^{-1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right)\right) \cdot \left(\frac{1}{\tan k} \cdot \ell\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\ell \cdot \frac{2}{\sin k}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\tan k}\right)\\

\end{array}
double f(double t, double l, double k) {
        double r123779 = 2.0;
        double r123780 = t;
        double r123781 = 3.0;
        double r123782 = pow(r123780, r123781);
        double r123783 = l;
        double r123784 = r123783 * r123783;
        double r123785 = r123782 / r123784;
        double r123786 = k;
        double r123787 = sin(r123786);
        double r123788 = r123785 * r123787;
        double r123789 = tan(r123786);
        double r123790 = r123788 * r123789;
        double r123791 = 1.0;
        double r123792 = r123786 / r123780;
        double r123793 = pow(r123792, r123779);
        double r123794 = r123791 + r123793;
        double r123795 = r123794 - r123791;
        double r123796 = r123790 * r123795;
        double r123797 = r123779 / r123796;
        return r123797;
}
↓
double f(double t, double l, double k) {
        double r123798 = t;
        double r123799 = 4.4219569794683185e-50;
        bool r123800 = r123798 <= r123799;
        double r123801 = 4.542715892992587e+84;
        bool r123802 = r123798 <= r123801;
        double r123803 = !r123802;
        bool r123804 = r123800 || r123803;
        double r123805 = l;
        double r123806 = k;
        double r123807 = sin(r123806);
        double r123808 = r123805 / r123807;
        double r123809 = 1.0;
        double r123810 = pow(r123798, r123809);
        double r123811 = 2.0;
        double r123812 = 2.0;
        double r123813 = r123811 / r123812;
        double r123814 = pow(r123806, r123813);
        double r123815 = r123810 * r123814;
        double r123816 = -1.0;
        double r123817 = pow(r123815, r123816);
        double r123818 = r123817 / r123814;
        double r123819 = pow(r123818, r123809);
        double r123820 = r123819 * r123811;
        double r123821 = r123808 * r123820;
        double r123822 = 1.0;
        double r123823 = tan(r123806);
        double r123824 = r123822 / r123823;
        double r123825 = r123824 * r123805;
        double r123826 = r123821 * r123825;
        double r123827 = 3.0;
        double r123828 = pow(r123798, r123827);
        double r123829 = r123822 / r123828;
        double r123830 = r123806 / r123798;
        double r123831 = pow(r123830, r123813);
        double r123832 = r123829 / r123831;
        double r123833 = r123811 / r123807;
        double r123834 = r123805 * r123833;
        double r123835 = r123834 / r123831;
        double r123836 = r123805 / r123823;
        double r123837 = r123835 * r123836;
        double r123838 = r123832 * r123837;
        double r123839 = r123804 ? r123826 : r123838;
        return r123839;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if t < 4.4219569794683185e-50 or 4.542715892992587e+84 < t

    1. Initial program 50.3

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

    2. Simplified40.2

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}}\]

    3. Taylor expanded around inf 16.2

      \[\leadsto \color{blue}{\left(2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)} \cdot \frac{\ell}{\tan k}\]

    4. Simplified16.1

      \[\leadsto \color{blue}{\left(\left(2 \cdot {\left(\frac{\frac{1}{{t}^{1}}}{{k}^{2}}\right)}^{1}\right) \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\ell}{\tan k}\]
    5. Using strategy rm

    6. Applied sqr-pow16.1

      \[\leadsto \left(\left(2 \cdot {\left(\frac{\frac{1}{{t}^{1}}}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot {k}^{\left(\frac{2}{2}\right)}}}\right)}^{1}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}\]

    7. Applied associate-/r*10.6

      \[\leadsto \left(\left(2 \cdot {\color{blue}{\left(\frac{\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}}^{1}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}\]

    8. Simplified10.6

      \[\leadsto \left(\left(2 \cdot {\left(\frac{\color{blue}{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}\]
    9. Using strategy rm

    10. Applied div-inv10.6

      \[\leadsto \left(\left(2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \color{blue}{\left(\ell \cdot \frac{1}{\tan k}\right)}\]
    11. Using strategy rm

    12. Applied inv-pow10.6

      \[\leadsto \left(\left(2 \cdot {\left(\frac{\color{blue}{{\left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}^{-1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \left(\ell \cdot \frac{1}{\tan k}\right)\]

    if 4.4219569794683185e-50 < t < 4.542715892992587e+84

    1. Initial program 30.4

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

    2. Simplified12.8

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}}\]
    3. Using strategy rm

    4. Applied sqr-pow12.8

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\ell}{\tan k}\]

    5. Applied *-un-lft-identity12.8

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{{t}^{3} \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\tan k}\]

    6. Applied times-frac12.7

      \[\leadsto \frac{\color{blue}{\frac{1}{{t}^{3}} \cdot \frac{2}{\frac{\sin k}{\ell}}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\tan k}\]

    7. Applied times-frac4.6

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right)} \cdot \frac{\ell}{\tan k}\]

    8. Applied associate-*l*6.7

      \[\leadsto \color{blue}{\frac{\frac{1}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\frac{2}{\frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\tan k}\right)}\]

    9. Simplified6.7

      \[\leadsto \frac{\frac{1}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\sin k} \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.421956979468318490074797826886651301589 \cdot 10^{-50} \lor \neg \left(t \le 4.542715892992587199784907751836612646733 \cdot 10^{84}\right):\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{{\left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right)}^{-1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right)\right) \cdot \left(\frac{1}{\tan k} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\ell \cdot \frac{2}{\sin k}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\tan k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))