Average Error: 47.4 → 8.5
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -2.3605873137923636 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{k}{t}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\ \mathbf{elif}\;k \le 4.168290049334843 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\frac{\sin k \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{k}{t}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\ \end{array}\]
\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}\;k \le -2.3605873137923636 \cdot 10^{+148}:\\
\;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{k}{t}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\

\mathbf{elif}\;k \le 4.168290049334843 \cdot 10^{+151}:\\
\;\;\;\;\frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\frac{\sin k \cdot \tan k}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{k}{t}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\

\end{array}
double f(double t, double l, double k) {
        double r2964007 = 2.0;
        double r2964008 = t;
        double r2964009 = 3.0;
        double r2964010 = pow(r2964008, r2964009);
        double r2964011 = l;
        double r2964012 = r2964011 * r2964011;
        double r2964013 = r2964010 / r2964012;
        double r2964014 = k;
        double r2964015 = sin(r2964014);
        double r2964016 = r2964013 * r2964015;
        double r2964017 = tan(r2964014);
        double r2964018 = r2964016 * r2964017;
        double r2964019 = 1.0;
        double r2964020 = r2964014 / r2964008;
        double r2964021 = pow(r2964020, r2964007);
        double r2964022 = r2964019 + r2964021;
        double r2964023 = r2964022 - r2964019;
        double r2964024 = r2964018 * r2964023;
        double r2964025 = r2964007 / r2964024;
        return r2964025;
}
↓
double f(double t, double l, double k) {
        double r2964026 = k;
        double r2964027 = -2.3605873137923636e+148;
        bool r2964028 = r2964026 <= r2964027;
        double r2964029 = -2.0;
        double r2964030 = t;
        double r2964031 = l;
        double r2964032 = r2964030 / r2964031;
        double r2964033 = cbrt(r2964032);
        double r2964034 = r2964033 * r2964033;
        double r2964035 = r2964026 / r2964030;
        double r2964036 = r2964033 * r2964035;
        double r2964037 = r2964034 * r2964036;
        double r2964038 = r2964037 * r2964032;
        double r2964039 = r2964038 * r2964035;
        double r2964040 = r2964030 * r2964039;
        double r2964041 = r2964029 / r2964040;
        double r2964042 = tan(r2964026);
        double r2964043 = sin(r2964026);
        double r2964044 = -r2964043;
        double r2964045 = r2964042 * r2964044;
        double r2964046 = r2964041 / r2964045;
        double r2964047 = 4.168290049334843e+151;
        bool r2964048 = r2964026 <= r2964047;
        double r2964049 = r2964026 * r2964026;
        double r2964050 = r2964049 / r2964031;
        double r2964051 = r2964030 * r2964050;
        double r2964052 = r2964029 / r2964051;
        double r2964053 = r2964043 * r2964042;
        double r2964054 = r2964053 / r2964031;
        double r2964055 = -r2964054;
        double r2964056 = r2964052 / r2964055;
        double r2964057 = r2964048 ? r2964056 : r2964046;
        double r2964058 = r2964028 ? r2964046 : r2964057;
        return r2964058;
}

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 k < -2.3605873137923636e+148 or 4.168290049334843e+151 < k

    1. Initial program 38.6

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

    2. Simplified17.2

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

    4. Applied frac-2neg17.2

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

    5. Simplified13.1

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

    7. Applied associate-*r*12.7

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

    9. Applied add-cube-cbrt12.8

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

    10. Applied associate-*l*12.8

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

    if -2.3605873137923636e+148 < k < 4.168290049334843e+151

    1. Initial program 53.6

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

    2. Simplified24.7

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

    4. Applied frac-2neg24.7

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

    5. Simplified20.1

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

    7. Applied associate-*r*19.2

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

    9. Applied associate-*l/19.2

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

    10. Applied associate-*l/19.2

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

    11. Applied associate-*r/17.3

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

    12. Applied associate-*l/13.3

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

    13. Applied associate-/r/13.0

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

    14. Applied associate-/l*12.1

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

    15. Taylor expanded around inf 5.4

      \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{{k}^{2}}{\ell}} \cdot t}}{\frac{-\sin k \cdot \tan k}{\ell}}\]

    16. Simplified5.4

      \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{k \cdot k}{\ell}} \cdot t}}{\frac{-\sin k \cdot \tan k}{\ell}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.3605873137923636 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{k}{t}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\ \mathbf{elif}\;k \le 4.168290049334843 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\frac{\sin k \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \sqrt[3]{\frac{t}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{k}{t}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \frac{k}{t}\right)}}{\tan k \cdot \left(-\sin k\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))