Average Error: 48.8 → 15.5
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -9.126743252136130554592578622359092123676 \cdot 10^{-80}:\\ \;\;\;\;\left(\left(\frac{\cos k}{\sin k} \cdot \ell\right) \cdot \left(2 \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1}\right)\right) \cdot \frac{1}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{\frac{\sin k}{\ell}}}{\frac{\sin k}{\ell}} \cdot \left(2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \end{array}\]
\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 -9.126743252136130554592578622359092123676 \cdot 10^{-80}:\\
\;\;\;\;\left(\left(\frac{\cos k}{\sin k} \cdot \ell\right) \cdot \left(2 \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1}\right)\right) \cdot \frac{1}{\frac{\sin k}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\cos k}{\frac{\sin k}{\ell}}}{\frac{\sin k}{\ell}} \cdot \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)\\

\end{array}
double f(double t, double l, double k) {
        double r10163012 = 2.0;
        double r10163013 = t;
        double r10163014 = 3.0;
        double r10163015 = pow(r10163013, r10163014);
        double r10163016 = l;
        double r10163017 = r10163016 * r10163016;
        double r10163018 = r10163015 / r10163017;
        double r10163019 = k;
        double r10163020 = sin(r10163019);
        double r10163021 = r10163018 * r10163020;
        double r10163022 = tan(r10163019);
        double r10163023 = r10163021 * r10163022;
        double r10163024 = 1.0;
        double r10163025 = r10163019 / r10163013;
        double r10163026 = pow(r10163025, r10163012);
        double r10163027 = r10163024 + r10163026;
        double r10163028 = r10163027 - r10163024;
        double r10163029 = r10163023 * r10163028;
        double r10163030 = r10163012 / r10163029;
        return r10163030;
}
↓
double f(double t, double l, double k) {
        double r10163031 = k;
        double r10163032 = -9.12674325213613e-80;
        bool r10163033 = r10163031 <= r10163032;
        double r10163034 = cos(r10163031);
        double r10163035 = sin(r10163031);
        double r10163036 = r10163034 / r10163035;
        double r10163037 = l;
        double r10163038 = r10163036 * r10163037;
        double r10163039 = 2.0;
        double r10163040 = 1.0;
        double r10163041 = t;
        double r10163042 = 1.0;
        double r10163043 = pow(r10163041, r10163042);
        double r10163044 = pow(r10163031, r10163039);
        double r10163045 = r10163043 * r10163044;
        double r10163046 = r10163040 / r10163045;
        double r10163047 = pow(r10163046, r10163042);
        double r10163048 = r10163039 * r10163047;
        double r10163049 = r10163038 * r10163048;
        double r10163050 = r10163035 / r10163037;
        double r10163051 = r10163040 / r10163050;
        double r10163052 = r10163049 * r10163051;
        double r10163053 = r10163034 / r10163050;
        double r10163054 = r10163053 / r10163050;
        double r10163055 = 2.0;
        double r10163056 = r10163039 / r10163055;
        double r10163057 = pow(r10163031, r10163056);
        double r10163058 = r10163057 * r10163043;
        double r10163059 = r10163040 / r10163058;
        double r10163060 = r10163059 / r10163057;
        double r10163061 = pow(r10163060, r10163042);
        double r10163062 = r10163039 * r10163061;
        double r10163063 = r10163054 * r10163062;
        double r10163064 = r10163033 ? r10163052 : r10163063;
        return r10163064;
}

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 < -9.12674325213613e-80

    1. Initial program 46.8

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

    2. Simplified38.1

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

    3. Taylor expanded around inf 20.2

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

    4. Simplified19.7

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

    6. Applied sqr-pow19.7

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

    7. Applied associate-*r*17.2

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

    9. Applied *-un-lft-identity17.2

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

    10. Applied times-frac17.2

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

    11. Applied associate-*l*11.0

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

    12. Simplified14.5

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

    if -9.12674325213613e-80 < k

    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. Simplified43.4

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

    3. Taylor expanded around inf 24.1

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

    4. Simplified21.3

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

    6. Applied sqr-pow21.3

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

    7. Applied associate-*r*16.6

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

    9. Applied associate-/r*16.4

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

    11. Applied associate-/r*16.2

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -9.126743252136130554592578622359092123676 \cdot 10^{-80}:\\ \;\;\;\;\left(\left(\frac{\cos k}{\sin k} \cdot \ell\right) \cdot \left(2 \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1}\right)\right) \cdot \frac{1}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k}{\frac{\sin k}{\ell}}}{\frac{\sin k}{\ell}} \cdot \left(2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))