Average Error: 48.1 → 5.3
Time: 1.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\ell}{\tan k} \cdot \left(2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\ell \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\sin k}\right)\right)\]
\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)}
↓
\frac{\ell}{\tan k} \cdot \left(2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\ell \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\sin k}\right)\right)
double f(double t, double l, double k) {
        double r9962844 = 2.0;
        double r9962845 = t;
        double r9962846 = 3.0;
        double r9962847 = pow(r9962845, r9962846);
        double r9962848 = l;
        double r9962849 = r9962848 * r9962848;
        double r9962850 = r9962847 / r9962849;
        double r9962851 = k;
        double r9962852 = sin(r9962851);
        double r9962853 = r9962850 * r9962852;
        double r9962854 = tan(r9962851);
        double r9962855 = r9962853 * r9962854;
        double r9962856 = 1.0;
        double r9962857 = r9962851 / r9962845;
        double r9962858 = pow(r9962857, r9962844);
        double r9962859 = r9962856 + r9962858;
        double r9962860 = r9962859 - r9962856;
        double r9962861 = r9962855 * r9962860;
        double r9962862 = r9962844 / r9962861;
        return r9962862;
}
↓
double f(double t, double l, double k) {
        double r9962863 = l;
        double r9962864 = k;
        double r9962865 = tan(r9962864);
        double r9962866 = r9962863 / r9962865;
        double r9962867 = 2.0;
        double r9962868 = 1.0;
        double r9962869 = t;
        double r9962870 = 1.0;
        double r9962871 = pow(r9962869, r9962870);
        double r9962872 = 2.0;
        double r9962873 = r9962867 / r9962872;
        double r9962874 = pow(r9962864, r9962873);
        double r9962875 = r9962871 * r9962874;
        double r9962876 = r9962868 / r9962875;
        double r9962877 = pow(r9962876, r9962870);
        double r9962878 = r9962868 / r9962874;
        double r9962879 = pow(r9962878, r9962870);
        double r9962880 = r9962863 * r9962879;
        double r9962881 = sin(r9962864);
        double r9962882 = r9962880 / r9962881;
        double r9962883 = r9962877 * r9962882;
        double r9962884 = r9962867 * r9962883;
        double r9962885 = r9962866 * r9962884;
        return r9962885;
}

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. Initial program 48.1

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

  2. Simplified37.2

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

  3. Taylor expanded around inf 15.8

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

  5. Applied sqr-pow15.8

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

  6. Applied associate-*r*11.1

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

  8. Applied *-un-lft-identity11.1

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

  9. Applied times-frac10.7

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

  10. Applied unpow-prod-down10.7

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

  11. Applied associate-*l*5.4

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

  13. Applied associate-*r/5.3

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

  14. Final simplification5.3

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

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))))