Average Error: 48.5 → 4.4
Time: 2.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \left(\left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}} \cdot \sqrt[3]{\frac{\sin k}{\ell}}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)\right)\]
\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)}
↓
2 \cdot \left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \left(\left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}} \cdot \sqrt[3]{\frac{\sin k}{\ell}}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\sqrt[3]{\frac{\sin k}{\ell}}}\right)\right)
double f(double t, double l, double k) {
        double r6646613 = 2.0;
        double r6646614 = t;
        double r6646615 = 3.0;
        double r6646616 = pow(r6646614, r6646615);
        double r6646617 = l;
        double r6646618 = r6646617 * r6646617;
        double r6646619 = r6646616 / r6646618;
        double r6646620 = k;
        double r6646621 = sin(r6646620);
        double r6646622 = r6646619 * r6646621;
        double r6646623 = tan(r6646620);
        double r6646624 = r6646622 * r6646623;
        double r6646625 = 1.0;
        double r6646626 = r6646620 / r6646614;
        double r6646627 = pow(r6646626, r6646613);
        double r6646628 = r6646625 + r6646627;
        double r6646629 = r6646628 - r6646625;
        double r6646630 = r6646624 * r6646629;
        double r6646631 = r6646613 / r6646630;
        return r6646631;
}
↓
double f(double t, double l, double k) {
        double r6646632 = 2.0;
        double r6646633 = k;
        double r6646634 = cos(r6646633);
        double r6646635 = sin(r6646633);
        double r6646636 = l;
        double r6646637 = r6646635 / r6646636;
        double r6646638 = r6646634 / r6646637;
        double r6646639 = 1.0;
        double r6646640 = 2.0;
        double r6646641 = r6646632 / r6646640;
        double r6646642 = pow(r6646633, r6646641);
        double r6646643 = r6646639 / r6646642;
        double r6646644 = 1.0;
        double r6646645 = pow(r6646643, r6646644);
        double r6646646 = cbrt(r6646637);
        double r6646647 = r6646646 * r6646646;
        double r6646648 = r6646645 / r6646647;
        double r6646649 = t;
        double r6646650 = pow(r6646649, r6646644);
        double r6646651 = r6646639 / r6646650;
        double r6646652 = pow(r6646651, r6646644);
        double r6646653 = r6646648 * r6646652;
        double r6646654 = r6646645 / r6646646;
        double r6646655 = r6646653 * r6646654;
        double r6646656 = r6646638 * r6646655;
        double r6646657 = r6646632 * r6646656;
        return r6646657;
}

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 48.5

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

  2. Simplified40.5

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

  3. Taylor expanded around inf 22.0

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

  5. Applied *-un-lft-identity22.0

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

  6. Applied times-frac21.9

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

  7. Applied unpow-prod-down21.9

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

  8. Applied associate-*l*22.7

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

  9. Simplified19.7

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

  11. Applied times-frac16.0

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

  12. Applied associate-*r*13.5

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

  14. Applied add-cube-cbrt13.7

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

  15. Applied sqr-pow13.7

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

  16. Applied *-un-lft-identity13.7

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

  17. Applied times-frac13.5

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

  18. Applied unpow-prod-down13.5

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

  19. Applied times-frac9.2

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

  20. Applied associate-*r*4.4

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

  21. Final simplification4.4

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

Reproduce

herbie shell --seed 2019192 
(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))))