Average Error: 32.2 → 13.8
Time: 47.4s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\sin k} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\left(\frac{1}{\frac{\ell}{t}} \cdot \sin k\right) \cdot t}\right)\]
\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)}
↓
\left(\frac{\ell}{t} \cdot \cos k\right) \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\sin k} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\left(\frac{1}{\frac{\ell}{t}} \cdot \sin k\right) \cdot t}\right)
double f(double t, double l, double k) {
        double r1687606 = 2.0;
        double r1687607 = t;
        double r1687608 = 3.0;
        double r1687609 = pow(r1687607, r1687608);
        double r1687610 = l;
        double r1687611 = r1687610 * r1687610;
        double r1687612 = r1687609 / r1687611;
        double r1687613 = k;
        double r1687614 = sin(r1687613);
        double r1687615 = r1687612 * r1687614;
        double r1687616 = tan(r1687613);
        double r1687617 = r1687615 * r1687616;
        double r1687618 = 1.0;
        double r1687619 = r1687613 / r1687607;
        double r1687620 = pow(r1687619, r1687606);
        double r1687621 = r1687618 + r1687620;
        double r1687622 = r1687621 + r1687618;
        double r1687623 = r1687617 * r1687622;
        double r1687624 = r1687606 / r1687623;
        return r1687624;
}
↓
double f(double t, double l, double k) {
        double r1687625 = l;
        double r1687626 = t;
        double r1687627 = r1687625 / r1687626;
        double r1687628 = k;
        double r1687629 = cos(r1687628);
        double r1687630 = r1687627 * r1687629;
        double r1687631 = 2.0;
        double r1687632 = sqrt(r1687631);
        double r1687633 = r1687628 / r1687626;
        double r1687634 = r1687633 * r1687633;
        double r1687635 = r1687634 + r1687631;
        double r1687636 = sqrt(r1687635);
        double r1687637 = r1687632 / r1687636;
        double r1687638 = sin(r1687628);
        double r1687639 = r1687637 / r1687638;
        double r1687640 = 1.0;
        double r1687641 = r1687640 / r1687627;
        double r1687642 = r1687641 * r1687638;
        double r1687643 = r1687642 * r1687626;
        double r1687644 = r1687637 / r1687643;
        double r1687645 = r1687639 * r1687644;
        double r1687646 = r1687630 * r1687645;
        return r1687646;
}

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 32.2

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

  2. Simplified24.5

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

  4. Applied associate-*l*20.4

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

  6. Applied *-un-lft-identity20.4

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

  7. Applied times-frac19.7

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

  8. Applied associate-*r*17.5

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

  10. Applied tan-quot17.5

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

  11. Applied associate-*l/17.5

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

  12. Applied frac-times16.6

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

  13. Applied associate-*r/15.4

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

  14. Applied associate-/r/14.0

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

  16. Applied add-sqr-sqrt14.1

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

  17. Applied add-sqr-sqrt14.1

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

  18. Applied times-frac14.1

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

  19. Applied times-frac13.8

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

  20. Final simplification13.8

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

Reproduce

herbie shell --seed 2019128 
(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))))