Average Error: 31.5 → 11.2
Time: 52.3s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{1}{\sin k \cdot \frac{t}{\ell}} \cdot \left(\left(\frac{2}{\frac{t}{\ell}} \cdot \frac{1}{\sin k}\right) \cdot \frac{\frac{\cos k}{t}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)\]
\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{1}{\sin k \cdot \frac{t}{\ell}} \cdot \left(\left(\frac{2}{\frac{t}{\ell}} \cdot \frac{1}{\sin k}\right) \cdot \frac{\frac{\cos k}{t}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)
double f(double t, double l, double k) {
        double r1680499 = 2.0;
        double r1680500 = t;
        double r1680501 = 3.0;
        double r1680502 = pow(r1680500, r1680501);
        double r1680503 = l;
        double r1680504 = r1680503 * r1680503;
        double r1680505 = r1680502 / r1680504;
        double r1680506 = k;
        double r1680507 = sin(r1680506);
        double r1680508 = r1680505 * r1680507;
        double r1680509 = tan(r1680506);
        double r1680510 = r1680508 * r1680509;
        double r1680511 = 1.0;
        double r1680512 = r1680506 / r1680500;
        double r1680513 = pow(r1680512, r1680499);
        double r1680514 = r1680511 + r1680513;
        double r1680515 = r1680514 + r1680511;
        double r1680516 = r1680510 * r1680515;
        double r1680517 = r1680499 / r1680516;
        return r1680517;
}
↓
double f(double t, double l, double k) {
        double r1680518 = 1.0;
        double r1680519 = k;
        double r1680520 = sin(r1680519);
        double r1680521 = t;
        double r1680522 = l;
        double r1680523 = r1680521 / r1680522;
        double r1680524 = r1680520 * r1680523;
        double r1680525 = r1680518 / r1680524;
        double r1680526 = 2.0;
        double r1680527 = r1680526 / r1680523;
        double r1680528 = r1680518 / r1680520;
        double r1680529 = r1680527 * r1680528;
        double r1680530 = cos(r1680519);
        double r1680531 = r1680530 / r1680521;
        double r1680532 = r1680519 / r1680521;
        double r1680533 = r1680532 * r1680532;
        double r1680534 = r1680533 + r1680526;
        double r1680535 = r1680531 / r1680534;
        double r1680536 = r1680529 * r1680535;
        double r1680537 = r1680525 * r1680536;
        return r1680537;
}

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 31.5

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

  2. Simplified20.0

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

  4. Applied associate-*r*18.3

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

  6. Applied *-un-lft-identity18.3

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

  7. Applied div-inv18.3

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

  8. Applied times-frac18.0

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

  9. Applied times-frac16.6

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

  10. Simplified14.0

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

  12. Applied *-un-lft-identity14.0

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

  13. Applied times-frac13.8

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

  14. Applied associate-*l*11.5

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

  16. Applied *-un-lft-identity11.5

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

  17. Applied *-un-lft-identity11.5

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

  18. Applied tan-quot11.5

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

  19. Applied associate-/r/11.5

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

  20. Applied times-frac11.5

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

  21. Applied times-frac11.3

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

  22. Applied associate-*r*11.2

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

  23. Simplified11.2

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

  24. Final simplification11.2

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

Reproduce

herbie shell --seed 2019155 
(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))))