Average Error: 47.9 → 0.5
Time: 2.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}}\right) \cdot 2\]
\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)}
↓
\left(\left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}}\right) \cdot 2
double f(double t, double l, double k) {
        double r7126397 = 2.0;
        double r7126398 = t;
        double r7126399 = 3.0;
        double r7126400 = pow(r7126398, r7126399);
        double r7126401 = l;
        double r7126402 = r7126401 * r7126401;
        double r7126403 = r7126400 / r7126402;
        double r7126404 = k;
        double r7126405 = sin(r7126404);
        double r7126406 = r7126403 * r7126405;
        double r7126407 = tan(r7126404);
        double r7126408 = r7126406 * r7126407;
        double r7126409 = 1.0;
        double r7126410 = r7126404 / r7126398;
        double r7126411 = pow(r7126410, r7126397);
        double r7126412 = r7126409 + r7126411;
        double r7126413 = r7126412 - r7126409;
        double r7126414 = r7126408 * r7126413;
        double r7126415 = r7126397 / r7126414;
        return r7126415;
}
↓
double f(double t, double l, double k) {
        double r7126416 = k;
        double r7126417 = cos(r7126416);
        double r7126418 = sin(r7126416);
        double r7126419 = l;
        double r7126420 = r7126418 / r7126419;
        double r7126421 = r7126417 / r7126420;
        double r7126422 = 1.0;
        double r7126423 = 2.0;
        double r7126424 = 2.0;
        double r7126425 = r7126423 / r7126424;
        double r7126426 = pow(r7126416, r7126425);
        double r7126427 = r7126422 / r7126426;
        double r7126428 = 1.0;
        double r7126429 = pow(r7126427, r7126428);
        double r7126430 = r7126421 * r7126429;
        double r7126431 = t;
        double r7126432 = pow(r7126431, r7126428);
        double r7126433 = r7126422 / r7126432;
        double r7126434 = pow(r7126433, r7126428);
        double r7126435 = r7126430 * r7126434;
        double r7126436 = r7126429 / r7126420;
        double r7126437 = r7126435 * r7126436;
        double r7126438 = r7126437 * r7126423;
        return r7126438;
}

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 47.9

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

  2. Simplified40.7

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

  3. Taylor expanded around inf 22.4

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

  5. Applied *-un-lft-identity22.4

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

  6. Applied times-frac22.4

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

  7. Applied unpow-prod-down22.4

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

  8. Applied associate-*l*23.3

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

  9. Simplified20.3

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

  11. Applied times-frac16.7

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

  12. Applied associate-*r*13.8

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

  14. Applied *-un-lft-identity13.8

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

  15. Applied *-un-lft-identity13.8

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

  16. Applied times-frac13.8

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

  17. Applied sqr-pow13.8

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

  18. Applied *-un-lft-identity13.8

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

  19. Applied times-frac13.6

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

  20. Applied unpow-prod-down13.6

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

  21. Applied times-frac9.1

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

  22. Applied associate-*r*4.4

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

  23. Simplified0.5

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

  24. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(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))))