Average Error: 48.4 → 0.5
Time: 2.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}} \cdot \left(\left(\left(\frac{\cos k}{\sin k} \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\right)\]
\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{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}} \cdot \left(\left(\left(\frac{\cos k}{\sin k} \cdot \ell\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\right)
double f(double t, double l, double k) {
        double r6465352 = 2.0;
        double r6465353 = t;
        double r6465354 = 3.0;
        double r6465355 = pow(r6465353, r6465354);
        double r6465356 = l;
        double r6465357 = r6465356 * r6465356;
        double r6465358 = r6465355 / r6465357;
        double r6465359 = k;
        double r6465360 = sin(r6465359);
        double r6465361 = r6465358 * r6465360;
        double r6465362 = tan(r6465359);
        double r6465363 = r6465361 * r6465362;
        double r6465364 = 1.0;
        double r6465365 = r6465359 / r6465353;
        double r6465366 = pow(r6465365, r6465352);
        double r6465367 = r6465364 + r6465366;
        double r6465368 = r6465367 - r6465364;
        double r6465369 = r6465363 * r6465368;
        double r6465370 = r6465352 / r6465369;
        return r6465370;
}
↓
double f(double t, double l, double k) {
        double r6465371 = 2.0;
        double r6465372 = 1.0;
        double r6465373 = k;
        double r6465374 = 2.0;
        double r6465375 = r6465371 / r6465374;
        double r6465376 = pow(r6465373, r6465375);
        double r6465377 = r6465372 / r6465376;
        double r6465378 = 1.0;
        double r6465379 = pow(r6465377, r6465378);
        double r6465380 = sin(r6465373);
        double r6465381 = l;
        double r6465382 = r6465380 / r6465381;
        double r6465383 = r6465379 / r6465382;
        double r6465384 = cos(r6465373);
        double r6465385 = r6465384 / r6465380;
        double r6465386 = r6465385 * r6465381;
        double r6465387 = r6465386 * r6465379;
        double r6465388 = t;
        double r6465389 = pow(r6465388, r6465378);
        double r6465390 = r6465372 / r6465389;
        double r6465391 = pow(r6465390, r6465378);
        double r6465392 = r6465387 * r6465391;
        double r6465393 = r6465383 * r6465392;
        double r6465394 = r6465371 * r6465393;
        return r6465394;
}

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 48.4

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

  2. Simplified40.2

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

  3. Taylor expanded around inf 21.8

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

  5. Applied *-un-lft-identity21.8

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

  6. Applied times-frac21.8

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

  7. Applied unpow-prod-down21.8

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

  8. Applied associate-*l*22.6

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

  9. Simplified20.1

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

  11. Applied times-frac16.3

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

  12. Applied associate-*r*13.6

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

  14. Applied *-un-lft-identity13.6

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

  15. Applied *-un-lft-identity13.6

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

  16. Applied times-frac13.6

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

  17. Applied sqr-pow13.6

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

  18. Applied *-un-lft-identity13.6

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

  19. Applied times-frac13.4

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

  20. Applied unpow-prod-down13.4

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

  21. Applied times-frac9.0

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

  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

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

  24. Final simplification0.5

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

Reproduce

herbie shell --seed 2019170 +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))))