Average Error: 48.1 → 8.3
Time: 2.2m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}}{\frac{\sin k}{\ell}} \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\]
\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{2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}}{\frac{\sin k}{\ell}} \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}
double f(double t, double l, double k) {
        double r7870485 = 2.0;
        double r7870486 = t;
        double r7870487 = 3.0;
        double r7870488 = pow(r7870486, r7870487);
        double r7870489 = l;
        double r7870490 = r7870489 * r7870489;
        double r7870491 = r7870488 / r7870490;
        double r7870492 = k;
        double r7870493 = sin(r7870492);
        double r7870494 = r7870491 * r7870493;
        double r7870495 = tan(r7870492);
        double r7870496 = r7870494 * r7870495;
        double r7870497 = 1.0;
        double r7870498 = r7870492 / r7870486;
        double r7870499 = pow(r7870498, r7870485);
        double r7870500 = r7870497 + r7870499;
        double r7870501 = r7870500 - r7870497;
        double r7870502 = r7870496 * r7870501;
        double r7870503 = r7870485 / r7870502;
        return r7870503;
}
↓
double f(double t, double l, double k) {
        double r7870504 = 2.0;
        double r7870505 = 1.0;
        double r7870506 = k;
        double r7870507 = 2.0;
        double r7870508 = r7870504 / r7870507;
        double r7870509 = pow(r7870506, r7870508);
        double r7870510 = r7870505 / r7870509;
        double r7870511 = t;
        double r7870512 = 1.0;
        double r7870513 = pow(r7870511, r7870512);
        double r7870514 = r7870510 / r7870513;
        double r7870515 = pow(r7870514, r7870512);
        double r7870516 = r7870504 * r7870515;
        double r7870517 = sin(r7870506);
        double r7870518 = l;
        double r7870519 = r7870517 / r7870518;
        double r7870520 = r7870516 / r7870519;
        double r7870521 = cos(r7870506);
        double r7870522 = r7870521 / r7870519;
        double r7870523 = r7870520 * r7870522;
        double r7870524 = pow(r7870510, r7870512);
        double r7870525 = r7870523 * r7870524;
        return r7870525;
}

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 48.1

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

  2. Simplified40.2

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

  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. Simplified20.2

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

  6. Applied sqr-pow20.2

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

  7. Applied associate-*l*16.5

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

  9. Applied *-un-lft-identity16.5

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

  10. Applied times-frac16.3

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

  11. Applied unpow-prod-down16.3

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

  12. Applied associate-*l*14.8

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

  13. Simplified14.7

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

  15. Applied *-un-lft-identity14.7

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

  16. Applied times-frac14.5

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

  17. Applied associate-*r*8.3

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

  18. Simplified8.3

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

  19. Final simplification8.3

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

Reproduce

herbie shell --seed 2019168 
(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))))