Average Error: 48.8 → 5.3
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\ell}{\tan k} \cdot \left(2 \cdot \left({\left(\frac{1}{{t}^{1}} \cdot \frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)\right)\]
\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)}
↓
\frac{\ell}{\tan k} \cdot \left(2 \cdot \left({\left(\frac{1}{{t}^{1}} \cdot \frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\ell}{\sin k}\right)\right)\right)
double f(double t, double l, double k) {
        double r9257513 = 2.0;
        double r9257514 = t;
        double r9257515 = 3.0;
        double r9257516 = pow(r9257514, r9257515);
        double r9257517 = l;
        double r9257518 = r9257517 * r9257517;
        double r9257519 = r9257516 / r9257518;
        double r9257520 = k;
        double r9257521 = sin(r9257520);
        double r9257522 = r9257519 * r9257521;
        double r9257523 = tan(r9257520);
        double r9257524 = r9257522 * r9257523;
        double r9257525 = 1.0;
        double r9257526 = r9257520 / r9257514;
        double r9257527 = pow(r9257526, r9257513);
        double r9257528 = r9257525 + r9257527;
        double r9257529 = r9257528 - r9257525;
        double r9257530 = r9257524 * r9257529;
        double r9257531 = r9257513 / r9257530;
        return r9257531;
}
↓
double f(double t, double l, double k) {
        double r9257532 = l;
        double r9257533 = k;
        double r9257534 = tan(r9257533);
        double r9257535 = r9257532 / r9257534;
        double r9257536 = 2.0;
        double r9257537 = 1.0;
        double r9257538 = t;
        double r9257539 = 1.0;
        double r9257540 = pow(r9257538, r9257539);
        double r9257541 = r9257537 / r9257540;
        double r9257542 = 2.0;
        double r9257543 = r9257536 / r9257542;
        double r9257544 = pow(r9257533, r9257543);
        double r9257545 = r9257537 / r9257544;
        double r9257546 = r9257541 * r9257545;
        double r9257547 = pow(r9257546, r9257539);
        double r9257548 = pow(r9257545, r9257539);
        double r9257549 = sin(r9257533);
        double r9257550 = r9257532 / r9257549;
        double r9257551 = r9257548 * r9257550;
        double r9257552 = r9257547 * r9257551;
        double r9257553 = r9257536 * r9257552;
        double r9257554 = r9257535 * r9257553;
        return r9257554;
}

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 48.8

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

  2. Simplified37.2

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

  3. Taylor expanded around inf 15.7

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

  5. Applied sqr-pow15.7

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

  6. Applied associate-*r*11.1

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

  8. Applied *-un-lft-identity11.1

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

  9. Applied times-frac10.7

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

  10. Applied unpow-prod-down10.7

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

  11. Applied associate-*l*5.4

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

  13. Applied *-un-lft-identity5.4

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

  14. Applied times-frac5.3

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

  15. Final simplification5.3

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

Reproduce

herbie shell --seed 2019171 +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))))