Average Error: 47.9 → 18.5
Time: 27.3s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{1}{\frac{{\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\right)\right)\]
\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({\left(\frac{\sqrt{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 \frac{1}{\frac{{\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\right)\right)
double f(double t, double l, double k) {
        double r91592 = 2.0;
        double r91593 = t;
        double r91594 = 3.0;
        double r91595 = pow(r91593, r91594);
        double r91596 = l;
        double r91597 = r91596 * r91596;
        double r91598 = r91595 / r91597;
        double r91599 = k;
        double r91600 = sin(r91599);
        double r91601 = r91598 * r91600;
        double r91602 = tan(r91599);
        double r91603 = r91601 * r91602;
        double r91604 = 1.0;
        double r91605 = r91599 / r91593;
        double r91606 = pow(r91605, r91592);
        double r91607 = r91604 + r91606;
        double r91608 = r91607 - r91604;
        double r91609 = r91603 * r91608;
        double r91610 = r91592 / r91609;
        return r91610;
}
↓
double f(double t, double l, double k) {
        double r91611 = 2.0;
        double r91612 = 1.0;
        double r91613 = sqrt(r91612);
        double r91614 = k;
        double r91615 = 2.0;
        double r91616 = r91611 / r91615;
        double r91617 = pow(r91614, r91616);
        double r91618 = r91613 / r91617;
        double r91619 = 1.0;
        double r91620 = pow(r91618, r91619);
        double r91621 = t;
        double r91622 = pow(r91621, r91619);
        double r91623 = r91617 * r91622;
        double r91624 = r91612 / r91623;
        double r91625 = pow(r91624, r91619);
        double r91626 = sin(r91614);
        double r91627 = pow(r91626, r91615);
        double r91628 = cos(r91614);
        double r91629 = l;
        double r91630 = pow(r91629, r91615);
        double r91631 = r91628 * r91630;
        double r91632 = r91627 / r91631;
        double r91633 = r91612 / r91632;
        double r91634 = r91625 * r91633;
        double r91635 = r91620 * r91634;
        double r91636 = r91611 * r91635;
        return r91636;
}

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. Simplified40.3

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

  3. Taylor expanded around inf 22.5

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

  5. Applied sqr-pow22.5

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

  6. Applied associate-*l*20.3

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

  8. Applied add-sqr-sqrt20.3

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

  9. Applied times-frac20.1

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

  10. Applied unpow-prod-down20.1

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

  11. Applied associate-*l*18.5

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

  12. Simplified18.5

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

  14. Applied clear-num18.5

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

  15. Final simplification18.5

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

Reproduce

herbie shell --seed 2019352 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))