Average Error: 48.6 → 16.9
Time: 1.4min
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.409768601945556 \cdot 10^{+288}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\ell \cdot \ell}{\sin k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;\ell \cdot \ell \leq 3.409768601945556 \cdot 10^{+288}:\\
\;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\ell \cdot \ell}{\sin k}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(\left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\

\end{array}
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 3.409768601945556e+288)
   (*
    2.0
    (*
     (pow (/ 1.0 (pow k (/ 2.0 2.0))) 1.0)
     (*
      (/ (cos k) (sin k))
      (*
       (pow (/ 1.0 (* (pow k (/ 2.0 2.0)) (pow t 1.0))) 1.0)
       (/ (* l l) (sin k))))))
   (/
    2.0
    (* (* (* (sin k) (/ (/ (pow t 3.0) l) l)) (tan k)) (pow (/ k t) 2.0)))))
double code(double t, double l, double k) {
	return (2.0 / ((double) (((double) (((double) ((((double) pow(t, 3.0)) / ((double) (l * l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow((k / t), 2.0)))) - 1.0)))));
}
↓
double code(double t, double l, double k) {
	double tmp;
	if ((((double) (l * l)) <= 3.409768601945556e+288)) {
		tmp = ((double) (2.0 * ((double) (((double) pow((1.0 / ((double) pow(k, (2.0 / 2.0)))), 1.0)) * ((double) ((((double) cos(k)) / ((double) sin(k))) * ((double) (((double) pow((1.0 / ((double) (((double) pow(k, (2.0 / 2.0))) * ((double) pow(t, 1.0))))), 1.0)) * (((double) (l * l)) / ((double) sin(k)))))))))));
	} else {
		tmp = (2.0 / ((double) (((double) (((double) (((double) sin(k)) * ((((double) pow(t, 3.0)) / l) / l))) * ((double) tan(k)))) * ((double) pow((k / t), 2.0)))));
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if (*.f64 l l) < 3.40976860194555606e288

    1. Initial program 45.6

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

    2. Simplified36.5

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

    3. Taylor expanded around inf 13.8

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

    4. Simplified13.8

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

    6. Applied sqr-pow_binary6413.8

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

    7. Applied associate-*l*_binary6411.5

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

    8. Simplified11.5

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

    10. Applied *-un-lft-identity_binary6411.5

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

    11. Applied times-frac_binary6411.4

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

    12. Applied unpow-prod-down_binary6411.4

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

    13. Applied associate-*l*_binary649.8

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

    14. Simplified9.8

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

    16. Applied unpow2_binary649.8

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

    17. Applied times-frac_binary649.5

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

    18. Applied associate-*l*_binary649.4

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

    19. Simplified9.4

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

    if 3.40976860194555606e288 < (*.f64 l l)

    1. Initial program 63.1

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

    2. Simplified62.7

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

    4. Applied associate-/r*_binary6453.5

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.409768601945556 \cdot 10^{+288}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\ell \cdot \ell}{\sin k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))