Average Error: 48.6 → 13.2
Time: 22.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}\]
\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 \frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \cos k}{k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}
(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
 (* 2.0 (/ (* (* l (/ l k)) (cos k)) (* k (* t (pow (sin k) 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) {
	return ((double) (2.0 * (((double) (((double) (l * (l / k))) * ((double) cos(k)))) / ((double) (k * ((double) (t * ((double) pow(((double) sin(k)), 2.0)))))))));
}

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 48.6

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

  2. Simplified40.9

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

  3. Taylor expanded around inf 22.9

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

  4. Simplified22.9

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

  6. Applied associate-*l*_binary6420.9

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

  8. Applied associate-/r*_binary6418.8

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

  9. Simplified18.8

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

  11. Applied *-un-lft-identity_binary6418.8

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

  12. Applied times-frac_binary6413.2

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

  13. Simplified13.2

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

  14. Final simplification13.2

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

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))))