Average Error: 48.4 → 10.1
Time: 22.0s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\ell \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{k}\right)\]
\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)}
↓
\ell \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{k}\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
 (* l (* (/ 2.0 (/ (* k (* t (pow (sin k) 2.0))) (cos k))) (/ l k))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}
↓
double code(double t, double l, double k) {
	return l * ((2.0 / ((k * (t * pow(sin(k), 2.0))) / cos(k))) * (l / k));
}

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 48.4

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

  2. Simplified41.0

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

  3. Taylor expanded around inf 23.7

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

  4. Simplified23.7

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

  6. Applied associate-*l*_binary6421.7

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

  8. Applied times-frac_binary6419.7

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

  9. Applied *-un-lft-identity_binary6419.7

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

  10. Applied times-frac_binary6419.7

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

  11. Simplified19.6

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

  13. Applied *-un-lft-identity_binary6419.6

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

  14. Applied times-frac_binary6414.3

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

  15. Applied associate-*l*_binary6410.1

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

  16. Simplified10.1

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

  17. Final simplification10.1

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

Reproduce

herbie shell --seed 2020224 
(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))))