Average Error: 48.5 → 2.6
Time: 35.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}}\right)\]
\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)}
↓
\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos 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 k) (* (/ l k) (/ 2.0 (/ (* (sin k) (* t (sin k))) (cos 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 / k) * ((l / k) * (2.0 / ((sin(k) * (t * sin(k))) / cos(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 48.5

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

  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 0 23.5

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

  4. Simplified23.5

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

  6. Applied times-frac_binary64_42522.4

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

  7. Applied *-un-lft-identity_binary64_41922.4

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

  8. Applied times-frac_binary64_42522.5

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

  9. Simplified22.3

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

  11. Applied times-frac_binary64_4259.9

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

  12. Applied associate-*l*_binary64_3604.3

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

  14. Applied unpow2_binary64_4844.3

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

  15. Applied associate-*r*_binary64_3592.6

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

  16. Final simplification2.6

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

Reproduce

herbie shell --seed 2021075 
(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))))