Average Error: 47.9 → 2.4
Time: 41.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\ell}{k} \cdot \frac{2}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k} \cdot \frac{k}{\ell}}\]
\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 \frac{2}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k} \cdot \frac{k}{\ell}}
(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) (/ 2.0 (* (/ (* (sin k) (* t (sin k))) (cos k)) (/ k l)))))
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) * (2.0 / (((sin(k) * (t * sin(k))) / cos(k)) * (k / l)));
}

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. Simplified40.1

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

  3. Taylor expanded around 0 22.8

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

  4. Simplified22.8

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

  6. Applied associate-/l*_binary64_36421.8

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

  7. Simplified21.8

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

  9. Applied *-un-lft-identity_binary64_41921.8

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

  10. Applied times-frac_binary64_42517.1

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

  11. Applied times-frac_binary64_4257.3

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

  12. Applied *-un-lft-identity_binary64_4197.3

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

  13. Applied times-frac_binary64_4257.0

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

  14. Simplified7.0

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

  15. Simplified3.9

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

  17. Applied unpow2_binary64_4843.9

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

  18. Applied associate-*r*_binary64_3592.4

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

  19. Final simplification2.4

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

Reproduce

herbie shell --seed 2021098 
(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))))