Average Error: 47.8 → 2.5
Time: 26.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{2}{\left(\sin k \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\ell}{\frac{1}{\cos k}}}}\]
\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{\frac{2}{\left(\sin k \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{k}{\ell}}}{\frac{k}{\frac{\ell}{\frac{1}{\cos k}}}}
(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 (* (* (sin k) (* t (sin k))) (/ k l))) (/ k (/ l (/ 1.0 (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 (2.0 / ((sin(k) * (t * sin(k))) * (k / l))) / (k / (l / (1.0 / 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 47.8

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

  2. Simplified40.7

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

  3. Taylor expanded around inf 22.9

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

  4. Simplified22.9

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

  6. Applied associate-/l*_binary6422.1

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

  7. Simplified22.1

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

  9. Applied div-inv_binary6422.1

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

  10. Applied times-frac_binary6417.6

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

  11. Applied times-frac_binary647.5

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

  12. Applied associate-/r*_binary647.2

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

  13. Simplified4.0

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

  15. Applied unpow2_binary644.0

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

  16. Applied associate-*r*_binary642.5

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

  17. Final simplification2.5

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

Reproduce

herbie shell --seed 2020260 
(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))))