Average Error: 47.8 → 3.4
Time: 23.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{2}{\frac{\frac{k}{\ell} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)\right)}{\cos k}}\]
\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{2}{\frac{\frac{k}{\ell} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)\right)}{\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 (/ (* (/ k l) (* t (* (/ k l) (pow (sin k) 2.0)))) (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 / (((k / l) * (t * ((k / l) * pow(sin(k), 2.0)))) / 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. Simplified40.2

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

  3. Taylor expanded around 0 22.3

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

  4. Simplified22.3

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

  6. Applied associate-/r*_binary6422.3

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

  7. Simplified21.4

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

  9. Applied times-frac_binary649.8

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

  10. Applied associate-*l*_binary644.2

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

  11. Simplified4.2

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

  13. Applied associate-*l*_binary643.4

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

  14. Final simplification3.4

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

Reproduce

herbie shell --seed 2021175 
(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))))