Average Error: 47.8 → 2.5
Time: 22.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\right)\right)\]
\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)}
↓
2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\right)\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
 (* 2.0 (* (/ l k) (* (/ l k) (/ (cos k) (* (sin k) (* t (sin 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 * ((l / k) * ((l / k) * (cos(k) / (sin(k) * (t * sin(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 inf 22.6

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

  4. Simplified22.6

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

  6. Applied times-frac_binary64_42022.0

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

  8. Applied times-frac_binary64_42010.1

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

  9. Applied associate-*l*_binary64_3574.0

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

  11. Applied unpow2_binary64_4764.0

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

  12. Applied associate-*r*_binary64_3562.5

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

  13. Final simplification2.5

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

Reproduce

herbie shell --seed 2020288 
(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))))