Average Error: 48.0 → 4.1
Time: 24.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{k}{\ell \cdot \cos k}}\]
\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}{\left(\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{k}{\ell \cdot \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 (pow (sin k) 2.0))) (/ k (* l (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 * pow(sin(k), 2.0))) * (k / (l * 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 48.0

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

  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 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*_binary64_36420.8

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

  8. Applied associate-/l*_binary64_36818.6

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

  9. Simplified18.6

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

  11. Applied div-inv_binary64_42018.6

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

  12. Applied times-frac_binary64_42912.2

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

  13. Applied *-un-lft-identity_binary64_42312.2

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

  14. Applied times-frac_binary64_4298.6

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

  15. Simplified4.2

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

  16. Simplified4.1

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

  17. Final simplification4.1

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

Reproduce

herbie shell --seed 2020277 
(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))))