Average Error: 47.8 → 1.0
Time: 47.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{t}\right)}{\sin k}\]
\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{\ell}{k}}{\sin k} \cdot \frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{t}\right)}{\sin 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
 (* (/ (/ l k) (sin k)) (/ (* (cos k) (* 2.0 (/ (/ l 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 ((l / k) / sin(k)) * ((cos(k) * (2.0 * ((l / 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. Simplified37.1

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

  3. Taylor expanded around inf 17.6

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

  4. Simplified11.0

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

  6. Applied frac-times_binary64_4298.5

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

  7. Applied associate-/r/_binary64_3658.5

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

  8. Applied associate-*r*_binary64_3596.8

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

  9. Simplified6.7

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

  11. Applied associate-*r/_binary64_3616.7

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

  12. Applied associate-*r/_binary64_3617.0

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

  13. Simplified3.5

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

  15. Applied unpow2_binary64_4843.5

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

  16. Applied times-frac_binary64_4251.0

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

  17. Final simplification1.0

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

Reproduce

herbie shell --seed 2021176 
(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))))