Average Error: 48.3 → 6.8
Time: 34.0s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\left(\frac{2}{k} \cdot \ell\right) \cdot \frac{\frac{1}{k} \cdot \cos k}{\frac{t \cdot {\sin k}^{2}}{\ell}} \]
\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)}
↓
\left(\frac{2}{k} \cdot \ell\right) \cdot \frac{\frac{1}{k} \cdot \cos k}{\frac{t \cdot {\sin k}^{2}}{\ell}}

(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) (/ (* (/ 1.0 k) (cos k)) (/ (* t (pow (sin k) 2.0)) l))))
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) * (((1.0 / k) * cos(k)) / ((t * pow(sin(k), 2.0)) / l));
}

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 48.3

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

  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 in t around 0 22.8

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

  4. Applied unpow2_binary6422.8

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

  5. Applied associate-*l*_binary6420.6

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

  6. Applied times-frac_binary6418.4

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

  7. Applied associate-/r*_binary6418.2

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

  8. Applied add-sqr-sqrt_binary6441.4

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

  9. Applied unpow-prod-down_binary6441.4

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

  10. Applied times-frac_binary6436.6

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

  11. Applied div-inv_binary6436.6

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

  12. Applied times-frac_binary6435.7

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

  13. Simplified35.7

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

  14. Simplified6.8

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

  15. Final simplification6.8

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

Reproduce

herbie shell --seed 2022087 
(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))))