Average Error: 47.6 → 8.8
Time: 21.8s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{2}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k} \cdot \frac{\sqrt[3]{k}}{\ell}\right)}\]
\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)}
↓
\frac{2}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k} \cdot \frac{\sqrt[3]{k}}{\ell}\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
  (*
   (/ (* (cbrt k) (cbrt k)) l)
   (* (/ (* k (* t (pow (sin k) 2.0))) (cos k)) (/ (cbrt k) 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 / (((cbrt(k) * cbrt(k)) / l) * (((k * (t * pow(sin(k), 2.0))) / cos(k)) * (cbrt(k) / 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 47.6

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

  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 23.1

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

  4. Simplified23.1

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

  6. Applied associate-*l*_binary64_36021.1

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

  8. Applied times-frac_binary64_42518.7

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

  10. Applied add-cube-cbrt_binary64_45418.9

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

  11. Applied times-frac_binary64_42513.9

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

  12. Applied associate-*l*_binary64_3608.8

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

  13. Simplified8.8

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

  14. Final simplification8.8

    \[\leadsto \frac{2}{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k} \cdot \frac{\sqrt[3]{k}}{\ell}\right)}\]

Reproduce

herbie shell --seed 2020356 
(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))))