Average Error: 47.4 → 2.7
Time: 1.3min
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\sin k}\right)}^{2}\\ \frac{\frac{1}{k}}{\frac{t_1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \left(\frac{\frac{\cos k}{k}}{\frac{t_1}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{t_1}{\frac{\ell}{\sqrt[3]{t}}}}\right) \end{array} \]
\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)}
↓
\begin{array}{l}
t_1 := {\left(\sqrt[3]{\sin k}\right)}^{2}\\
\frac{\frac{1}{k}}{\frac{t_1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \left(\frac{\frac{\cos k}{k}}{\frac{t_1}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{t_1}{\frac{\ell}{\sqrt[3]{t}}}}\right)
\end{array}

(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
 (let* ((t_1 (pow (cbrt (sin k)) 2.0)))
   (*
    (/ (/ 1.0 k) (/ t_1 (/ (* (cbrt l) (cbrt l)) (cbrt t))))
    (*
     (/ (/ (cos k) k) (/ t_1 (/ (cbrt l) (cbrt t))))
     (/ 2.0 (/ t_1 (/ l (cbrt t))))))))
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) {
	double t_1 = pow(cbrt(sin(k)), 2.0);
	return ((1.0 / k) / (t_1 / ((cbrt(l) * cbrt(l)) / cbrt(t)))) * (((cos(k) / k) / (t_1 / (cbrt(l) / cbrt(t)))) * (2.0 / (t_1 / (l / cbrt(t)))));
}

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 47.4

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

  2. Simplified39.4

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

  3. Taylor expanded in t around 0 22.7

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

  4. Simplified22.7

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

  6. Applied times-frac_binary6421.6

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

  7. Applied *-un-lft-identity_binary6421.6

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

  8. Applied times-frac_binary6421.7

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

  9. Simplified21.7

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

  10. Simplified22.5

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

  12. Applied add-cube-cbrt_binary6422.6

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

  13. Applied times-frac_binary6418.5

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

  14. Applied add-cube-cbrt_binary6418.7

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

  15. Applied unpow-prod-down_binary6418.7

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

  16. Applied times-frac_binary6416.4

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

  17. Applied *-un-lft-identity_binary6416.4

    \[\leadsto \frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{1 \cdot 2}}{\frac{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  18. Applied times-frac_binary6416.2

    \[\leadsto \frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{\frac{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}}\right)} \]

  19. Applied associate-*r*_binary6413.3

    \[\leadsto \color{blue}{\left(\frac{\cos k}{k \cdot k} \cdot \frac{1}{\frac{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}\right) \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}}} \]

  20. Simplified13.3

    \[\leadsto \color{blue}{\frac{\frac{\cos k}{k \cdot k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\frac{\ell}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]
  21. Using strategy rm

  22. Applied add-cube-cbrt_binary6413.3

    \[\leadsto \frac{\frac{\cos k}{k \cdot k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  23. Applied times-frac_binary6413.3

    \[\leadsto \frac{\frac{\cos k}{k \cdot k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\color{blue}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  24. Applied sqr-pow_binary6413.3

    \[\leadsto \frac{\frac{\cos k}{k \cdot k}}{\frac{\color{blue}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)} \cdot {\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  25. Applied times-frac_binary6412.9

    \[\leadsto \frac{\frac{\cos k}{k \cdot k}}{\color{blue}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}} \cdot \frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  26. Applied *-un-lft-identity_binary6412.9

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \cos k}}{k \cdot k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}} \cdot \frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  27. Applied times-frac_binary6412.6

    \[\leadsto \frac{\color{blue}{\frac{1}{k} \cdot \frac{\cos k}{k}}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}} \cdot \frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  28. Applied times-frac_binary646.5

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{\frac{\cos k}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}}\right)} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}} \]

  29. Applied associate-*l*_binary642.7

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \left(\frac{\frac{\cos k}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}}\right)} \]

  30. Simplified2.7

    \[\leadsto \frac{\frac{1}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \color{blue}{\left(\frac{\frac{\cos k}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}}\right)} \]

  31. Final simplification2.7

    \[\leadsto \frac{\frac{1}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \left(\frac{\frac{\cos k}{k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\frac{\ell}{\sqrt[3]{t}}}}\right) \]

Reproduce

herbie shell --seed 2021206 
(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))))