Average Error: 48.3 → 7.9
Time: 52.3s
Precision: binary64
Cost: 1280
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\]
\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)}
↓
2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\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 (/ (* (* (/ l k) (/ l k)) (cos k)) (* (sin 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 2.0 * ((((l / k) * (l / k)) * cos(k)) / (sin(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
Alternative 1
Accuracy9.4
Cost1216
\[2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}\]
Alternative 2
Accuracy26.4
Cost1796
\[\begin{array}{l} \mathbf{if}\;k \leq -2.2403952311595194 \cdot 10^{-35} \lor \neg \left(k \leq 48053594659.906425\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}{t \cdot \log \left(e^{{\sin k}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \sin k\right) \cdot {\left(\sqrt{\sin k}\right)}^{2}\right)}\\ \end{array}\]
Alternative 3
Accuracy29.8
Cost1344
\[2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}{t \cdot \log \left(e^{{\sin k}^{2}}\right)}\]

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. Simplified40.8

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

  3. Taylor expanded around inf 22.7

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

  4. Simplified22.7

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

  6. Applied associate-/r*_binary64_70421.7

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

  7. Simplified21.7

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

  9. Applied add-sqr-sqrt_binary64_78221.7

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

  10. Simplified21.7

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

  11. Simplified9.4

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

  13. Applied unpow2_binary64_8259.4

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

  14. Applied associate-*r*_binary64_7007.9

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

  15. Final simplification7.9

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

Reproduce

herbie shell --seed 2020322 
(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))))