Average Error: 48.4 → 7.8
Time: 23.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 2.0248506160887325 \cdot 10^{-154} \lor \neg \left(k \leq 4.21563689246568 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{\sqrt[3]{k}}{\ell}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt{k}}{\ell} \cdot \frac{{k}^{1.5}}{\frac{\ell}{t \cdot {\sin k}^{2}}}}{\cos k}}\\ \end{array}\]
\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}
\mathbf{if}\;k \leq 2.0248506160887325 \cdot 10^{-154} \lor \neg \left(k \leq 4.21563689246568 \cdot 10^{+170}\right):\\
\;\;\;\;\frac{2}{\frac{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{\sqrt[3]{k}}{\ell}\right)}{\cos k}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\frac{\sqrt{k}}{\ell} \cdot \frac{{k}^{1.5}}{\frac{\ell}{t \cdot {\sin k}^{2}}}}{\cos k}}\\

\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
 (if (or (<= k 2.0248506160887325e-154) (not (<= k 4.21563689246568e+170)))
   (/
    2.0
    (/
     (*
      (/ (* (cbrt k) (cbrt k)) l)
      (* (* k (* t (pow (sin k) 2.0))) (/ (cbrt k) l)))
     (cos k)))
   (/
    2.0
    (/
     (* (/ (sqrt k) l) (/ (pow k 1.5) (/ l (* t (pow (sin k) 2.0)))))
     (cos 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) {
	double tmp;
	if ((k <= 2.0248506160887325e-154) || !(k <= 4.21563689246568e+170)) {
		tmp = 2.0 / ((((cbrt(k) * cbrt(k)) / l) * ((k * (t * pow(sin(k), 2.0))) * (cbrt(k) / l))) / cos(k));
	} else {
		tmp = 2.0 / (((sqrt(k) / l) * (pow(k, 1.5) / (l / (t * pow(sin(k), 2.0))))) / cos(k));
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if k < 2.0248506160887325e-154 or 4.2156368924656801e170 < k

    1. Initial program 46.4

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

    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 around 0 23.7

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

    4. Simplified23.7

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}}}\]
    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 associate-/r*_binary64_36321.1

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

    9. Simplified18.7

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

    11. Applied add-cube-cbrt_binary64_45418.9

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

    12. Applied times-frac_binary64_42512.5

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

    13. Applied associate-*l*_binary64_3608.7

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

    14. Simplified8.7

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

    if 2.0248506160887325e-154 < k < 4.2156368924656801e170

    1. Initial program 53.2

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

    2. Simplified42.6

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

    3. Taylor expanded around 0 21.0

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

    4. Simplified21.0

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

    6. Applied associate-*l*_binary64_36020.1

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

    8. Applied associate-/r*_binary64_36320.1

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

    9. Simplified18.7

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

    11. Applied add-sqr-sqrt_binary64_44118.8

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

    12. Applied times-frac_binary64_42515.1

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

    13. Applied associate-*l*_binary64_3607.8

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

    14. Simplified5.7

      \[\leadsto \frac{2}{\frac{\frac{\sqrt{k}}{\ell} \cdot \color{blue}{\frac{{k}^{1.5}}{\frac{\ell}{t \cdot {\sin k}^{2}}}}}{\cos k}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.0248506160887325 \cdot 10^{-154} \lor \neg \left(k \leq 4.21563689246568 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{\sqrt[3]{k}}{\ell}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt{k}}{\ell} \cdot \frac{{k}^{1.5}}{\frac{\ell}{t \cdot {\sin k}^{2}}}}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(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))))