Average Error: 48.0 → 7.3
Time: 30.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -4.240753838228914 \cdot 10^{-17} \lor \neg \left(t \leq 2.2676346131408102 \cdot 10^{-188}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)\\ \end{array}\]
\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}
\mathbf{if}\;t \leq -4.240753838228914 \cdot 10^{-17} \lor \neg \left(t \leq 2.2676346131408102 \cdot 10^{-188}\right):\\
\;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\

\mathbf{else}:\\
\;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\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
 (if (or (<= t -4.240753838228914e-17) (not (<= t 2.2676346131408102e-188)))
   (* (* (/ l k) (/ l k)) (/ 2.0 (/ (* t (pow (sin k) 2.0)) (cos k))))
   (* l (* (/ l k) (* 2.0 (/ (cos k) (* k (* t (pow (sin k) 2.0)))))))))
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 ((t <= -4.240753838228914e-17) || !(t <= 2.2676346131408102e-188)) {
		tmp = ((l / k) * (l / k)) * (2.0 / ((t * pow(sin(k), 2.0)) / cos(k)));
	} else {
		tmp = l * ((l / k) * (2.0 * (cos(k) / (k * (t * pow(sin(k), 2.0))))));
	}
	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 t < -4.24075383822891424e-17 or 2.2676346131408102e-188 < t

    1. Initial program 45.3

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

    2. Simplified34.5

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

    3. Taylor expanded around 0 21.4

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

    4. Simplified21.4

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

    6. Applied associate-*l*_binary64_36020.9

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

    8. Applied times-frac_binary64_42518.3

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

    9. Applied *-un-lft-identity_binary64_41918.3

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

    10. Applied times-frac_binary64_42518.3

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

    11. Simplified18.2

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

    13. Applied *-un-lft-identity_binary64_41918.2

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

    14. Applied times-frac_binary64_42518.2

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

    15. Applied *-un-lft-identity_binary64_41918.2

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

    16. Applied times-frac_binary64_42518.1

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

    17. Applied associate-*r*_binary64_35917.9

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

    18. Simplified7.3

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

    if -4.24075383822891424e-17 < t < 2.2676346131408102e-188

    1. Initial program 55.1

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

    2. Simplified54.8

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

    3. Taylor expanded around 0 26.3

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

    4. Simplified26.3

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

    6. Applied associate-*l*_binary64_36019.6

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

    8. Applied times-frac_binary64_42518.2

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

    9. Applied *-un-lft-identity_binary64_41918.2

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

    10. Applied times-frac_binary64_42518.2

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

    11. Simplified17.9

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

    13. Applied *-un-lft-identity_binary64_41917.9

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

    14. Applied times-frac_binary64_42511.7

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

    15. Applied associate-*l*_binary64_3607.5

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

    16. Simplified7.5

      \[\leadsto \frac{\ell}{1} \cdot \color{blue}{\left(\left(2 \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\right) \cdot \frac{\ell}{k}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.240753838228914 \cdot 10^{-17} \lor \neg \left(t \leq 2.2676346131408102 \cdot 10^{-188}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021046 
(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))))