Average Error: 33.6 → 12.9
Time: 14.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.192768194971382 \cdot 10^{-191} \lor \neg \left(t \leq 9.599619134737473 \cdot 10^{-91}\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{t}^{3}}{\frac{\ell \cdot \ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \end{array}\]
\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}\;t \leq -1.192768194971382 \cdot 10^{-191} \lor \neg \left(t \leq 9.599619134737473 \cdot 10^{-91}\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{t}^{3}}{\frac{\ell \cdot \ell}{\frac{{\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 (<= t -1.192768194971382e-191) (not (<= t 9.599619134737473e-91)))
   (*
    2.0
    (*
     (/ 1.0 (* t (* (sin k) (/ t l))))
     (/ (/ (/ l t) (tan k)) (+ 2.0 (pow (/ k t) 2.0)))))
   (/
    2.0
    (+
     (/ (* (* k k) (* t (pow (sin k) 2.0))) (* (* l l) (cos k)))
     (* 2.0 (/ (pow t 3.0) (/ (* l l) (/ (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 ((t <= -1.192768194971382e-191) || !(t <= 9.599619134737473e-91)) {
		tmp = 2.0 * ((1.0 / (t * (sin(k) * (t / l)))) * (((l / t) / tan(k)) / (2.0 + pow((k / t), 2.0))));
	} else {
		tmp = 2.0 / ((((k * k) * (t * pow(sin(k), 2.0))) / ((l * l) * cos(k))) + (2.0 * (pow(t, 3.0) / ((l * l) / (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 t < -1.1927681949713819e-191 or 9.59961913473747316e-91 < t

    1. Initial program 26.9

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

    2. Simplified26.9

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

    4. Applied cube-mult_binary6426.9

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

    5. Applied times-frac_binary6419.3

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

    6. Applied associate-*l*_binary6417.2

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

    8. Applied *-un-lft-identity_binary6417.2

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

    9. Applied times-frac_binary6411.8

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

    10. Applied associate-*l*_binary6410.7

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{1} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
    11. Using strategy rm

    12. Applied div-inv_binary6410.7

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

    13. Simplified8.7

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

    15. Applied *-un-lft-identity_binary648.7

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

    16. Applied *-un-lft-identity_binary648.7

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

    17. Applied *-un-lft-identity_binary648.7

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

    18. Applied times-frac_binary648.7

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

    19. Applied times-frac_binary647.6

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

    20. Applied times-frac_binary646.2

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

    21. Simplified6.2

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

    if -1.1927681949713819e-191 < t < 9.59961913473747316e-91

    1. Initial program 62.2

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

    2. Simplified62.2

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

    4. Applied cube-mult_binary6462.2

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

    5. Applied times-frac_binary6456.7

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

    6. Applied associate-*l*_binary6456.6

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

    8. Applied *-un-lft-identity_binary6456.6

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

    9. Applied times-frac_binary6447.8

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

    10. Applied associate-*l*_binary6447.8

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

    11. Taylor expanded around inf 41.0

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

    12. Simplified41.0

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.192768194971382 \cdot 10^{-191} \lor \neg \left(t \leq 9.599619134737473 \cdot 10^{-91}\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{t}^{3}}{\frac{\ell \cdot \ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020262 
(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))))