Average Error: 46.8 → 11.4
Time: 5.0m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -5.102484297178739 \cdot 10^{+304}:\\ \;\;\;\;\left(\ell \cdot \left(\left(\frac{\cos k}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{1}{t}}{\sin k}\right) \cdot \frac{\ell}{t}\right)\right) \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\\ \mathbf{elif}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -1.7772874778501266 \cdot 10^{-209}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{2}{t}}{\sin k} \cdot \ell}{t \cdot \left|\frac{k}{t}\right|}\\ \end{array}\]

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 3 regimes

  2. if (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2))))) < -5.102484297178739e+304

    1. Initial program 62.0

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

    2. Initial simplification48.8

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt48.8

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    5. Applied times-frac48.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac47.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified47.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    8. Simplified27.9

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

    10. Applied *-un-lft-identity27.9

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

    11. Applied *-un-lft-identity27.9

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

    12. Applied div-inv27.9

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

    13. Applied times-frac27.9

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

    14. Applied times-frac27.9

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

    15. Applied associate-*l*27.9

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

    16. Simplified27.9

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

    18. Applied *-un-lft-identity27.9

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

    19. Applied tan-quot27.9

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

    20. Applied associate-/r/27.9

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

    21. Applied times-frac28.2

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

    22. Simplified28.2

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

    if -5.102484297178739e+304 < (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2))))) < -1.7772874778501266e-209

    1. Initial program 50.3

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

    2. Initial simplification39.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]

    3. Taylor expanded around inf 1.8

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

    if -1.7772874778501266e-209 < (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2)))))

    1. Initial program 45.9

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

    2. Initial simplification28.8

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt28.8

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    5. Applied times-frac28.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac25.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified25.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    8. Simplified13.5

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

    10. Applied *-un-lft-identity13.5

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

    11. Applied *-un-lft-identity13.5

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

    12. Applied div-inv13.5

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

    13. Applied times-frac13.5

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

    14. Applied times-frac13.5

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

    15. Applied associate-*l*13.5

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

    16. Simplified13.5

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

    18. Applied pow113.5

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

    19. Applied pow113.5

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

    20. Applied pow-prod-down13.5

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

    21. Simplified12.2

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

    23. Applied frac-times11.6

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -5.102484297178739 \cdot 10^{+304}:\\ \;\;\;\;\left(\ell \cdot \left(\left(\frac{\cos k}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{1}{t}}{\sin k}\right) \cdot \frac{\ell}{t}\right)\right) \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\\ \mathbf{elif}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -1.7772874778501266 \cdot 10^{-209}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{2}{t}}{\sin k} \cdot \ell}{t \cdot \left|\frac{k}{t}\right|}\\ \end{array}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))