Average Error: 47.6 → 11.9
Time: 7.9m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -2.067936593063351 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t}\\ \mathbf{elif}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -1.463223388342217 \cdot 10^{-266}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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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))))) < -2.067936593063351e+246

    1. Initial program 61.2

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

    2. Initial simplification47.3

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

    4. Applied add-sqr-sqrt47.3

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

    5. Applied times-frac47.0

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

    6. Applied times-frac43.8

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

    7. Simplified43.8

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

    8. Simplified23.7

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

    10. Applied add-sqr-sqrt23.7

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

    11. Applied add-cube-cbrt24.1

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

    12. Applied add-cube-cbrt24.2

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

    13. Applied times-frac24.2

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

    14. Applied times-frac23.7

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

    16. Applied associate-*r*26.7

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

    if -2.067936593063351e+246 < (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2))))) < -1.463223388342217e-266

    1. Initial program 51.3

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

    2. Initial simplification42.3

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

    3. Taylor expanded around inf 1.2

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

    if -1.463223388342217e-266 < (* 2 (/ (* (pow l 2) (cos k)) (* t (* (pow k 2) (pow (sin k) 2)))))

    1. Initial program 46.5

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

    2. Initial simplification28.6

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

    4. Applied add-sqr-sqrt28.6

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

    5. Applied times-frac28.6

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

    6. Applied times-frac25.4

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

    7. Simplified25.4

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

    8. Simplified12.7

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

    10. Applied *-un-lft-identity12.7

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

    11. Applied add-cube-cbrt12.8

      \[\leadsto \frac{\frac{\frac{2}{t}}{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}}{1 \cdot \left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]

    12. Applied add-cube-cbrt12.9

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \sqrt[3]{\frac{2}{t}}}}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}{1 \cdot \left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]

    13. Applied times-frac12.9

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}}{1 \cdot \left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]

    14. Applied times-frac12.2

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{1} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\left|\frac{k}{t}\right|}\right)} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\]

    15. Simplified12.2

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -2.067936593063351 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t}\\ \mathbf{elif}\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2 \le -1.463223388342217 \cdot 10^{-266}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \end{array}\]

Runtime

Time bar (total: 7.9m)Debug logProfile

herbie shell --seed 2018216 
(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))))