Average Error: 47.4 → 11.8
Time: 3.7m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.313676741445256 \cdot 10^{-267}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\right) \cdot \left(\ell \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{elif}\;t \le 8.706404020968043 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\ell}{t}}{\sin k}\right) \cdot \cos k\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\\ \end{array}\]

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 t < -3.313676741445256e-267

    1. Initial program 47.1

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

    2. Initial simplification30.4

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

    4. Applied add-sqr-sqrt30.4

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

    5. Applied times-frac30.4

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

    6. Applied times-frac27.5

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

    7. Simplified27.5

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

    8. Simplified13.0

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

    10. Applied associate-*r*11.3

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

    12. Applied add-sqr-sqrt11.4

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

    13. Applied div-inv11.4

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

    14. Applied times-frac11.3

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

    15. Applied associate-*l*10.9

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

    17. Applied *-un-lft-identity10.9

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

    18. Applied div-inv10.9

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

    19. Applied times-frac10.9

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

    20. Simplified10.9

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

    if -3.313676741445256e-267 < t < 8.706404020968043e-145

    1. Initial program 62.6

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

    2. Initial simplification50.2

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

    3. Taylor expanded around -inf 26.0

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

    if 8.706404020968043e-145 < t

    1. Initial program 42.6

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

    2. Initial simplification25.0

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

    4. Applied add-sqr-sqrt25.0

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

    5. Applied times-frac24.8

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

    6. Applied times-frac23.1

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

    7. Simplified23.1

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

    8. Simplified9.8

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

    10. Applied associate-*r*8.1

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

    12. Applied tan-quot8.1

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

    13. Applied associate-/r/8.2

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

    14. Applied associate-*r*8.2

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.313676741445256 \cdot 10^{-267}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\right) \cdot \left(\ell \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{elif}\;t \le 8.706404020968043 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\ell}{t}}{\sin k}\right) \cdot \cos k\right) \cdot \frac{\frac{\ell}{t}}{\left|\frac{k}{t}\right|}\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(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))))