Average Error: 47.5 → 12.3
Time: 3.6m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -9.8738982995312 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{elif}\;k \le 4.555909137596045 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\frac{k}{t}}}{\frac{\tan k}{\frac{\frac{\ell}{t}}{\frac{k}{t}}}} \cdot \frac{\frac{2}{t}}{\sin k}\\ \mathbf{elif}\;k \le 2.298108769312118 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\left(\frac{\frac{1}{\tan k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{t}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\\ \end{array}\]

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 k < -9.8738982995312e-38 or 4.555909137596045e-87 < k < 2.298108769312118e+152

    1. Initial program 47.6

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

    2. Initial simplification28.4

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

    4. Applied add-sqr-sqrt28.4

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

    5. Applied times-frac28.3

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

    6. Applied times-frac24.9

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

    7. Simplified24.9

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

    8. Simplified9.4

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

    10. Applied associate-*r*8.4

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

    if -9.8738982995312e-38 < k < 4.555909137596045e-87

    1. Initial program 61.4

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

    2. Initial simplification53.0

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

    4. Applied *-un-lft-identity53.0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}}\]

    5. Applied times-frac52.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}\]

    6. Applied times-frac51.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{1} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    7. Simplified51.3

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

    8. Simplified31.2

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

    if 2.298108769312118e+152 < k

    1. Initial program 38.0

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

    2. Initial simplification23.2

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

    4. Applied add-sqr-sqrt23.2

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

    5. Applied times-frac23.2

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

    6. Applied times-frac19.2

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

    7. Simplified19.2

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

    8. Simplified12.4

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

    10. Applied add-sqr-sqrt12.4

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

    11. Applied div-inv12.4

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

    12. Applied times-frac12.4

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

    13. Applied associate-*l*12.4

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -9.8738982995312 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{elif}\;k \le 4.555909137596045 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{\frac{k}{t}}}{\frac{\tan k}{\frac{\frac{\ell}{t}}{\frac{k}{t}}}} \cdot \frac{\frac{2}{t}}{\sin k}\\ \mathbf{elif}\;k \le 2.298108769312118 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\left(\frac{\frac{1}{\tan k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{\ell}{t}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018234 
(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))))