Average Error: 31.3 → 4.8
Time: 6.8m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.3679061397530498 \cdot 10^{-53} \lor \neg \left(k \le 8.538781939858447 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{\left(\frac{k}{\ell} \cdot \frac{k}{\ell} + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\sin k}{\cos k}}{\frac{1}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{\ell}{t}}{\sin k}}{t}}{\sqrt[3]{\tan k}}}{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{2 \cdot \frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}\\ \end{array}\]

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 2 regimes

  2. if k < -1.3679061397530498e-53 or 8.538781939858447e-78 < k

    1. Initial program 30.5

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

    2. Initial simplification18.4

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

    4. Applied times-frac18.2

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

    5. Applied associate-/l*16.4

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

    6. Taylor expanded around inf 21.9

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

    7. Simplified5.1

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

    9. Applied div-inv5.1

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

    10. Applied associate-/l*5.2

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

    if -1.3679061397530498e-53 < k < 8.538781939858447e-78

    1. Initial program 33.5

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

    2. Initial simplification41.3

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

    4. Applied times-frac17.0

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

    5. Applied associate-/l*16.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt16.9

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

    8. Applied times-frac7.4

      \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\color{blue}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}}\]

    9. Applied *-un-lft-identity7.4

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

    10. Applied times-frac7.2

      \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}}\]

    11. Applied *-un-lft-identity7.2

      \[\leadsto \frac{\frac{\frac{2}{t}}{\color{blue}{1 \cdot \sin k}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]

    12. Applied div-inv7.2

      \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{1}{t}}}{1 \cdot \sin k}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]

    13. Applied times-frac7.2

      \[\leadsto \frac{\color{blue}{\frac{2}{1} \cdot \frac{\frac{1}{t}}{\sin k}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]

    14. Applied times-frac4.6

      \[\leadsto \color{blue}{\frac{\frac{2}{1}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}} \cdot \frac{\frac{\frac{1}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}}\]

    15. Simplified4.6

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot 2}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{\frac{1}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]

    16. Simplified3.6

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.3679061397530498 \cdot 10^{-53} \lor \neg \left(k \le 8.538781939858447 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{\left(\frac{k}{\ell} \cdot \frac{k}{\ell} + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{\sin k}{\cos k}}{\frac{1}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{\ell}{t}}{\sin k}}{t}}{\sqrt[3]{\tan k}}}{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{2 \cdot \frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}\\ \end{array}\]

Runtime

Time bar (total: 6.8m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes8.24.82.75.463.1%
herbie shell --seed 2018290 
(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))))