Average Error: 31.8 → 10.1
Time: 5.4m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.2963031154875214 \cdot 10^{-126} \lor \neg \left(t \le 4.2415807531278995 \cdot 10^{-186}\right):\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{1}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sin k} \cdot \frac{\left(\cos k \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot \ell}{\frac{k \cdot k}{t} + 2 \cdot t}\\ \end{array}\]

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 t < -2.2963031154875214e-126 or 4.2415807531278995e-186 < t

    1. Initial program 25.9

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

    2. Initial simplification20.1

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

    4. Applied *-un-lft-identity20.1

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

    5. Applied tan-quot20.1

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

    6. Applied associate-*r/20.1

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

    7. Applied associate-/r/20.1

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

    8. Applied times-frac20.1

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

    9. Simplified9.4

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

    11. Applied div-inv9.4

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

    12. Applied times-frac7.2

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

    13. Applied associate-*l*6.0

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

    if -2.2963031154875214e-126 < t < 4.2415807531278995e-186

    1. Initial program 62.7

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

    2. Initial simplification49.4

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

    4. Applied *-un-lft-identity49.4

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

    5. Applied tan-quot49.4

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

    6. Applied associate-*r/49.4

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

    7. Applied associate-/r/49.4

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

    8. Applied times-frac49.4

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

    9. Simplified49.1

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

    11. Applied add-cube-cbrt49.2

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

    12. Applied times-frac49.2

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

    13. Applied associate-*l*40.4

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

    15. Applied associate-/r/40.4

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

    16. Applied associate-*l*38.0

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

    18. Applied associate-*r/38.0

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

    19. Applied frac-times38.0

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

    20. Simplified31.7

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2963031154875214 \cdot 10^{-126} \lor \neg \left(t \le 4.2415807531278995 \cdot 10^{-186}\right):\\ \;\;\;\;\frac{2}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(\frac{\frac{1}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sin k} \cdot \frac{\left(\cos k \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot \ell}{\frac{k \cdot k}{t} + 2 \cdot t}\\ \end{array}\]

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018278 
(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))))