Average Error: 32.3 → 10.2
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.532904452950893 \cdot 10^{+62} \lor \neg \left(t \le 1.9083106523976526 \cdot 10^{-83}\right):\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\cos k}{t}\right) \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)}{\left(t \cdot t + k \cdot k\right) + t \cdot t}\\ \end{array}\]

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 < -7.532904452950893e+62 or 1.9083106523976526e-83 < t

    1. Initial program 22.7

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

    2. Initial simplification17.6

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity17.6

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

    5. Applied tan-quot17.6

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

    6. Applied associate-*r/17.6

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

    7. Applied associate-/r/17.6

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

    8. Applied times-frac17.6

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

    9. Simplified6.2

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

    11. Applied add-cube-cbrt6.4

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

    12. Applied times-frac3.8

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

    if -7.532904452950893e+62 < t < 1.9083106523976526e-83

    1. Initial program 48.2

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

    2. Initial simplification38.0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity38.0

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

    5. Applied tan-quot38.0

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

    6. Applied associate-*r/38.0

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

    7. Applied associate-/r/38.0

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

    8. Applied times-frac38.0

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

    9. Simplified34.0

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

    11. Applied associate-*l/34.7

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

    12. Applied associate-/r/35.5

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

    13. Applied associate-*l*32.7

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

    15. Applied div-inv32.7

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

    16. Applied associate-*l*27.7

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

    18. Applied frac-times27.7

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

    19. Applied associate-*r/28.5

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

    20. Applied frac-times28.3

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

    21. Simplified28.8

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

    22. Simplified20.7

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.532904452950893 \cdot 10^{+62} \lor \neg \left(t \le 1.9083106523976526 \cdot 10^{-83}\right):\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\cos k}{t}\right) \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)}{\left(t \cdot t + k \cdot k\right) + t \cdot t}\\ \end{array}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018230 
(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))))