Average Error: 47.5 → 6.0
Time: 4.9m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.0588982186288542 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k \cdot \ell}{\left(k \cdot \sin k\right) \cdot t}}}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \sin k}{\cos k \cdot \ell}}{\frac{\frac{\ell}{t}}{k \cdot \sin k}}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if t < 1.0588982186288542e+17

    1. Initial program 47.8

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

    2. Initial simplification32.5

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

    4. Applied associate-*l/32.7

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

    5. Applied tan-quot32.7

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

    6. Applied associate-*r/33.1

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

    7. Applied frac-times33.1

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

    8. Applied frac-times27.5

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

    9. Simplified25.3

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

    10. Simplified21.4

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

    12. Applied associate-/l*16.1

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

    14. Applied associate-*l/16.1

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

    15. Applied associate-/l/9.0

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

    17. Applied associate-*r/11.4

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

    18. Applied associate-/r/11.4

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

    19. Applied times-frac6.1

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

    20. Applied associate-/r*5.8

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

    if 1.0588982186288542e+17 < t

    1. Initial program 46.9

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

    2. Initial simplification25.0

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

    4. Applied associate-*l/25.0

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

    5. Applied tan-quot25.0

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

    6. Applied associate-*r/26.0

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

    7. Applied frac-times26.0

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

    8. Applied frac-times21.4

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

    9. Simplified19.6

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

    10. Simplified17.2

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

    12. Applied associate-/l*12.8

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

    14. Applied associate-*l/12.8

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

    15. Applied associate-/l/7.8

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

    17. Applied div-inv7.8

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

    18. Applied associate-/r*6.8

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

    19. Simplified6.5

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.0588982186288542 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{2}{\frac{\sin k}{\frac{\cos k \cdot \ell}{\left(k \cdot \sin k\right) \cdot t}}}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \sin k}{\cos k \cdot \ell}}{\frac{\frac{\ell}{t}}{k \cdot \sin k}}}\\ \end{array}\]

Runtime

Time bar (total: 4.9m)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))))