Average Error: 32.1 → 7.7
Time: 5.5m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.551848039283683 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{t \cdot \sin k}\\ \mathbf{elif}\;t \le 3.2418813369402593 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{1}{k \cdot k + 2 \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{t \cdot \sin k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -7.551848039283683e-47 or 3.2418813369402593e-69 < t

    1. Initial program 22.9

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

    2. Simplified12.8

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

    4. Applied *-un-lft-identity12.8

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

    5. Applied div-inv12.8

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

    6. Applied times-frac12.8

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

    7. Simplified12.8

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

    8. Simplified8.1

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

    10. Applied *-un-lft-identity8.1

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

    11. Applied *-un-lft-identity8.1

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

    12. Applied div-inv8.1

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

    13. Applied times-frac8.5

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

    14. Applied times-frac7.2

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

    15. Applied associate-*r*3.8

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

    16. Simplified3.4

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

    18. Applied frac-times3.4

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

    19. Simplified3.4

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

    20. Simplified3.4

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

    if -7.551848039283683e-47 < t < 3.2418813369402593e-69

    1. Initial program 55.2

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

    2. Simplified41.0

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

    4. Applied *-un-lft-identity41.0

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

    5. Applied div-inv41.0

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

    6. Applied times-frac40.1

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

    7. Simplified40.1

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

    8. Simplified34.8

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

    10. Applied associate-/r/34.8

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

    11. Applied *-un-lft-identity34.8

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

    12. Applied div-inv34.8

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

    13. Applied times-frac30.3

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

    14. Applied times-frac31.2

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

    15. Simplified31.2

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

    16. Simplified18.6

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.551848039283683 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{t \cdot \sin k}\\ \mathbf{elif}\;t \le 3.2418813369402593 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{1}{k \cdot k + 2 \cdot \left(t \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{\tan k} \cdot \frac{\frac{\frac{\ell}{t}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}{t \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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))))