Average Error: 31.6 → 12.9
Time: 9.2m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 7.744247477931036 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{\frac{t}{\ell}}{\sqrt[3]{\frac{2}{\sin k}}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{\tan k}{\frac{\ell}{t}}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{1}{\frac{1}{\frac{\frac{\ell}{t}}{\tan k}}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{t} \cdot \left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{t} \cdot \ell\right)\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < 7.744247477931036e-139

    1. Initial program 36.0

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

    2. Simplified23.2

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

    4. Applied *-un-lft-identity23.2

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

    5. Applied times-frac22.1

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

    6. Applied add-cube-cbrt22.2

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

    7. Applied times-frac21.9

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

    8. Applied times-frac19.8

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

    9. Simplified18.0

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

    11. Applied div-inv18.0

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

    13. Applied associate-*l/18.4

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

    14. Applied associate-*r/18.4

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

    15. Applied associate-*l/17.2

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

    16. Simplified17.1

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

    if 7.744247477931036e-139 < t

    1. Initial program 25.1

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

    2. Simplified11.8

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

    4. Applied *-un-lft-identity11.8

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

    5. Applied times-frac10.4

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

    6. Applied add-cube-cbrt10.6

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

    7. Applied times-frac10.4

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

    8. Applied times-frac9.4

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

    9. Simplified6.6

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

    11. Applied div-inv6.6

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

    13. Applied *-un-lft-identity6.6

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

    14. Applied associate-/l*6.6

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 7.744247477931036 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{\frac{t}{\ell}}{\sqrt[3]{\frac{2}{\sin k}}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{\tan k}{\frac{\ell}{t}}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{1}{\frac{1}{\frac{\frac{\ell}{t}}{\tan k}}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{t} \cdot \left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{t} \cdot \ell\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(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))))