Average Error: 31.9 → 12.5
Time: 4.2m
Precision: 64
Internal Precision: 384
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.5905127574298983 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \frac{\frac{\frac{\ell + \ell}{t}}{t \cdot \sin k}}{\tan k}\\ \mathbf{if}\;t \le 1.725710249328264 \cdot 10^{-191}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4}} \cdot \left(\frac{2}{3} \cdot t + \frac{2}{t}\right) - \frac{\frac{1}{3}}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{if}\;t \le 1.9758591388594481 \cdot 10^{-47}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\tan k \cdot \frac{t}{\ell}}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*} \cdot \frac{\frac{\frac{\ell + \ell}{t}}{t \cdot \sin k}}{\tan k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -2.5905127574298983e-121 or 1.9758591388594481e-47 < t

    1. Initial program 23.5

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

    3. Applied unpow323.5

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

    4. Applied times-frac16.7

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

    5. Applied associate-*l*14.5

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

    6. Taylor expanded around 0 14.5

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

    7. Applied simplify9.0

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

    9. Applied div-inv9.0

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

    10. Applied *-un-lft-identity9.0

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

    11. Applied times-frac9.1

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

    12. Applied div-inv9.2

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

    13. Applied *-un-lft-identity9.2

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

    14. Applied times-frac9.2

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

    15. Applied times-frac8.0

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

    16. Applied times-frac5.4

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

    17. Applied simplify5.3

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

    18. Applied simplify5.2

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

    if -2.5905127574298983e-121 < t < 1.725710249328264e-191

    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. Using strategy rm

    3. Applied unpow362.7

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

    4. Applied times-frac57.8

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

    5. Applied associate-*l*57.8

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

    6. Taylor expanded around 0 57.8

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

    7. Applied simplify47.9

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

    8. Taylor expanded around 0 40.6

      \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot \frac{{\ell}^{2} \cdot t}{{k}^{4}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - \frac{1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}\]

    9. Applied simplify38.7

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{{k}^{4}} \cdot \left(\frac{2}{3} \cdot t + \frac{2}{t}\right) - \frac{\frac{1}{3}}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\]

    if 1.725710249328264e-191 < t < 1.9758591388594481e-47

    1. Initial program 46.6

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

    3. Applied unpow346.6

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

    4. Applied times-frac32.5

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

    5. Applied associate-*l*31.4

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

    6. Taylor expanded around 0 31.4

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

    7. Applied simplify27.9

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

    9. Applied div-inv27.9

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

    10. Applied *-un-lft-identity27.9

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

    11. Applied add-cube-cbrt28.1

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

    12. Applied times-frac28.1

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

    13. Applied times-frac30.0

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

    14. Applied times-frac27.4

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

    15. Applied simplify27.3

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

    16. Applied simplify25.9

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

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))))