Average Error: 46.9 → 21.7
Time: 3.9m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.118742858483706 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot t\right) \cdot \frac{{k}^{3}}{\ell}}{\ell \cdot \cos k}}\\ \mathbf{if}\;\ell \cdot \ell \le 3.2313996782473684 \cdot 10^{+238}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \left(\sin k \cdot t\right)}{\left(\ell \cdot \cos k\right) \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (* l l) < 9.118742858483706e-58

    1. Initial program 43.6

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

    3. Applied add-cbrt-cube44.7

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

    4. Applied simplify32.0

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/35.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/35.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot35.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/35.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times35.3

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div36.4

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div36.1

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify26.6

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify22.9

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]
    15. Using strategy rm

    16. Applied associate-*l*22.5

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot t\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}\right)}}{\ell \cdot \cos k}}\]

    17. Taylor expanded around 0 14.4

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

    if 9.118742858483706e-58 < (* l l) < 3.2313996782473684e+238

    1. Initial program 44.3

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

    3. Applied add-cbrt-cube47.8

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

    4. Applied simplify41.2

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/41.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/41.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot41.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/41.4

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times41.2

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div43.2

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div42.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify33.8

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify31.2

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]
    15. Using strategy rm

    16. Applied associate-*r/29.9

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \left(\sin k \cdot t\right)}{\frac{\ell}{t}}}}{\ell \cdot \cos k}}\]

    17. Applied associate-/l/29.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \left(\sin k \cdot t\right)}{\left(\ell \cdot \cos k\right) \cdot \frac{\ell}{t}}}}\]

    if 3.2313996782473684e+238 < (* l l)

    1. Initial program 59.7

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

    3. Applied add-cbrt-cube60.3

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

    4. Applied simplify52.1

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied associate-*l/54.5

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \left(\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell}} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\]

    7. Applied associate-*l/54.5

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}}\right)}^{3}}}\]

    8. Applied tan-quot54.5

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    9. Applied associate-*r/54.5

      \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k}{\cos k}} \cdot \frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\ell}\right)}^{3}}}\]

    10. Applied frac-times54.2

      \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)}{\cos k \cdot \ell}\right)}}^{3}}}\]

    11. Applied cube-div62.3

      \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    12. Applied cbrt-div62.3

      \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}}\]

    13. Applied simplify62.0

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}}{\sqrt[3]{{\left(\cos k \cdot \ell\right)}^{3}}}}\]

    14. Applied simplify41.6

      \[\leadsto \frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\color{blue}{\ell \cdot \cos k}}}\]
    15. Using strategy rm

    16. Applied associate-*l*40.7

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sin k \cdot t\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}\right)}}{\ell \cdot \cos k}}\]
    17. Using strategy rm

    18. Applied times-frac34.7

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\cos k}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))))