Average Error: 47.1 → 9.2
Time: 4.7m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\ell} \le 5.871156910429544 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\\ \mathbf{if}\;\frac{1}{\ell} \le 6.7032019549394435 \cdot 10^{+124}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}\\ \mathbf{if}\;\frac{1}{\ell} \le +\infty:\\ \;\;\;\;\frac{\cos k + \cos k}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k + \cos k}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 l) < 5.871156910429544e-85

    1. Initial program 49.5

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

    3. Applied add-cbrt-cube50.9

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

    4. Applied simplify40.9

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

    6. Applied unpow-prod-down47.7

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

    7. Applied cbrt-prod47.4

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

    8. Applied simplify41.8

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

    9. Applied simplify30.8

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

    10. Taylor expanded around inf 28.3

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

    11. Taylor expanded around -inf 62.7

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

    12. Applied simplify9.5

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

    if 5.871156910429544e-85 < (/ 1 l) < 6.7032019549394435e+124

    1. Initial program 42.0

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

    3. Applied add-cbrt-cube45.0

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

    4. Applied simplify35.8

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

    6. Applied unpow-prod-down44.6

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

    7. Applied cbrt-prod44.2

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

    8. Applied simplify38.3

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

    9. Applied simplify30.1

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

    10. Taylor expanded around inf 8.7

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

    12. Applied unpow28.7

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

    13. Applied associate-*l*5.1

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

    if 6.7032019549394435e+124 < (/ 1 l)

    1. Initial program 44.1

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

    3. Applied add-cbrt-cube44.6

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

    4. Applied simplify31.7

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

    6. Applied unpow-prod-down38.6

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

    7. Applied cbrt-prod38.3

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

    8. Applied simplify31.9

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

    9. Applied simplify22.6

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

    10. Taylor expanded around inf 17.5

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

    11. Taylor expanded around -inf 62.9

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

    12. Applied simplify11.9

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

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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))))