Average Error: 46.8 → 2.7
Time: 6.8m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2 \cdot \cos k}{t \cdot k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le -1.1910273397120728 \cdot 10^{-300}:\\ \;\;\;\;\frac{2 \cdot \cos k}{t \cdot k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{if}\;\frac{2 \cdot \cos k}{t \cdot k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le 0.0:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{\frac{k}{\ell}}{\cos k}} \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{\sin k}\\ \mathbf{if}\;\frac{2 \cdot \cos k}{t \cdot k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \le 9.461363920748954 \cdot 10^{+275}:\\ \;\;\;\;\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{\frac{k}{\ell}}{\cos k}} \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{\sin k}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (* 2 (cos k)) (* t k)) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < -1.1910273397120728e-300

    1. Initial program 58.1

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

    3. Applied associate-*l*58.1

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

    4. Applied simplify55.3

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

    5. Taylor expanded around inf 36.7

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

    7. Applied add-cbrt-cube50.7

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

    8. Applied simplify41.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\right)}^{3}}}\]
    9. Using strategy rm

    10. Applied div-inv40.9

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

    11. Applied div-inv40.9

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

    12. Applied times-frac42.4

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

    13. Applied times-frac42.0

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

    14. Applied unpow-prod-down55.3

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

    15. Applied cbrt-prod53.5

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

    16. Applied simplify41.7

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

    17. Applied simplify9.1

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

    if -1.1910273397120728e-300 < (* (/ (* 2 (cos k)) (* t k)) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < 0.0 or 9.461363920748954e+275 < (* (/ (* 2 (cos k)) (* t k)) (* (/ l (sin k)) (/ (/ l k) (sin k))))

    1. Initial program 39.2

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

    3. Applied associate-*l*39.2

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

    4. Applied simplify29.0

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

    5. Taylor expanded around inf 12.3

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

    7. Applied add-cbrt-cube13.6

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

    8. Applied simplify10.5

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\right)}^{3}}}\]
    9. Using strategy rm

    10. Applied div-inv10.5

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

    11. Applied times-frac9.9

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

    12. Applied unpow-prod-down11.7

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

    13. Applied cbrt-prod11.5

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

    14. Applied simplify6.5

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

    15. Applied simplify0.7

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

    if 0.0 < (* (/ (* 2 (cos k)) (* t k)) (* (/ l (sin k)) (/ (/ l k) (sin k)))) < 9.461363920748954e+275

    1. Initial program 56.0

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

    3. Applied associate-*l*56.0

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

    4. Applied simplify52.5

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

    5. Taylor expanded around inf 35.3

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

    7. Applied add-cbrt-cube49.0

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

    8. Applied simplify38.8

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\right)}^{3}}}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity38.8

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

    11. Applied associate-/r/38.8

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

    12. Applied times-frac40.2

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

    13. Applied times-frac39.7

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

    14. Applied unpow-prod-down53.3

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

    15. Applied cbrt-prod50.9

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

    16. Applied simplify38.1

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

    17. Applied simplify1.0

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

Runtime

Time bar (total: 6.8m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(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))))