Average Error: 47.4 → 2.6
Time: 5.5m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{k}}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 2}{\frac{k}{\ell}} \cdot \cos k}} \le -2.974875642858391 \cdot 10^{+306}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{k}{\ell}}}{\sin k \cdot \sin k}\\ \mathbf{if}\;\frac{\frac{1}{k}}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 2}{\frac{k}{\ell}} \cdot \cos k}} \le -4.3422026466553 \cdot 10^{-317}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 2}{\frac{k}{\ell}} \cdot \cos k}}\\ \mathbf{if}\;\frac{\frac{1}{k}}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 2}{\frac{k}{\ell}} \cdot \cos k}} \le 8.378076423337497 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \left(\frac{\frac{\ell}{\sin k}}{\sin k} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}\right)\\ \mathbf{if}\;\frac{\frac{1}{k}}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 2}{\frac{k}{\ell}} \cdot \cos k}} \le 9.635942338644337 \cdot 10^{+266}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\frac{\ell \cdot 2}{\frac{k}{\ell}} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \cos k\right) \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \end{array}\]

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 4 regimes

  2. if (/ (/ 1 k) (/ (* (sin k) (* t (sin k))) (* (/ (* l 2) (/ k l)) (cos k)))) < -2.974875642858391e+306

    1. Initial program 63.1

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

    3. Applied add-cbrt-cube63.1

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

    4. Applied simplify56.4

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

    5. Taylor expanded around inf 62.0

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

    6. Applied simplify21.5

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

    8. Applied *-un-lft-identity21.5

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

    9. Applied add-cube-cbrt22.0

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

    10. Applied times-frac22.0

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

    11. Applied times-frac5.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{1}}{\frac{t}{\cos k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]

    12. Applied simplify5.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{t}{\cos k}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{k}{\ell}}}{\sin k \cdot \sin k}\]

    if -2.974875642858391e+306 < (/ (/ 1 k) (/ (* (sin k) (* t (sin k))) (* (/ (* l 2) (/ k l)) (cos k)))) < -4.3422026466553e-317 or 8.378076423337497e-308 < (/ (/ 1 k) (/ (* (sin k) (* t (sin k))) (* (/ (* l 2) (/ k l)) (cos k)))) < 9.635942338644337e+266

    1. Initial program 55.2

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

    3. Applied add-cbrt-cube59.3

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

    4. Applied simplify54.1

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

    5. Taylor expanded around inf 47.9

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

    6. Applied simplify9.1

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

    8. Applied div-inv9.1

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

    9. Applied *-un-lft-identity9.1

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

    10. Applied times-frac9.3

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

    11. Applied associate-/l*4.0

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

    12. Applied simplify1.5

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

    if -4.3422026466553e-317 < (/ (/ 1 k) (/ (* (sin k) (* t (sin k))) (* (/ (* l 2) (/ k l)) (cos k)))) < 8.378076423337497e-308

    1. Initial program 38.2

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

    3. Applied add-cbrt-cube38.2

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

    4. Applied simplify17.3

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

    5. Taylor expanded around inf 8.0

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

    6. Applied simplify4.9

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

    8. Applied div-inv4.9

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

    9. Applied add-cube-cbrt4.9

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

    10. Applied times-frac5.2

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

    11. Applied times-frac2.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{1}{\ell}}}{\sin k \cdot \sin k}}\]

    12. Applied simplify0.9

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

    if 9.635942338644337e+266 < (/ (/ 1 k) (/ (* (sin k) (* t (sin k))) (* (/ (* l 2) (/ k l)) (cos k))))

    1. Initial program 60.4

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

    3. Applied add-cbrt-cube60.5

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

    4. Applied simplify54.4

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

    5. Taylor expanded around inf 60.4

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

    6. Applied simplify31.9

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

    8. Applied *-un-lft-identity31.9

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

    9. Applied div-inv31.9

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

    10. Applied times-frac31.9

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

    11. Applied times-frac24.1

      \[\leadsto \color{blue}{\frac{\frac{2}{1}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]

    12. Applied simplify24.1

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

    13. Applied simplify16.9

      \[\leadsto \left(\frac{2}{t} \cdot \cos k\right) \cdot \color{blue}{\left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed 2018193 +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))))