Average Error: 32.8 → 14.2
Time: 2.6m
Precision: 64
Internal Precision: 320
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\tan k}\right) \le -1.6603180324295 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{2}{t}}{\tan k} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\tan k}\right) \le 1.937923838861096 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\left(\frac{k}{t} \cdot k\right) \cdot \left(\left(1 - 1\right) + 1\right) + t \cdot \left({1}^{3} + {1}^{3}\right)} \cdot t\right) \cdot \frac{\frac{2}{t}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\tan k}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (* (cbrt (/ 2 t)) (cbrt (/ 2 t))) (* (/ (cbrt (/ 2 t)) (tan k)) (* (/ l t) (/ (/ (/ l t) (sin k)) (+ (* (/ k t) (/ k t)) (+ 1 1)))))) < -1.6603180324295e-311

    1. Initial program 47.5

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

    2. Initial simplification29.6

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

    4. Applied *-un-lft-identity29.6

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

    5. Applied times-frac17.0

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

    6. Applied times-frac15.1

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

    7. Simplified15.1

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\tan k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity15.1

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

    10. Applied *-un-lft-identity15.1

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

    11. Applied times-frac6.3

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

    12. Applied times-frac3.6

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

    13. Simplified3.6

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

    if -1.6603180324295e-311 < (* (* (cbrt (/ 2 t)) (cbrt (/ 2 t))) (* (/ (cbrt (/ 2 t)) (tan k)) (* (/ l t) (/ (/ (/ l t) (sin k)) (+ (* (/ k t) (/ k t)) (+ 1 1)))))) < 1.937923838861096e-111

    1. Initial program 25.9

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

    2. Initial simplification20.9

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

    4. Applied *-un-lft-identity20.9

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

    5. Applied times-frac16.3

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

    6. Applied times-frac14.9

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

    7. Simplified14.9

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\tan k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\]
    8. Using strategy rm

    9. Applied flip3-+14.9

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

    10. Applied associate-*r/15.2

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

    11. Applied frac-add15.2

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

    12. Applied associate-/r/14.2

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

    13. Simplified14.2

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

    if 1.937923838861096e-111 < (* (* (cbrt (/ 2 t)) (cbrt (/ 2 t))) (* (/ (cbrt (/ 2 t)) (tan k)) (* (/ l t) (/ (/ (/ l t) (sin k)) (+ (* (/ k t) (/ k t)) (+ 1 1))))))

    1. Initial program 50.0

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

    2. Initial simplification40.1

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

    4. Applied *-un-lft-identity40.1

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

    5. Applied times-frac31.6

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

    6. Applied times-frac29.9

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

    7. Simplified29.9

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\tan k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity29.9

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

    10. Applied *-un-lft-identity29.9

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

    11. Applied times-frac24.7

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

    12. Applied times-frac22.4

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

    13. Simplified22.4

      \[\leadsto \frac{\frac{2}{t}}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{t}} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\right)\]
    14. Using strategy rm

    15. Applied *-un-lft-identity22.4

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

    16. Applied add-cube-cbrt22.9

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

    17. Applied times-frac22.9

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

    18. Applied associate-*l*23.9

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

    19. Simplified23.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right)} \cdot \left(\frac{\sqrt[3]{\frac{2}{t}}}{\tan k} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)}\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\tan k}\right) \le -1.6603180324295 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{2}{t}}{\tan k} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\tan k}\right) \le 1.937923838861096 \cdot 10^{-111}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}}{\left(\frac{k}{t} \cdot k\right) \cdot \left(\left(1 - 1\right) + 1\right) + t \cdot \left({1}^{3} + {1}^{3}\right)} \cdot t\right) \cdot \frac{\frac{2}{t}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\sin k}}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\tan k}\right)\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018215 
(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))))