Average Error: 31.4 → 12.3
Time: 4.7m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \le 9.259664538197736 \cdot 10^{-264}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right) \cdot \left(\sin k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{if}\;\frac{2}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \le 1.4455572240426125 \cdot 10^{+293}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(\sin k \cdot t\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t}}}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]

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 (* (* (* (* (* (/ t (/ l t)) (/ t l)) (sin k)) (tan k)) (sqrt (+ (+ 1 (pow (/ k t) 2)) 1))) (sqrt (+ (+ 1 (pow (/ k t) 2)) 1)))) < 9.259664538197736e-264

    1. Initial program 22.6

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

    3. Applied add-cube-cbrt22.7

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

    4. Applied times-frac19.5

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

    5. Applied simplify19.5

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

    6. Applied simplify11.0

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

    8. Applied tan-quot11.0

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

    9. Applied associate-*l/11.0

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

    10. Applied associate-*l/7.6

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

    11. Applied frac-times7.8

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

    12. Applied associate-*l/7.4

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

    14. Applied *-un-lft-identity7.4

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

    15. Applied associate-*r*7.4

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

    16. Applied simplify4.0

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

    18. Applied associate-*l*3.9

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

    if 9.259664538197736e-264 < (/ 2 (* (* (* (* (* (/ t (/ l t)) (/ t l)) (sin k)) (tan k)) (sqrt (+ (+ 1 (pow (/ k t) 2)) 1))) (sqrt (+ (+ 1 (pow (/ k t) 2)) 1)))) < 1.4455572240426125e+293

    1. Initial program 37.7

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

    3. Applied add-cube-cbrt38.0

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

    4. Applied times-frac31.6

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

    5. Applied simplify31.3

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

    6. Applied simplify1.0

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

    8. Applied add-sqr-sqrt1.4

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

    9. Applied associate-*r*1.3

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

    if 1.4455572240426125e+293 < (/ 2 (* (* (* (* (* (/ t (/ l t)) (/ t l)) (sin k)) (tan k)) (sqrt (+ (+ 1 (pow (/ k t) 2)) 1))) (sqrt (+ (+ 1 (pow (/ k t) 2)) 1))))

    1. Initial program 61.9

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

    3. Applied add-cube-cbrt61.9

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

    4. Applied times-frac61.5

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

    5. Applied simplify61.5

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

    6. Applied simplify61.3

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

    8. Applied tan-quot61.3

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

    9. Applied associate-*l/61.3

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

    10. Applied associate-*l/61.3

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

    11. Applied frac-times61.9

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

    12. Applied associate-*l/56.3

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

    14. Applied *-un-lft-identity56.3

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

    15. Applied associate-*r*56.3

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

    16. Applied simplify56.3

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

    18. Applied associate-*l/56.3

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

    19. Applied associate-*l/56.4

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

    20. Applied associate-*l/49.9

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

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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))))