Average Error: 47.0 → 19.2
Time: 6.2m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.747891034753405 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{\left|\frac{k}{t}\right| \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left(\sin k \cdot \left(\tan k \cdot t\right)\right)\right)}\\ \mathbf{if}\;t \le -3.343425587628506 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|}{\cos k \cdot \ell} \cdot \left|\frac{k}{t}\right|}\\ \mathbf{if}\;t \le 2.2613754421181447 \cdot 10^{-69} \lor \neg \left(t \le 5.373468861171188 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{2}{\left|\frac{k}{t}\right| \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left(\sin k \cdot \left(\tan k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{t \cdot t}}{\sin k \cdot \tan k}}{\frac{\left|\frac{k}{t}\right|}{\ell} \cdot \frac{\left|\frac{k}{t}\right|}{\ell}}\\ \end{array}\]

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 t < -2.747891034753405e+69 or -3.343425587628506e-54 < t < 2.2613754421181447e-69 or 5.373468861171188e+103 < t

    1. Initial program 52.7

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

    3. Applied add-sqr-sqrt52.7

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

    4. Applied simplify52.7

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

    5. Applied simplify45.5

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

    7. Applied add-cube-cbrt45.5

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

    8. Applied times-frac44.0

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

    9. Applied simplify43.9

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

    10. Applied simplify33.1

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

    12. Applied associate-*r*27.5

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

    14. Applied *-un-lft-identity27.5

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

    15. Applied associate-*r*27.5

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

    16. Applied simplify22.8

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

    if -2.747891034753405e+69 < t < -3.343425587628506e-54

    1. Initial program 29.6

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

    3. Applied add-sqr-sqrt29.6

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

    4. Applied simplify29.6

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

    5. Applied simplify21.4

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

    7. Applied add-cube-cbrt21.5

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

    8. Applied times-frac19.4

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

    9. Applied simplify19.3

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

    10. Applied simplify19.2

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

    12. Applied associate-*r*16.3

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

    14. Applied tan-quot16.3

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

    15. Applied associate-*r/16.3

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

    16. Applied associate-*l/14.4

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

    17. Applied frac-times12.9

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

    18. Applied associate-*l/7.7

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

    if 2.2613754421181447e-69 < t < 5.373468861171188e+103

    1. Initial program 29.7

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

    3. Applied add-sqr-sqrt29.7

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

    4. Applied simplify29.7

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

    5. Applied simplify21.5

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

    6. Taylor expanded around -inf 62.5

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

    7. Applied simplify8.2

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

  4. Applied simplify19.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;t \le -2.747891034753405 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{\left|\frac{k}{t}\right| \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left(\sin k \cdot \left(\tan k \cdot t\right)\right)\right)}\\ \mathbf{if}\;t \le -3.343425587628506 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(t \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|}{\cos k \cdot \ell} \cdot \left|\frac{k}{t}\right|}\\ \mathbf{if}\;t \le 2.2613754421181447 \cdot 10^{-69} \lor \neg \left(t \le 5.373468861171188 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{2}{\left|\frac{k}{t}\right| \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left(\sin k \cdot \left(\tan k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{t \cdot t}}{\sin k \cdot \tan k}}{\frac{\left|\frac{k}{t}\right|}{\ell} \cdot \frac{\left|\frac{k}{t}\right|}{\ell}}\\ \end{array}}\]

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed 2019053 
(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))))