Average Error: 31.4 → 15.6
Time: 6.0m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \le 3.8176941047138654 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\left(\left(\log \left(t \cdot \sin k\right) + \log \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) + \log \left(\tan k\right)\right) + \log \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\\ \end{array}\]

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 2 regimes

  2. if (/ 2 (* (* (* t (sin k)) (* (* (tan k) (/ t l)) (/ t l))) (+ (+ 1 (pow (/ k t) 2)) 1))) < 3.8176941047138654e+245

    1. Initial program 24.2

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

    3. Applied add-cube-cbrt24.3

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

    4. Applied times-frac21.0

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

    5. Applied simplify20.9

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

    6. Applied simplify10.3

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

    8. Applied *-un-lft-identity10.3

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

    9. Applied associate-*r*10.3

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

    10. Applied simplify8.8

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

    12. Applied associate-*r*6.4

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

    14. Applied *-un-lft-identity6.4

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

    15. Applied associate-*r*6.4

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

    16. Applied simplify4.8

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

    if 3.8176941047138654e+245 < (/ 2 (* (* (* t (sin k)) (* (* (tan k) (/ t l)) (/ t l))) (+ (+ 1 (pow (/ k t) 2)) 1)))

    1. Initial program 60.9

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

    3. Applied add-cube-cbrt60.9

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

    4. Applied times-frac59.9

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

    5. Applied simplify59.8

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

    6. Applied simplify58.4

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

    8. Applied *-un-lft-identity58.4

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

    9. Applied associate-*r*58.4

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

    10. Applied simplify59.5

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

    12. Applied add-exp-log59.5

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

    13. Applied add-exp-log61.3

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

    14. Applied add-exp-log61.3

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

    15. Applied add-exp-log61.3

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

    16. Applied prod-exp61.3

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

    17. Applied prod-exp61.3

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

    18. Applied prod-exp59.3

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

Runtime

Time bar (total: 6.0m)Debug logProfile

herbie shell --seed 2018296 +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))))