Average Error: 31.7 → 12.2
Time: 4.5m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(t \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \le -9.660347629688947 \cdot 10^{-279}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{if}\;\left(t \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \le 3.104615549635093 \cdot 10^{-285}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\cos k \cdot \ell}}\\ \mathbf{if}\;\left(t \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \le 2.5865529068730424 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\frac{k \cdot t}{\ell} + \frac{1}{3} \cdot \frac{{k}^{3} \cdot t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

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 3 regimes

  2. if (* (* t (* (* (tan k) (/ t l)) (* (/ t l) (sin k)))) (+ (+ 1 (pow (/ k t) 2)) 1)) < -9.660347629688947e-279 or 3.104615549635093e-285 < (* (* t (* (* (tan k) (/ t l)) (* (/ t l) (sin k)))) (+ (+ 1 (pow (/ k t) 2)) 1)) < 2.5865529068730424e+305

    1. Initial program 29.1

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

    3. Applied unpow329.1

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

    4. Applied times-frac19.6

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

    5. Applied associate-*l*16.5

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

    7. Applied *-un-lft-identity16.5

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

    8. Applied associate-*r*16.5

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

    9. Applied simplify8.1

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

    11. Applied *-un-lft-identity8.1

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

    12. Applied associate-*r*8.1

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

    13. Applied simplify2.6

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

    if -9.660347629688947e-279 < (* (* t (* (* (tan k) (/ t l)) (* (/ t l) (sin k)))) (+ (+ 1 (pow (/ k t) 2)) 1)) < 3.104615549635093e-285

    1. Initial program 57.5

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

    3. Applied unpow357.5

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

    4. Applied times-frac44.6

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

    5. Applied associate-*l*44.0

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

    7. Applied *-un-lft-identity44.0

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

    8. Applied associate-*r*44.0

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

    9. Applied simplify27.4

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

    11. Applied *-un-lft-identity27.4

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

    12. Applied associate-*r*27.4

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

    13. Applied simplify56.6

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

    15. Applied associate-*l/57.4

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

    16. Applied tan-quot57.4

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

    17. Applied associate-*l/57.4

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

    18. Applied frac-times58.3

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

    19. Applied associate-*r/26.8

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

    20. Applied associate-*l/18.5

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

    if 2.5865529068730424e+305 < (* (* t (* (* (tan k) (/ t l)) (* (/ t l) (sin k)))) (+ (+ 1 (pow (/ k t) 2)) 1))

    1. Initial program 31.8

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

    3. Applied unpow331.8

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

    4. Applied times-frac28.8

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

    5. Applied associate-*l*28.8

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

    7. Applied *-un-lft-identity28.8

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

    8. Applied associate-*r*28.8

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

    9. Applied simplify27.0

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

    11. Applied *-un-lft-identity27.0

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

    12. Applied associate-*r*27.0

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

    13. Applied simplify26.9

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

    14. Taylor expanded around 0 23.2

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

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2020178 
(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))))