Average Error: 47.2 → 0.6
Time: 4.0m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.7604726680248956 \cdot 10^{+70} \lor \neg \left(t \le 2.434594445758215 \cdot 10^{-200}\right):\\ \;\;\;\;\left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{2}{t}}{\sin k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t} \cdot \frac{\ell}{k}}{\sin k}}{\tan k}\\ \end{array}\]

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 2 regimes

  2. if t < -4.7604726680248956e+70 or 2.434594445758215e-200 < t

    1. Initial program 46.0

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

    2. Initial simplification26.4

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

    4. Applied times-frac26.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    5. Applied times-frac16.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

    6. Simplified16.0

      \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]

    9. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]

    10. Applied times-frac15.1

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]

    11. Applied times-frac10.0

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

    12. Applied associate-*r*8.6

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

    13. Simplified6.5

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

    15. Applied associate-*l/6.0

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}}{\sin k}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]

    16. Simplified1.3

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{2}{t}}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
    17. Using strategy rm

    18. Applied *-un-lft-identity1.3

      \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\color{blue}{1 \cdot \sin k}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]

    19. Applied times-frac0.4

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

    20. Applied associate-*l*0.6

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

    21. Simplified0.6

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

    if -4.7604726680248956e+70 < t < 2.434594445758215e-200

    1. Initial program 49.6

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

    2. Initial simplification37.7

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

    4. Applied times-frac37.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

    5. Applied times-frac28.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]

    6. Simplified26.2

      \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity26.2

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]

    9. Applied *-un-lft-identity26.2

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]

    10. Applied times-frac25.5

      \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]

    11. Applied times-frac18.1

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

    12. Applied associate-*r*16.1

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

    13. Simplified8.9

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

    15. Applied associate-*l/8.1

      \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}}{\sin k}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]

    16. Simplified0.5

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{2}{t}}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
    17. Using strategy rm

    18. Applied associate-*r/0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.7604726680248956 \cdot 10^{+70} \lor \neg \left(t \le 2.434594445758215 \cdot 10^{-200}\right):\\ \;\;\;\;\left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{2}{t}}{\sin k}\right) \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t} \cdot \frac{\ell}{k}}{\sin k}}{\tan k}\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018346 +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))))