Average Error: 47.4 → 13.5
Time: 3.7m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.3226220525186276 \cdot 10^{-287}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{1}{\left|\frac{k}{t}\right|}\right)\right)\\ \mathbf{elif}\;t \le 5.276403233956927 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{1}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\ell}{t}\right)\\ \end{array}\]

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 < -4.3226220525186276e-287

    1. Initial program 47.5

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

    2. Initial simplification31.1

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt31.1

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    5. Applied times-frac31.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac28.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified28.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    8. Simplified13.7

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

    10. Applied add-sqr-sqrt13.7

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

    11. Applied div-inv13.7

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

    12. Applied times-frac13.3

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

    14. Applied div-inv13.5

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

    if -4.3226220525186276e-287 < t < 5.276403233956927e-162

    1. Initial program 62.5

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

    2. Initial simplification51.2

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]

    3. Taylor expanded around inf 26.1

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

    if 5.276403233956927e-162 < t

    1. Initial program 43.2

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

    2. Initial simplification25.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt25.4

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    5. Applied times-frac25.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac23.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified23.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    8. Simplified10.0

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

    10. Applied add-sqr-sqrt10.1

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

    11. Applied div-inv10.1

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

    12. Applied times-frac10.1

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

    14. Applied add-sqr-sqrt10.1

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

    15. Applied *-un-lft-identity10.1

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

    16. Applied times-frac10.1

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.3226220525186276 \cdot 10^{-287}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{1}{\left|\frac{k}{t}\right|}\right)\right)\\ \mathbf{elif}\;t \le 5.276403233956927 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{1}{\sin k}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \left(\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{1}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\ell}{t}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018230 
(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))))