Average Error: 47.5 → 6.9
Time: 4.7m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -9.947179733255817 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \sin k}{\cos k} \cdot \left(\frac{1}{\ell} \cdot \left(\frac{\frac{k}{\ell}}{\frac{1}{t}} \cdot \sin k\right)\right)}\\ \mathbf{elif}\;\ell \le 5.035464579739297 \cdot 10^{-250}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \sin k}{\cos k} \cdot \left(\frac{1}{\ell} \cdot \left(\frac{\frac{k}{\ell}}{\frac{1}{t}} \cdot \sin k\right)\right)}\\ \end{array}\]

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 l < -9.947179733255817e-164 or 5.035464579739297e-250 < l

    1. Initial program 47.8

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

    2. Initial simplification32.0

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

    4. Applied tan-quot32.0

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

    5. Applied frac-times31.8

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

    6. Applied associate-*l/32.5

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

    7. Applied frac-times26.4

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

    8. Simplified24.7

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

    9. Simplified19.9

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

    11. Applied times-frac15.1

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

    12. Applied associate-*l*15.1

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

    14. Applied associate-/r*8.6

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

    16. Applied div-inv8.6

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

    17. Applied *-un-lft-identity8.6

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

    18. Applied times-frac7.1

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

    19. Applied associate-*l*6.1

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

    if -9.947179733255817e-164 < l < 5.035464579739297e-250

    1. Initial program 46.6

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

    2. Initial simplification24.4

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

    4. Applied tan-quot24.4

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

    5. Applied frac-times25.1

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

    6. Applied associate-*l/25.1

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

    7. Applied frac-times19.5

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

    8. Simplified16.4

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

    9. Simplified16.4

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

    11. Applied times-frac15.4

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

    12. Applied associate-*l*15.4

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

    14. Applied associate-/r*13.2

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

    16. Applied associate-*l/10.6

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

    17. Applied frac-times9.6

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -9.947179733255817 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \sin k}{\cos k} \cdot \left(\frac{1}{\ell} \cdot \left(\frac{\frac{k}{\ell}}{\frac{1}{t}} \cdot \sin k\right)\right)}\\ \mathbf{elif}\;\ell \le 5.035464579739297 \cdot 10^{-250}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \sin k}{\cos k} \cdot \left(\frac{1}{\ell} \cdot \left(\frac{\frac{k}{\ell}}{\frac{1}{t}} \cdot \sin k\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2018234 +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))))