Average Error: 32.2 → 13.5
Time: 4.1m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(\left(\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}\right) \cdot t\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k} \le 4.3400588385362695 \cdot 10^{+247}:\\ \;\;\;\;\frac{2}{\frac{\left(\sqrt[3]{\left(\left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \sqrt[3]{\left(\left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \cdot \sqrt[3]{\left(\left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(\sin k \cdot t\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t}}}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]

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 (/ (* (* (* (* (* (cbrt (/ (sin k) (/ l t))) (cbrt (/ (sin k) (/ l t)))) (cbrt (/ (sin k) (/ l t)))) t) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k))) < 4.3400588385362695e+247

    1. Initial program 32.0

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

    3. Applied add-cube-cbrt32.1

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

    4. Applied times-frac28.2

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

    5. Applied simplify28.1

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

    6. Applied simplify15.3

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

    8. Applied tan-quot15.3

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

    9. Applied associate-*l/15.3

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

    10. Applied associate-*l/11.5

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

    11. Applied frac-times12.8

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

    12. Applied associate-*l/10.9

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

    14. Applied *-un-lft-identity10.9

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

    15. Applied associate-*r*10.9

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

    16. Applied simplify6.6

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

    18. Applied add-cube-cbrt6.9

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

    if 4.3400588385362695e+247 < (/ (* (* (* (* (* (cbrt (/ (sin k) (/ l t))) (cbrt (/ (sin k) (/ l t)))) (cbrt (/ (sin k) (/ l t)))) t) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))

    1. Initial program 32.5

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

    3. Applied add-cube-cbrt32.5

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

    4. Applied times-frac31.0

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

    5. Applied simplify31.0

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

    6. Applied simplify26.5

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

    8. Applied tan-quot26.5

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

    9. Applied associate-*l/26.5

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

    10. Applied associate-*l/26.4

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

    11. Applied frac-times26.1

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

    12. Applied associate-*l/25.3

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

    14. Applied *-un-lft-identity25.3

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

    15. Applied associate-*r*25.3

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

    16. Applied simplify24.6

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

    18. Applied associate-*l/24.6

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

    19. Applied associate-*l/25.1

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

    20. Applied associate-*l/22.9

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

Runtime

Time bar (total: 4.1m)Debug logProfile

herbie shell --seed 2018214 
(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))))