Average Error: 31.5 → 13.5
Time: 4.9m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right)}{\frac{\ell}{t} \cdot \cos k} \le -7.891159574845611 \cdot 10^{-259}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{if}\;\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right)}{\frac{\ell}{t} \cdot \cos k} \le 1.1056528181896636 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \left(1 + 1\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]

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 (/ (* (* (* (sin k) t) (* (/ t l) (sin k))) (+ (* (/ k t) (/ k t)) (+ 1 1))) (* (/ l t) (cos k))) < -7.891159574845611e-259 or 1.1056528181896636e-182 < (/ (* (* (* (sin k) t) (* (/ t l) (sin k))) (+ (* (/ k t) (/ k t)) (+ 1 1))) (* (/ l t) (cos k)))

    1. Initial program 30.6

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

    3. Applied add-cube-cbrt30.6

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

    4. Applied times-frac28.1

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

    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 simplify19.1

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

    8. Applied associate-*l*19.1

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

    10. Applied add-sqr-sqrt19.2

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

    11. Applied associate-*r*19.1

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

    13. Applied tan-quot19.1

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

    14. Applied associate-*l/19.1

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

    15. Applied associate-*l/19.1

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

    16. Applied associate-*l/19.1

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

    17. Applied associate-*l/17.0

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

    18. Applied frac-times15.6

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

    19. Applied simplify13.1

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

    if -7.891159574845611e-259 < (/ (* (* (* (sin k) t) (* (/ t l) (sin k))) (+ (* (/ k t) (/ k t)) (+ 1 1))) (* (/ l t) (cos k))) < 1.1056528181896636e-182

    1. Initial program 48.9

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

    3. Applied add-cube-cbrt49.0

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

    4. Applied times-frac41.1

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

    5. Applied simplify40.9

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

    6. Applied simplify24.0

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

    8. Applied associate-*l*20.7

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

    10. Applied associate-/r*21.2

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

Runtime

Time bar (total: 4.9m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(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))))