Average Error: 47.3 → 23.8
Time: 3.2m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot \left({t}^{3} \cdot \sin k\right)}{\left(\left(t \cdot \ell\right) \cdot \left(t \cdot \ell\right)\right) \cdot \cos k} \le -2.060940217645377 \cdot 10^{-233}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot \sin k\right) \cdot \left({t}^{3} \cdot \sin k\right)}{\left(\left(t \cdot \ell\right) \cdot \left(t \cdot \ell\right)\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{\sqrt[3]{\frac{k}{t}}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\frac{k}{t}}}}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\\ \end{array}\]

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 (/ (* (* (* k k) (sin k)) (* (pow t 3) (sin k))) (* (* (* t l) (* t l)) (cos k))) < -2.060940217645377e-233

    1. Initial program 30.8

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

    3. Applied add-exp-log31.2

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

    4. Applied simplify18.2

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot e^{\color{blue}{\log \left(\frac{k}{t} \cdot \frac{k}{t}\right)}}}\]
    5. Using strategy rm

    6. Applied frac-times29.6

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

    7. Applied log-div29.7

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot e^{\color{blue}{\log \left(k \cdot k\right) - \log \left(t \cdot t\right)}}}\]

    8. Applied exp-diff29.6

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

    9. Applied tan-quot29.6

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

    10. Applied associate-*l/29.4

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

    11. Applied frac-times29.1

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

    12. Applied frac-times27.1

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

    13. Applied simplify27.0

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

    14. Applied simplify8.6

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

    if -2.060940217645377e-233 < (/ (* (* (* k k) (sin k)) (* (pow t 3) (sin k))) (* (* (* t l) (* t l)) (cos k)))

    1. Initial program 51.3

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

    3. Applied add-exp-log51.5

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

    4. Applied simplify45.2

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot e^{\color{blue}{\log \left(\frac{k}{t} \cdot \frac{k}{t}\right)}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt45.2

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

    7. Applied associate-*r*45.2

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

    8. Applied simplify30.0

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

    9. Taylor expanded around 0 54.0

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

    10. Applied simplify29.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{\frac{t}{\frac{\ell}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}\]
    11. Using strategy rm

    12. Applied cbrt-prod29.7

      \[\leadsto \frac{\frac{\frac{\frac{2}{\frac{t}{\frac{\ell}{t}}}}{\color{blue}{\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}}}}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\]

    13. Applied associate-/r/29.6

      \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{2}{t} \cdot \frac{\ell}{t}}}{\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}}}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t}}}\]

    14. Applied times-frac27.5

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2018206 
(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))))