Average Error: 16.7 → 8.5
Time: 1.1m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
\[\begin{array}{l} \mathbf{if}\;\ell \le -4.2064016145675135 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}{\frac{F}{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}}\\ \mathbf{elif}\;\ell \le 4.27640551226906 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + \frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right) - \left({\ell}^{2} \cdot {\pi}^{2}\right) \cdot \frac{1}{2}\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)}{F \cdot F}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if l < -4.2064016145675135e+153

    1. Initial program 21.4

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Initial simplification21.4

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]
    3. Using strategy rm
    4. Applied associate-/r*21.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
    5. Using strategy rm
    6. Applied add-cube-cbrt21.4

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}}{F}\]
    7. Applied associate-/l*21.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}{\frac{F}{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}}}\]

    if -4.2064016145675135e+153 < l < 4.27640551226906e+153

    1. Initial program 15.1

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Initial simplification14.7

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]
    3. Using strategy rm
    4. Applied associate-/r*9.7

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
    5. Using strategy rm
    6. Applied tan-quot9.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F}}{F}\]
    7. Applied associate-/l/9.7

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}}{F}\]
    8. Taylor expanded around 0 3.9

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}}{F}\]

    if 4.27640551226906e+153 < l

    1. Initial program 20.2

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Initial simplification20.2

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]
    3. Using strategy rm
    4. Applied add-cube-cbrt20.2

      \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)}}{F \cdot F}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.2064016145675135 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}{\frac{F}{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}}\\ \mathbf{elif}\;\ell \le 4.27640551226906 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + \frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right) - \left({\ell}^{2} \cdot {\pi}^{2}\right) \cdot \frac{1}{2}\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)}{F \cdot F}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.78.58.04.889%
herbie shell --seed 2018274 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))