Average Error: 16.7 → 11.3
Time: 2.2m
Precision: 64
Internal Precision: 3136
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(-\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F}\right) \cdot \left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{(\left(\frac{1}{24} \cdot {\pi}^{4}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{2}}}\right) + \left(\pi \cdot \ell\right))_* \le -4.809834010199066 \cdot 10^{+130}:\\ \;\;\;\;(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{if}\;(\left(-\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F}\right) \cdot \left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{(\left(\frac{1}{24} \cdot {\pi}^{4}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{2}}}\right) + \left(\pi \cdot \ell\right))_* \le 2.364884909448289 \cdot 10^{+143}:\\ \;\;\;\;(\left(-\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F}\right) \cdot \left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{(\left(\frac{1}{24} \cdot {\pi}^{4}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{2}}}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}\right) + \left(\ell \cdot \pi\right))_*\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (fma (- (* (/ (cbrt (tan (* PI l))) F) (/ (cbrt (tan (* PI l))) F))) (cbrt (/ (sin (* PI l)) (- (fma (* 1/24 (pow PI 4)) (pow l 4) 1) (* (* (* PI l) (* PI l)) 1/2)))) (* PI l)) < -4.809834010199066e+130 or 2.364884909448289e+143 < (fma (- (* (/ (cbrt (tan (* PI l))) F) (/ (cbrt (tan (* PI l))) F))) (cbrt (/ (sin (* PI l)) (- (fma (* 1/24 (pow PI 4)) (pow l 4) 1) (* (* (* PI l) (* PI l)) 1/2)))) (* PI l))

    1. Initial program 30.3

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt30.3

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)}\]

    4. Applied associate-*r*30.3

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F \cdot F} \cdot \left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}\]

    5. Applied simplify25.7

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

    7. Applied add-cube-cbrt25.7

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

    8. Taylor expanded around inf 25.7

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

    9. Applied simplify22.8

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

    if -4.809834010199066e+130 < (fma (- (* (/ (cbrt (tan (* PI l))) F) (/ (cbrt (tan (* PI l))) F))) (cbrt (/ (sin (* PI l)) (- (fma (* 1/24 (pow PI 4)) (pow l 4) 1) (* (* (* PI l) (* PI l)) 1/2)))) (* PI l)) < 2.364884909448289e+143

    1. Initial program 5.6

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt5.8

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \color{blue}{\left(\left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)}\]

    4. Applied associate-*r*5.8

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{F \cdot F} \cdot \left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}\]

    5. Applied simplify4.4

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

    7. Applied add-cube-cbrt4.4

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

    8. Taylor expanded around inf 4.4

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

    9. Applied simplify4.4

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

    10. Taylor expanded around 0 1.9

      \[\leadsto (\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\color{blue}{\left(1 + \frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}}\right) + \left(\ell \cdot \pi\right))_*\]

    11. Applied simplify1.9

      \[\leadsto \color{blue}{(\left(-\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F}\right) \cdot \left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{(\left(\frac{1}{24} \cdot {\pi}^{4}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{2}}}\right) + \left(\pi \cdot \ell\right))_*}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))