Average Error: 16.2 → 13.2
Time: 1.2m
Precision: 64
Internal Precision: 2624
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right)} \le -3.2080101628163186 \cdot 10^{+180}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (\left(\log F\right) \cdot 2 + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{elif}\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right)} \le 4.158770752654795 \cdot 10^{+144}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\log_* (1 + (e^{\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}} - 1)^*)}{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot {F}^{2}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (- (* PI l) (/ (sin (* PI l)) (* (pow F 2) (- (fma 1/24 (* (pow PI 4) (pow l 4)) 1) (* (* (* PI l) (* PI l)) 1/2))))) < -3.2080101628163186e+180

    1. Initial program 53.9

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

    2. Initial simplification53.9

      \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]

    3. Taylor expanded around inf 52.9

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
    4. Using strategy rm

    5. Applied add-exp-log52.9

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{e^{\log \left(\cos \left(\pi \cdot \ell\right)\right)}}}\]

    6. Applied pow-to-exp59.0

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{e^{\log F \cdot 2}} \cdot e^{\log \left(\cos \left(\pi \cdot \ell\right)\right)}}\]

    7. Applied prod-exp59.0

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{e^{\log F \cdot 2 + \log \left(\cos \left(\pi \cdot \ell\right)\right)}}}\]

    8. Applied add-exp-log59.0

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{e^{\log \left(\sin \left(\pi \cdot \ell\right)\right)}}}{e^{\log F \cdot 2 + \log \left(\cos \left(\pi \cdot \ell\right)\right)}}\]

    9. Applied div-exp43.6

      \[\leadsto \pi \cdot \ell - \color{blue}{e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - \left(\log F \cdot 2 + \log \left(\cos \left(\pi \cdot \ell\right)\right)\right)}}\]

    10. Simplified43.6

      \[\leadsto \pi \cdot \ell - e^{\color{blue}{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (\left(\log F\right) \cdot 2 + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}}\]

    if -3.2080101628163186e+180 < (- (* PI l) (/ (sin (* PI l)) (* (pow F 2) (- (fma 1/24 (* (pow PI 4) (pow l 4)) 1) (* (* (* PI l) (* PI l)) 1/2))))) < 4.158770752654795e+144

    1. Initial program 5.6

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

    2. Initial simplification5.6

      \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]

    3. Taylor expanded around inf 5.3

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

    4. Taylor expanded around 0 1.6

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

    5. Simplified1.6

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

    if 4.158770752654795e+144 < (- (* PI l) (/ (sin (* PI l)) (* (pow F 2) (- (fma 1/24 (* (pow PI 4) (pow l 4)) 1) (* (* (* PI l) (* PI l)) 1/2)))))

    1. Initial program 28.4

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

    2. Initial simplification28.4

      \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]

    3. Taylor expanded around inf 28.2

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt28.2

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right)}}\]
    6. Using strategy rm

    7. Applied log1p-expm1-u28.2

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\log_* (1 + (e^{\sin \left(\pi \cdot \ell\right)} - 1)^*)}}{{F}^{2} \cdot \left(\left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt28.2

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right)} \le -3.2080101628163186 \cdot 10^{+180}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (\left(\log F\right) \cdot 2 + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{elif}\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right)} \le 4.158770752654795 \cdot 10^{+144}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\log_* (1 + (e^{\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}} - 1)^*)}{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot {F}^{2}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))