Average Error: 16.6 → 11.4
Time: 1.3m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -4.853950121481714 \cdot 10^{-70}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \mathbf{elif}\;F \le -3.037055494784001 \cdot 10^{-156}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(\left(\pi \cdot \ell\right) \cdot {F}^{2}\right)}\\ \mathbf{elif}\;F \le 7.877727659061987 \cdot 10^{-159}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \mathbf{elif}\;F \le 1.5430490039204164 \cdot 10^{-74}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}{\sin \left(\pi \cdot \ell\right)} \cdot \left(F \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 4 regimes

  2. if F < -4.853950121481714e-70 or -3.037055494784001e-156 < F < 7.877727659061987e-159

    1. Initial program 23.7

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

    2. Taylor expanded around -inf 23.4

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

    4. Applied *-un-lft-identity23.4

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

    5. Applied associate-/l*23.4

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

    6. Taylor expanded around -inf 62.4

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{\cos \left(\pi \cdot \ell\right) \cdot e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{F}\right)\right)}}{\sin \left(\pi \cdot \ell\right)}}}\]

    7. Simplified23.4

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(F \cdot F\right) \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
    8. Using strategy rm

    9. Applied associate-*l*16.3

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{F \cdot \left(F \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right)}}\]

    if -4.853950121481714e-70 < F < -3.037055494784001e-156

    1. Initial program 23.0

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

    2. Taylor expanded around -inf 22.2

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

    4. Applied *-un-lft-identity22.2

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

    5. Applied associate-/l*22.2

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

    6. Taylor expanded around 0 10.6

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left({F}^{2} \cdot \left(\pi \cdot \ell\right)\right)}}\]

    if 7.877727659061987e-159 < F < 1.5430490039204164e-74

    1. Initial program 24.7

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

    2. Taylor expanded around -inf 23.6

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

    4. Applied *-un-lft-identity23.6

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

    5. Applied associate-/l*23.6

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

    6. Taylor expanded around -inf 62.4

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{\cos \left(\pi \cdot \ell\right) \cdot e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{F}\right)\right)}}{\sin \left(\pi \cdot \ell\right)}}}\]

    7. Simplified23.7

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(F \cdot F\right) \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]

    8. Taylor expanded around 0 19.0

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

    9. Simplified19.0

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

    if 1.5430490039204164e-74 < F

    1. Initial program 1.9

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

    2. Taylor expanded around -inf 1.9

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

    4. Applied *-un-lft-identity1.9

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

    5. Applied associate-/l*1.9

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

    6. Taylor expanded around -inf 62.5

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{\cos \left(\pi \cdot \ell\right) \cdot e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{F}\right)\right)}}{\sin \left(\pi \cdot \ell\right)}}}\]

    7. Simplified1.9

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(F \cdot F\right) \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]

    8. Taylor expanded around -inf 1.9

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -4.853950121481714 \cdot 10^{-70}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \mathbf{elif}\;F \le -3.037055494784001 \cdot 10^{-156}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(\left(\pi \cdot \ell\right) \cdot {F}^{2}\right)}\\ \mathbf{elif}\;F \le 7.877727659061987 \cdot 10^{-159}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \mathbf{elif}\;F \le 1.5430490039204164 \cdot 10^{-74}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}{\sin \left(\pi \cdot \ell\right)} \cdot \left(F \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))