Average Error: 16.7 → 10.8
Time: 1.3m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.2462332527342774 \cdot 10^{+139}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;\ell \le -1.2421861132526175 \cdot 10^{+73}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(F \cdot F\right) \cdot \frac{(\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}{\sin \left(\pi \cdot \ell\right)}}\\ \mathbf{elif}\;\ell \le 8213514867.838334:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \mathbf{elif}\;\ell \le 3.2316646001877984 \cdot 10^{+147}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(F \cdot F\right) \cdot \frac{(\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}{\sin \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*} \cdot \sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 4 regimes

  2. if l < -1.2462332527342774e+139

    1. Initial program 21.7

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

    2. Initial simplification21.7

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

    4. Applied associate-/r*21.7

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

    if -1.2462332527342774e+139 < l < -1.2421861132526175e+73 or 8213514867.838334 < l < 3.2316646001877984e+147

    1. Initial program 27.6

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

    2. Initial simplification27.6

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

    3. Taylor expanded around -inf 27.6

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

    5. Applied *-un-lft-identity27.6

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

    6. Applied associate-/l*27.6

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

    8. Applied *-un-lft-identity27.6

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

    9. Applied times-frac27.6

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

    10. Simplified27.6

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

    11. Taylor expanded around 0 17.0

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

    12. Simplified17.0

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

    if -1.2421861132526175e+73 < l < 8213514867.838334

    1. Initial program 10.6

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

    2. Initial simplification10.6

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

    3. Taylor expanded around -inf 10.1

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

    5. Applied *-un-lft-identity10.1

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

    6. Applied associate-/l*10.1

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

    8. Applied *-un-lft-identity10.1

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

    9. Applied times-frac10.1

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

    10. Simplified10.1

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

    12. Applied associate-*l*3.3

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

    if 3.2316646001877984e+147 < l

    1. Initial program 20.5

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

    2. Initial simplification20.5

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

    4. Applied add-sqr-sqrt20.1

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.2462332527342774 \cdot 10^{+139}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;\ell \le -1.2421861132526175 \cdot 10^{+73}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(F \cdot F\right) \cdot \frac{(\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}{\sin \left(\pi \cdot \ell\right)}}\\ \mathbf{elif}\;\ell \le 8213514867.838334:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \mathbf{elif}\;\ell \le 3.2316646001877984 \cdot 10^{+147}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(F \cdot F\right) \cdot \frac{(\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}{\sin \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*} \cdot \sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))