Average Error: 16.5 → 14.3
Time: 1.6m
Precision: 64
Internal Precision: 3136
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -9.143174873302084 \cdot 10^{+146}:\\ \;\;\;\;\pi \cdot \ell - \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\ell}\right)\right) \cdot {F}^{2}}} \cdot \left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\ell}\right)\right) \cdot {F}^{2}}} \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\ell}\right)\right) \cdot {F}^{2}}}\right)\\ \mathbf{elif}\;\ell \le 3.864430687282477 \cdot 10^{+150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot (\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*} \cdot \sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if l < -9.143174873302084e+146

    1. Initial program 21.8

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

    2. Initial simplification21.8

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

    3. Taylor expanded around inf 21.8

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

    5. Applied add-cube-cbrt21.9

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

    7. Applied cbrt-prod21.9

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

    9. Applied add-cube-cbrt21.9

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

    if -9.143174873302084e+146 < l < 3.864430687282477e+150

    1. Initial program 14.9

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

    2. Initial simplification14.9

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

    3. Taylor expanded around inf 14.5

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

    5. Applied add-cube-cbrt14.5

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

    7. Applied cbrt-prod14.5

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

    8. Taylor expanded around 0 11.9

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

    9. Simplified11.9

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

    if 3.864430687282477e+150 < l

    1. Initial program 19.8

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

    2. Initial simplification19.8

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

    4. Applied add-sqr-sqrt19.4

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

  4. Final simplification14.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -9.143174873302084 \cdot 10^{+146}:\\ \;\;\;\;\pi \cdot \ell - \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\ell}\right)\right) \cdot {F}^{2}}} \cdot \left(\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\ell}\right)\right) \cdot {F}^{2}}} \cdot \sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\ell}\right)\right) \cdot {F}^{2}}}\right)\\ \mathbf{elif}\;\ell \le 3.864430687282477 \cdot 10^{+150}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot (\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*} \cdot \sqrt{(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))