Average Error: 8.7 → 7.3
Time: 39.4s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le 2.893215181645407 \cdot 10^{-231}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;F \le 1.2908308852125874 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (2 \cdot \left(\log F\right) + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if F < 2.893215181645407e-231

    1. Initial program 10.3

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

    2. Simplified10.3

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

    if 2.893215181645407e-231 < F < 1.2908308852125874e-162

    1. Initial program 60.1

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

    2. Simplified60.1

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

    3. Taylor expanded around inf 60.1

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

    5. Applied add-exp-log60.1

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

    6. Applied add-exp-log60.1

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

    7. Applied prod-exp60.1

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

    8. Applied add-exp-log60.1

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

    9. Applied div-exp60.1

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

    10. Simplified34.4

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

    if 1.2908308852125874e-162 < F

    1. Initial program 1.8

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

    2. Simplified1.8

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

    3. Taylor expanded around inf 1.1

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le 2.893215181645407 \cdot 10^{-231}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;F \le 1.2908308852125874 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (2 \cdot \left(\log F\right) + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))