Average Error: 16.2 → 13.9
Time: 3.6m
Precision: 64
Internal Precision: 4416
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.402581417879336 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)}{\left(F \cdot F\right) \cdot \cos \left(\pi \cdot \ell\right)}\\ \mathbf{elif}\;\ell \le 5.835043802231026 \cdot 10^{+152}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right) \cdot \left(F \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \tan \left(\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell}\right) \cdot \frac{1}{F \cdot F}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if l < -3.402581417879336e+153

    1. Initial program 20.2

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied tan-quot20.2

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

    4. Applied frac-times20.2

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

    5. Simplified20.2

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

    7. Applied add-cube-cbrt20.3

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

    if -3.402581417879336e+153 < l < 5.835043802231026e+152

    1. Initial program 15.0

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied tan-quot15.0

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

    4. Applied frac-times14.6

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

    5. Simplified14.6

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

    6. Taylor expanded around 0 11.7

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

    7. Simplified11.7

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

    if 5.835043802231026e+152 < l

    1. Initial program 18.8

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt18.8

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.402581417879336 \cdot 10^{+153}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)}{\left(F \cdot F\right) \cdot \cos \left(\pi \cdot \ell\right)}\\ \mathbf{elif}\;\ell \le 5.835043802231026 \cdot 10^{+152}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\left((\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)\right) \cdot \left(F \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \tan \left(\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell}\right) \cdot \frac{1}{F \cdot F}\\ \end{array}\]

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))