Average Error: 16.5 → 7.1
Time: 2.1m
Precision: 64
Internal Precision: 3712
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.7983481742941125 \cdot 10^{+148}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\pi \cdot \ell} - \left(\pi \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot F\right)}\\ \mathbf{if}\;\pi \cdot \ell \le 4.6251105439013473 \cdot 10^{+117}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F}}{(\frac{1}{24} \cdot \left({\ell}^{4} \cdot {\pi}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\pi \cdot \ell} - \left(\pi \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot F\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < -1.7983481742941125e+148 or 4.6251105439013473e+117 < (* PI l)

    1. Initial program 20.9

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

    2. Applied simplify20.9

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

    4. Applied associate-/r*20.9

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

    6. Applied clear-num20.9

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

    7. Taylor expanded around 0 29.4

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

    8. Applied simplify8.8

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

    if -1.7983481742941125e+148 < (* PI l) < 4.6251105439013473e+117

    1. Initial program 14.5

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

    2. Applied simplify14.1

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

    4. Applied associate-/r*8.7

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

    6. Applied tan-quot8.7

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

    7. Taylor expanded around 0 4.1

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

    8. Applied simplify6.3

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F}}{(\frac{1}{24} \cdot \left({\ell}^{4} \cdot {\pi}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))