Average Error: 16.2 → 13.4
Time: 1.8m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \le -5.125297963519208 \cdot 10^{+305}:\\ \;\;\;\;(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left((\left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \left(\left(\left(\ell \cdot \pi\right) \cdot \ell\right) \cdot \left(\pi \cdot \frac{1}{9}\right) + \frac{13}{405} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right)\right) + \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right))_*\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{if}\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \le 8.138020319217507 \cdot 10^{+209}:\\ \;\;\;\;\pi \cdot \ell - {F}^{\left(-2\right)} \cdot \tan \left(\pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{\tan \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\tan \left(\ell \cdot \pi\right)}\right) \cdot \sqrt[3]{\tan \left(\ell \cdot \pi\right)}}}{F}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))) < -5.125297963519208e+305

    1. Initial program 60.3

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

    3. Applied add-cube-cbrt60.3

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

    4. Applied associate-*r*60.3

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

    5. Applied simplify47.8

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

    6. Taylor expanded around 0 51.0

      \[\leadsto \pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\color{blue}{e^{\frac{1}{3} \cdot \left(\log \pi + \log \ell\right)} + \left(\frac{1}{9} \cdot \left(e^{\frac{1}{3} \cdot \left(\log \pi + \log \ell\right)} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) + \frac{13}{405} \cdot \left(e^{\frac{1}{3} \cdot \left(\log \pi + \log \ell\right)} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right)\right)}}{F}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\]

    7. Applied simplify45.3

      \[\leadsto \color{blue}{(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left((\left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \left(\left(\left(\ell \cdot \pi\right) \cdot \ell\right) \cdot \left(\pi \cdot \frac{1}{9}\right) + \frac{13}{405} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right)\right) + \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right))_*\right) + \left(\ell \cdot \pi\right))_*}\]

    if -5.125297963519208e+305 < (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))) < 8.138020319217507e+209

    1. Initial program 4.5

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

    3. Applied pow24.5

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

    4. Applied pow-flip4.5

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

    if 8.138020319217507e+209 < (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l))))

    1. Initial program 39.7

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

    3. Applied add-cube-cbrt39.7

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

    4. Applied associate-*r*39.7

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

    5. Applied simplify32.5

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

    7. Applied add-cube-cbrt32.5

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

    9. Applied add-cube-cbrt32.5

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018207 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))