Average Error: 16.1 → 13.2
Time: 2.6m
Precision: 64
Internal Precision: 3904
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell} - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) = -\infty:\\ \;\;\;\;(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left((\left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \left(\left(\left(\ell \cdot \pi\right) \cdot \ell\right) \cdot \left(\pi \cdot \frac{1}{9}\right) + \frac{13}{405} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right)\right) + \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right))_*\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{if}\;\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell} - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \le 1.3366843831581759 \cdot 10^{+159}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left((e^{\log_* (1 + \pi \cdot \ell)} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (- (* (sqrt (* PI l)) (sqrt (* PI l))) (* (/ 1 (* F F)) (tan (* PI l)))) < -inf.0

    1. Initial program 62.0

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

    3. Applied add-cube-cbrt62.0

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

    4. Applied associate-*r*62.0

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

    5. Applied simplify44.5

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

    6. Taylor expanded around 0 45.7

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

    7. Applied simplify37.2

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

    if -inf.0 < (- (* (sqrt (* PI l)) (sqrt (* PI l))) (* (/ 1 (* F F)) (tan (* PI l)))) < 1.3366843831581759e+159

    1. Initial program 4.7

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

    3. Applied expm1-log1p-u4.7

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

    if 1.3366843831581759e+159 < (- (* (sqrt (* PI l)) (sqrt (* PI l))) (* (/ 1 (* F F)) (tan (* PI l))))

    1. Initial program 16.6

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

    3. Applied add-cube-cbrt16.7

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

    4. Applied associate-*r*16.7

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

    5. Applied simplify14.6

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))