Average Error: 16.9 → 1.4
Time: 2.4m
Precision: 64
Internal Precision: 3456
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -7.147344906003675 \cdot 10^{+39}:\\ \;\;\;\;\ell \cdot \pi - 0\\ \mathbf{if}\;\pi \cdot \ell \le 2.4200320987840002 \cdot 10^{-12}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\tan \left(\ell \cdot \pi\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi - 0\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < -7.147344906003675e+39 or 2.4200320987840002e-12 < (* PI l)

    1. Initial program 23.1

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

    2. Applied simplify23.1

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

    4. Applied associate-/r*23.1

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

    6. Applied div-inv23.1

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

    8. Applied add-cube-cbrt23.1

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

    9. Applied associate-*r*23.1

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

    10. Taylor expanded around 0 1.0

      \[\leadsto \ell \cdot \pi - \color{blue}{0}\]

    if -7.147344906003675e+39 < (* PI l) < 2.4200320987840002e-12

    1. Initial program 10.5

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

    2. Applied simplify10.0

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

    4. Applied associate-/r*1.8

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))