Average Error: 15.6 → 5.7
Time: 2.0m
Precision: 64
Internal Precision: 3968
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.2391563621069606 \cdot 10^{+121}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}} \cdot \left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right)}{\frac{F}{\sqrt[3]{\sqrt[3]{\sin \left(\frac{\pi}{\ell}\right) \cdot \frac{F}{\cos \left(\frac{\pi}{\ell}\right)}}}}}\\ \mathbf{if}\;\pi \cdot \ell \le 2.8029951490048387 \cdot 10^{+50}:\\ \;\;\;\;\ell \cdot \pi - \frac{1}{F} \cdot \frac{\tan \left(\ell \cdot \pi\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}} \cdot \left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right)}{\frac{F}{\sqrt[3]{\sqrt[3]{\sin \left(\frac{\pi}{\ell}\right) \cdot \frac{F}{\cos \left(\frac{\pi}{\ell}\right)}}}}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < -1.2391563621069606e+121 or 2.8029951490048387e+50 < (* PI l)

    1. Initial program 21.7

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

    2. Applied simplify21.7

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

    4. Applied *-un-lft-identity21.7

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

    5. Applied times-frac21.7

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

    7. Applied add-cube-cbrt21.7

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

    9. Applied add-cube-cbrt21.7

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

    10. Applied cbrt-prod21.7

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

    11. Taylor expanded around inf 50.0

      \[\leadsto \ell \cdot \pi - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{\tan \left(\ell \cdot \pi\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\ell \cdot \pi\right)}{F}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\tan \left(\ell \cdot \pi\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\ell \cdot \pi\right)}{F}}} \cdot \sqrt[3]{\color{blue}{e^{\frac{1}{3} \cdot \left(\log \left(\frac{\sin \left(\frac{\pi}{\ell}\right)}{\cos \left(\frac{\pi}{\ell}\right)}\right) + \log F\right)}}}\right)\right)\]

    12. Applied simplify5.7

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

    if -1.2391563621069606e+121 < (* PI l) < 2.8029951490048387e+50

    1. Initial program 11.8

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

    2. Applied simplify11.3

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

    4. Applied *-un-lft-identity11.3

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

    5. Applied times-frac5.8

      \[\leadsto \ell \cdot \pi - \color{blue}{\frac{1}{F} \cdot \frac{\tan \left(\ell \cdot \pi\right)}{F}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))