Average Error: 16.1 → 11.4
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\pi}^{4} \cdot \frac{1}{24}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right)} \le -5.105385596280217 \cdot 10^{+150}:\\ \;\;\;\;(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left((\left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \left(\left(\left(\ell \cdot \pi\right) \cdot \ell\right) \cdot \left(\pi \cdot \frac{1}{9}\right) + \frac{13}{405} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right)\right) + \left(\left({\ell}^{\frac{1}{3}} \cdot {\pi}^{\frac{1}{3}}\right) \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right))_*\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{if}\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\pi}^{4} \cdot \frac{1}{24}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right)} \le 2.3204759534125063 \cdot 10^{+245}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\pi}^{4} \cdot \frac{1}{24}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (- (* l PI) (/ (/ (sin (* l PI)) (* F F)) (- (fma (* (pow PI 4) 1/24) (pow l 4) 1) (* 1/2 (* (* l PI) (* l PI)))))) < -5.105385596280217e+150

    1. Initial program 47.4

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

    3. Applied add-cube-cbrt47.5

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

    4. Applied associate-*r*47.5

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

    5. Applied simplify28.0

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

    6. Taylor expanded around 0 34.0

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

    7. Applied simplify23.8

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

    if -5.105385596280217e+150 < (- (* l PI) (/ (/ (sin (* l PI)) (* F F)) (- (fma (* (pow PI 4) 1/24) (pow l 4) 1) (* 1/2 (* (* l PI) (* l PI)))))) < 2.3204759534125063e+245

    1. Initial program 5.5

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

    3. Applied tan-quot5.5

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

    4. Applied frac-times5.3

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

    5. Applied simplify5.3

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

    6. Taylor expanded around 0 1.9

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

    7. Applied simplify1.9

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

    if 2.3204759534125063e+245 < (- (* l PI) (/ (/ (sin (* l PI)) (* F F)) (- (fma (* (pow PI 4) 1/24) (pow l 4) 1) (* 1/2 (* (* l PI) (* l PI))))))

    1. Initial program 28.8

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

    3. Applied add-cube-cbrt28.8

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

    4. Applied associate-*r*28.8

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

    5. Applied simplify25.7

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018206 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))