Average Error: 16.2 → 13.4
Time: 2.8m
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}\;(\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))_* \le -1.2294582952786017 \cdot 10^{+287}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \mathbf{if}\;(\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))_* \le -1.9025460193008453 \cdot 10^{+230}:\\ \;\;\;\;(\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{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]{\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 (fma (- (/ (cbrt (tan (* l PI))) F)) (fma (/ (cbrt (tan (* l PI))) F) (* (* (pow l 1/3) (pow PI 1/3)) (+ (* (* (* l PI) l) (* PI 1/9)) (* 13/405 (* (pow PI 4) (pow l 4))))) (* (* (pow l 1/3) (pow PI 1/3)) (/ (cbrt (tan (* l PI))) F))) (* l PI)) < -1.2294582952786017e+287

    1. Initial program 26.8

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

    3. Applied add-cube-cbrt26.7

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

    if -1.2294582952786017e+287 < (fma (- (/ (cbrt (tan (* l PI))) F)) (fma (/ (cbrt (tan (* l PI))) F) (* (* (pow l 1/3) (pow PI 1/3)) (+ (* (* (* l PI) l) (* PI 1/9)) (* 13/405 (* (pow PI 4) (pow l 4))))) (* (* (pow l 1/3) (pow PI 1/3)) (/ (cbrt (tan (* l PI))) F))) (* l PI)) < -1.9025460193008453e+230

    1. Initial program 49.5

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

    3. Applied add-cube-cbrt49.6

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

    4. Applied associate-*r*49.7

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

    5. Applied simplify30.9

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

    6. Taylor expanded around 0 35.4

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

    7. Applied simplify14.4

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

    if -1.9025460193008453e+230 < (fma (- (/ (cbrt (tan (* l PI))) F)) (fma (/ (cbrt (tan (* l PI))) F) (* (* (pow l 1/3) (pow PI 1/3)) (+ (* (* (* l PI) l) (* PI 1/9)) (* 13/405 (* (pow PI 4) (pow l 4))))) (* (* (pow l 1/3) (pow PI 1/3)) (/ (cbrt (tan (* l PI))) F))) (* l PI))

    1. Initial program 12.2

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

    3. Applied add-cube-cbrt12.4

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

    4. Applied associate-*r*12.4

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

    5. Applied simplify9.1

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

    7. Applied add-cube-cbrt9.1

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018198 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))