Average Error: 16.0 → 10.4
Time: 2.0m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\frac{-1}{2} \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) + 1)_*\right))_*} \le -2.7126055391685166 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\cos \left(\pi \cdot \ell\right) \cdot F}{\frac{\sin \left(\pi \cdot \ell\right)}{F}}}\\ \mathbf{if}\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\frac{-1}{2} \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) + 1)_*\right))_*} \le 2.174789653348066 \cdot 10^{+100}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\frac{-1}{2} \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) + 1)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{\cos \left(\pi \cdot \ell\right) \cdot F}\\ \end{array}\]

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 l 4) (* (pow PI 4) 1/24) (fma -1/2 (* (* l PI) (* l PI)) 1)))) < -2.7126055391685166e+162

    1. Initial program 52.3

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

    2. Applied simplify52.3

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

    4. Applied associate-/r*52.3

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

    6. Applied add-cube-cbrt52.4

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

    7. Taylor expanded around inf 52.4

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(e^{\log \pi - \log \left(\frac{1}{\ell}\right)}\right)}{{F}^{2} \cdot \cos \left(e^{\log \pi - \log \left(\frac{1}{\ell}\right)}\right)}}\]

    8. Applied simplify16.4

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

    10. Applied clear-num16.4

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

    if -2.7126055391685166e+162 < (- (* l PI) (/ (/ (sin (* l PI)) (* F F)) (fma (pow l 4) (* (pow PI 4) 1/24) (fma -1/2 (* (* l PI) (* l PI)) 1)))) < 2.174789653348066e+100

    1. Initial program 4.4

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

    2. Applied simplify4.4

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

    4. Applied associate-/r*4.4

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

    6. Applied add-cube-cbrt4.6

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

    7. Taylor expanded around inf 35.5

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(e^{\log \pi - \log \left(\frac{1}{\ell}\right)}\right)}{{F}^{2} \cdot \cos \left(e^{\log \pi - \log \left(\frac{1}{\ell}\right)}\right)}}\]

    8. Applied simplify4.2

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

    9. Taylor expanded around 0 1.5

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

    10. Applied simplify1.6

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

    if 2.174789653348066e+100 < (- (* l PI) (/ (/ (sin (* l PI)) (* F F)) (fma (pow l 4) (* (pow PI 4) 1/24) (fma -1/2 (* (* l PI) (* l PI)) 1))))

    1. Initial program 27.0

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

    2. Applied simplify27.0

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

    4. Applied associate-/r*27.0

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

    6. Applied add-cube-cbrt27.0

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

    7. Taylor expanded around inf 44.4

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(e^{\log \pi - \log \left(\frac{1}{\ell}\right)}\right)}{{F}^{2} \cdot \cos \left(e^{\log \pi - \log \left(\frac{1}{\ell}\right)}\right)}}\]

    8. Applied simplify21.8

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

    10. Applied div-inv21.8

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))