Average Error: 16.9 → 11.0
Time: 2.6m
Precision: 64
Internal Precision: 3456
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.5669810766673902 \cdot 10^{+148}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\frac{\sin \left(\ell \cdot \pi\right)}{F}}}}{\cos \left(\pi \cdot \ell\right)}\\ \mathbf{if}\;\pi \cdot \ell \le -3.0457547662165865 \cdot 10^{-79}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\pi}^{4} \cdot \frac{1}{24}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) \cdot \frac{1}{2}}\\ \mathbf{if}\;\pi \cdot \ell \le 9.951617466633533 \cdot 10^{-141}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\frac{\sin \left(\ell \cdot \pi\right)}{F}}}}{\cos \left(\pi \cdot \ell\right)}\\ \mathbf{if}\;\pi \cdot \ell \le 2.082373362494963 \cdot 10^{+147}:\\ \;\;\;\;\ell \cdot \pi - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{F \cdot F}}{(\left({\pi}^{4} \cdot \frac{1}{24}\right) \cdot \left({\ell}^{4}\right) + 1)_* - \left(\left(\ell \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\frac{\sin \left(\ell \cdot \pi\right)}{F}}}}{\cos \left(\pi \cdot \ell\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -1.5669810766673902e+148 or 2.082373362494963e+147 < (* PI l)

    1. Initial program 21.1

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

    3. Applied tan-quot21.1

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

    4. Applied associate-*r/21.1

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

    5. Applied simplify21.1

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

    7. Applied associate-/r*21.1

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

    9. Applied clear-num21.1

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

    if -1.5669810766673902e+148 < (* PI l) < -3.0457547662165865e-79 or 9.951617466633533e-141 < (* PI l) < 2.082373362494963e+147

    1. Initial program 17.9

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

    3. Applied tan-quot17.9

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

    4. Applied associate-*r/17.9

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

    5. Applied simplify17.7

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

    6. Taylor expanded around 0 12.5

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

    7. Applied simplify12.5

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

    if -3.0457547662165865e-79 < (* PI l) < 9.951617466633533e-141

    1. Initial program 12.0

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

    3. Applied tan-quot12.0

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

    4. Applied associate-*r/12.0

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

    5. Applied simplify11.5

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

    7. Applied associate-/r*0.3

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

    9. Applied clear-num0.4

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

Runtime

Time bar (total: 2.6m)Debug logProfile

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