Average Error: 13.5 → 8.3
Time: 54.2s
Precision: 64
Ground Truth: 128
\[\pi \cdot \ell - \frac1{{F}^2} \cdot \tan \left(\pi \cdot \ell\right)\]
⬇
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.0207950218338678 \cdot 10^{-25}:\\ \;\;\;\;\ell \cdot \pi - \frac{\sin \left(-\frac{\pi}{\ell}\right)}{\left(F \cdot F\right) \cdot \cos \left(\frac{\pi}{\ell}\right)}\\ \mathbf{if}\;\ell \le 0.00014906890311918458:\\ \;\;\;\;\pi \cdot \ell - \left(\frac{1}{3} \cdot \frac{{\pi}^{3} \cdot {\ell}^{3}}{{F}^2} + \frac{\pi \cdot \ell}{{F}^2}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi - \frac{-\frac1{F \cdot F}}{\frac{\cos \left(\frac{\pi}{\ell}\right)}{\sin \left(\frac{\pi}{\ell}\right)}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes.

  2. if l < -2.0207950218338678e-25

    1. Initial program 18.1

      \[\pi \cdot \ell - \frac1{{F}^2} \cdot \tan \left(\pi \cdot \ell\right)\]

    2. Applied taylor 8.6

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

    3. Taylor expanded around -inf 8.6

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

    4. Applied simplify 8.6

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

    if -2.0207950218338678e-25 < l < 0.00014906890311918458

    1. Initial program 8.3

      \[\pi \cdot \ell - \frac1{{F}^2} \cdot \tan \left(\pi \cdot \ell\right)\]

    2. Applied taylor 8.0

      \[\leadsto \pi \cdot \ell - \left(\frac{1}{3} \cdot \frac{{\pi}^{3} \cdot {\ell}^{3}}{{F}^2} + \frac{\pi \cdot \ell}{{F}^2}\right)\]

    3. Taylor expanded around 0 8.0

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{1}{3} \cdot \frac{{\pi}^{3} \cdot {\ell}^{3}}{{F}^2} + \frac{\pi \cdot \ell}{{F}^2}\right)}\]

    if 0.00014906890311918458 < l

    1. Initial program 18.0

      \[\pi \cdot \ell - \frac1{{F}^2} \cdot \tan \left(\pi \cdot \ell\right)\]

    2. Applied taylor 8.5

      \[\leadsto \pi \cdot \ell - \frac1{{F}^2} \cdot \frac{\sin \left(-1 \cdot \frac{\pi}{\ell}\right)}{\cos \left(-1 \cdot \frac{\pi}{\ell}\right)}\]

    3. Taylor expanded around -inf 8.5

      \[\leadsto \pi \cdot \ell - \frac1{{F}^2} \cdot \color{blue}{\frac{\sin \left(-1 \cdot \frac{\pi}{\ell}\right)}{\cos \left(-1 \cdot \frac{\pi}{\ell}\right)}}\]

    4. Applied simplify 8.5

      \[\leadsto \color{blue}{\ell \cdot \pi - \frac{-\frac1{F \cdot F}}{\frac{\cos \left(\frac{\pi}{\ell}\right)}{\sin \left(\frac{\pi}{\ell}\right)}}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 54.2s Debug log

Please include this information when filing a bug report:

herbie --seed '#(3567249050 296211338 1013834924 488656093 3062917770 2150790285)'
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ (sqr F)) (tan (* PI l)))))