Average Error: 16.4 → 8.3
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -5.838429968357496 \cdot 10^{+157}:\\ \;\;\;\;\ell \cdot \pi - \frac{\tan \left(\left(\sqrt[3]{\ell \cdot \pi} \cdot \sqrt[3]{\ell \cdot \pi}\right) \cdot \sqrt[3]{\ell \cdot \pi}\right)}{F \cdot F}\\ \mathbf{if}\;\pi \cdot \ell \le 1.0378322572942908 \cdot 10^{+129}:\\ \;\;\;\;\ell \cdot \pi - \frac{1}{F} \cdot \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\left(1 + \frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right)\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi - \frac{\tan \left(\left(\sqrt[3]{\ell \cdot \pi} \cdot \sqrt[3]{\ell \cdot \pi}\right) \cdot \sqrt[3]{\ell \cdot \pi}\right)}{F \cdot F}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -5.838429968357496e+157

    1. Initial program 19.7

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

    2. Applied simplify19.7

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

    4. Applied add-cube-cbrt19.7

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

    if -5.838429968357496e+157 < (* PI l) < 1.0378322572942908e+129

    1. Initial program 15.2

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

    2. Applied simplify14.7

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

    4. Applied *-un-lft-identity14.7

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

    5. Applied times-frac9.3

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

    7. Applied tan-quot9.3

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

    8. Taylor expanded around 0 4.0

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

    if 1.0378322572942908e+129 < (* PI l)

    1. Initial program 18.8

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

    2. Applied simplify18.8

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

    4. Applied add-cube-cbrt18.7

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))