Average Error: 16.0 → 13.2
Time: 1.5m
Precision: 64
Internal Precision: 2944
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \pi \le -2.9273824472590256 \cdot 10^{-99}:\\ \;\;\;\;\ell \cdot \pi - \frac{\tan \left(\ell \cdot \pi\right)}{F \cdot F}\\ \mathbf{if}\;\ell \cdot \pi \le 2.3386421997926985 \cdot 10^{-172}:\\ \;\;\;\;(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + (\left(\frac{2}{15} \cdot {\pi}^{5}\right) \cdot \left({\ell}^{5}\right) + \left(\ell \cdot \pi\right))_*}}\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi - \tan \left((e^{\log_* (1 + \ell \cdot \pi)} - 1)^*\right) \cdot \frac{1}{F \cdot F}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -2.9273824472590256e-99

    1. Initial program 18.7

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

    3. Applied pow118.7

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

    4. Applied pow118.7

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

    5. Applied pow-prod-down18.7

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

    6. Applied simplify18.6

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

    if -2.9273824472590256e-99 < (* PI l) < 2.3386421997926985e-172

    1. Initial program 11.0

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

    3. Applied add-cube-cbrt11.2

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

    4. Applied associate-*r*11.2

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

    5. Applied simplify3.7

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

    7. Applied add-cube-cbrt3.9

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

    9. Applied add-cube-cbrt3.9

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

    10. Taylor expanded around 0 3.9

      \[\leadsto \pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\color{blue}{\frac{1}{3} \cdot \left({\pi}^{3} \cdot {\ell}^{3}\right) + \left(\frac{2}{15} \cdot \left({\pi}^{5} \cdot {\ell}^{5}\right) + \pi \cdot \ell\right)}}}}{F}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}\]

    11. Applied simplify1.0

      \[\leadsto \color{blue}{(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right)\right) \cdot \sqrt[3]{\sqrt[3]{\left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) + (\left(\frac{2}{15} \cdot {\pi}^{5}\right) \cdot \left({\ell}^{5}\right) + \left(\ell \cdot \pi\right))_*}}\right) + \left(\ell \cdot \pi\right))_*}\]

    if 2.3386421997926985e-172 < (* PI l)

    1. Initial program 17.5

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

    3. Applied expm1-log1p-u17.5

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \color{blue}{\left((e^{\log_* (1 + \pi \cdot \ell)} - 1)^*\right)}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify13.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\ell \cdot \pi \le -2.9273824472590256 \cdot 10^{-99}:\\ \;\;\;\;\ell \cdot \pi - \frac{\tan \left(\ell \cdot \pi\right)}{F \cdot F}\\ \mathbf{if}\;\ell \cdot \pi \le 2.3386421997926985 \cdot 10^{-172}:\\ \;\;\;\;(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right) \cdot \left(\left(\ell \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + (\left(\frac{2}{15} \cdot {\pi}^{5}\right) \cdot \left({\ell}^{5}\right) + \left(\ell \cdot \pi\right))_*}}\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi - \tan \left((e^{\log_* (1 + \ell \cdot \pi)} - 1)^*\right) \cdot \frac{1}{F \cdot F}\\ \end{array}}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))