VandenBroeck and Keller, Equation (6)

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e22 or 1.00000000000000005e-4 < (*.f64 (PI.f64) l)

   1. Initial program 62.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/62.8%

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

      2. *-lft-identity62.8%

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

   3. Simplified62.8%

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

   4. Taylor expanded in l around 0 46.7%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow246.7%

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

   6. Simplified46.7%

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

   7. Taylor expanded in F around inf 99.7%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -2e22 < (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 89.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/90.8%

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

      2. *-un-lft-identity90.8%

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

      3. associate-/r*99.3%

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

   3. Applied egg-rr99.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+22} \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -4e9 or 1.00000000000000005e-4 < (*.f64 (PI.f64) l)

   1. Initial program 63.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/63.7%

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

      2. *-lft-identity63.7%

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

   3. Simplified63.7%

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

   4. Taylor expanded in l around 0 47.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow247.2%

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

   6. Simplified47.2%

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

   7. Taylor expanded in F around inf 99.5%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -4e9 < (*.f64 (PI.f64) l) < 1.00000000000000005e-4

   1. Initial program 89.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/90.6%

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

      2. *-un-lft-identity90.6%

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

      3. associate-/r*99.3%

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

   3. Applied egg-rr99.3%

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

   4. Taylor expanded in l around 0 97.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]
   5. Step-by-step derivation

      1. associate-/l*97.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell}{\frac{F}{\pi}}}}{F} \]

   6. Simplified97.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell}{\frac{F}{\pi}}}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4000000000 \lor \neg \left(\pi \cdot \ell \leq 0.0001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -2.4e5 or 5.8e-5 < l

   1. Initial program 63.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/63.7%

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

      2. *-lft-identity63.7%

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

   3. Simplified63.7%

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

   4. Taylor expanded in l around 0 47.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow247.2%

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

   6. Simplified47.2%

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

   7. Taylor expanded in F around inf 99.5%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -2.4e5 < l < 5.8e-5

   1. Initial program 89.6%

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

   2. Taylor expanded in l around 0 89.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l*89.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

      2. associate-/r/89.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

      3. unpow289.2%

         \[\leadsto \pi \cdot \ell - \frac{\ell}{\color{blue}{F \cdot F}} \cdot \pi \]

   4. Simplified89.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F \cdot F} \cdot \pi} \]
   5. Step-by-step derivation

      1. *-commutative89.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\pi \cdot \frac{\ell}{F \cdot F}} \]

      2. distribute-lft-out--89.2%

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

      3. associate-/r*97.9%

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

   6. Applied egg-rr97.9%

      \[\leadsto \color{blue}{\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -240000 \lor \neg \left(\ell \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if F < -3.99999999999999983e-233 or 5.0000000000000001e-188 < F

   1. Initial program 82.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/83.1%

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

      2. *-lft-identity83.1%

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

   3. Simplified83.1%

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

   4. Taylor expanded in l around 0 72.3%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow272.3%

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

   6. Simplified72.3%

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

   7. Taylor expanded in F around inf 82.1%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -3.99999999999999983e-233 < F < 5.0000000000000001e-188

   1. Initial program 50.1%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/50.1%

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

      2. *-lft-identity50.1%

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

   3. Simplified50.1%

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

   4. Taylor expanded in l around 0 46.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow246.9%

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

   6. Simplified46.9%

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

   7. Taylor expanded in F around 0 46.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   8. Step-by-step derivation

      1. mul-1-neg46.9%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

      2. unpow246.9%

         \[\leadsto -\frac{\ell \cdot \pi}{\color{blue}{F \cdot F}} \]

      3. associate-/l*46.9%

         \[\leadsto -\color{blue}{\frac{\ell}{\frac{F \cdot F}{\pi}}} \]

   9. Simplified46.9%

      \[\leadsto \color{blue}{-\frac{\ell}{\frac{F \cdot F}{\pi}}} \]
   10. Step-by-step derivation

       1. associate-/l*46.9%

          \[\leadsto -\frac{\ell}{\color{blue}{\frac{F}{\frac{\pi}{F}}}} \]

       2. associate-/r/66.6%

          \[\leadsto -\color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]

   11. Applied egg-rr66.6%

       \[\leadsto -\color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-233} \lor \neg \left(F \leq 5 \cdot 10^{-188}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;-\frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

1. Split input into 2 regimes

2. if F < -1.7999999999999999e-234 or 6e-189 < F

   1. Initial program 82.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. associate-*l/83.1%

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

      2. *-lft-identity83.1%

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

   3. Simplified83.1%

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

   4. Taylor expanded in l around 0 72.3%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow272.3%

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

   6. Simplified72.3%

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

   7. Taylor expanded in F around inf 82.1%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -1.7999999999999999e-234 < F < 6e-189

   1. Initial program 50.1%

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

   2. Taylor expanded in l around 0 46.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l*46.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

      2. associate-/r/46.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

      3. unpow246.9%

         \[\leadsto \pi \cdot \ell - \frac{\ell}{\color{blue}{F \cdot F}} \cdot \pi \]

   4. Simplified46.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F \cdot F} \cdot \pi} \]

   5. Taylor expanded in F around 0 46.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   6. Step-by-step derivation

      1. mul-1-neg46.9%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

      2. unpow246.9%

         \[\leadsto -\frac{\ell \cdot \pi}{\color{blue}{F \cdot F}} \]

      3. *-commutative46.9%

         \[\leadsto -\frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      4. distribute-frac-neg46.9%

         \[\leadsto \color{blue}{\frac{-\pi \cdot \ell}{F \cdot F}} \]

      5. associate-/r*66.6%

         \[\leadsto \color{blue}{\frac{\frac{-\pi \cdot \ell}{F}}{F}} \]

      6. distribute-rgt-neg-in66.6%

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

      7. associate-/l*66.7%

         \[\leadsto \frac{\color{blue}{\frac{\pi}{\frac{F}{-\ell}}}}{F} \]

   7. Simplified66.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{\frac{F}{-\ell}}}{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-234} \lor \neg \left(F \leq 6 \cdot 10^{-189}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{F}{-\ell}}}{F}\\ \end{array} \]

1. Initial program 77.1%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. associate-*l/77.7%

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

   2. *-lft-identity77.7%

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

3. Simplified77.7%

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

4. Taylor expanded in l around 0 68.2%

   \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]
5. Step-by-step derivation

   1. unpow268.2%

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

6. Simplified68.2%

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

7. Taylor expanded in F around inf 73.9%

   \[\leadsto \color{blue}{\ell \cdot \pi} \]

8. Final simplification73.9%

   \[\leadsto \pi \cdot \ell \]

herbie shell --seed 2023166 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))