VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 97.7%

Time: 8.0s

Alternatives: 10

Speedup: 4.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \ell\\ t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (PI) l))) (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))

Initial Program: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \ell\\ t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (PI) l))) (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))

Alternative 1: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;t\_0 - \frac{l\_m \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{elif}\;l\_m \leq 3.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(F \cdot F\right) \cdot t\_0 - \tan t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 5.6e-11)
      (- t_0 (/ (* l_m (/ (PI) F)) F))
      (if (<= l_m 3.35e+19) (/ (- (* (* F F) t_0) (tan t_0)) (* F F)) t_0)))))

Derivation

1. Split input into 3 regimes

2. if l < 5.6e-11

   1. Initial program 74.9%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 80.0

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F} \]

   10. Applied rewrites 80.0%

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

   if 5.6e-11 < l < 3.35e19

   1. Initial program 99.4%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in l around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}} \]

      3. associate-/l* N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      7. pow2 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F} \]

      8. lift-*.f64 37.6

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F} \]

   8. Applied rewrites 37.6%

      \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F} \]

   9. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{{F}^{2} \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right) - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{{F}^{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

   if 3.35e19 < l

   1. Initial program 60.9%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 99.6

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{{F}^{-1} \cdot \sin t\_0}{F \cdot \sin \left(\left(-\mathsf{PI}\left(\right)\right) \cdot l\_m + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 3.2e+15)
      (-
       t_0
       (/
        (* (pow F -1.0) (sin t_0))
        (* F (sin (+ (* (- (PI)) l_m) (/ (PI) 2.0))))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 3.2e15

   1. Initial program 75.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 85.4

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

   4. Applied rewrites 85.4%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \ell\right)}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\ell \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\ell \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\ell \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \ell}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \ell}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \ell\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 85.1

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

   6. Applied rewrites 85.1%

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

   if 3.2e15 < l

   1. Initial program 61.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 99.6

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot \sin \left(\left(-\mathsf{PI}\left(\right)\right) \cdot \ell + \frac{\mathsf{PI}\left(\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{{F}^{-1} \cdot \sin t\_0}{F \cdot \cos t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 2.5e+15)
      (- t_0 (/ (* (pow F -1.0) (sin t_0)) (* F (cos t_0))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 2.5e15

   1. Initial program 75.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 85.4

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

   4. Applied rewrites 85.4%

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

   if 2.5e15 < l

   1. Initial program 61.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 99.6

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+221} \lor \neg \left(t\_1 \leq -4 \cdot 10^{-213}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)) (t_1 (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))
   (*
    l_s
    (if (or (<= t_1 -2e+221) (not (<= t_1 -4e-213)))
      t_0
      (* (- l_m) (/ (PI) (* F F)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -2.0000000000000001e221 or -3.9999999999999998e-213 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 64.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 70.7

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]

   if -2.0000000000000001e221 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -3.9999999999999998e-213

   1. Initial program 93.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 32.5

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in l around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}} \]

      3. associate-/l* N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{{F}^{2}} \]

      7. pow2 N/A

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F} \]

      8. lift-*.f64 31.9

         \[\leadsto -\ell \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F} \]

   8. Applied rewrites 31.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -2 \cdot 10^{+221} \lor \neg \left(\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -4 \cdot 10^{-213}\right):\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;t\_0 - \frac{l\_m \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{elif}\;l\_m \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(\left(F \cdot F\right) \cdot l\_m\right) \cdot \mathsf{PI}\left(\right) - \tan t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 5.6e-11)
      (- t_0 (/ (* l_m (/ (PI) F)) F))
      (if (<= l_m 2.5e+15)
        (/ (- (* (* (* F F) l_m) (PI)) (tan t_0)) (* F F))
        t_0)))))

Derivation

1. Split input into 3 regimes

2. if l < 5.6e-11

   1. Initial program 74.9%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 80.0

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F} \]

   10. Applied rewrites 80.0%

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

   if 5.6e-11 < l < 2.5e15

   1. Initial program 99.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{{F}^{2} \cdot \left(\ell \cdot \mathsf{PI}\left(\right)\right) - \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)}}{{F}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if 2.5e15 < l

   1. Initial program 61.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 99.6

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0 + \frac{-1}{F} \cdot \frac{\tan t\_0}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (* l_s (if (<= l_m 2.5e+15) (+ t_0 (* (/ -1.0 F) (/ (tan t_0) F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 2.5e15

   1. Initial program 75.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{\color{blue}{{F}^{2}}} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1}{{F}^{2}}} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      5. lift-tan.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{{F}^{2}} \cdot \color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      6. lift-PI.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{{F}^{2}} \cdot \tan \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \ell\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{1}{{F}^{2}} \cdot \tan \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{{F}^{2}}} \]

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 85.4

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

   4. Applied rewrites 85.4%

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

   if 2.5e15 < l

   1. Initial program 61.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 99.6

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell + \frac{-1}{F} \cdot \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.15 \cdot 10^{-28}:\\ \;\;\;\;t\_0 - \frac{l\_m \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{elif}\;l\_m \leq 2.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan t\_0, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 1.15e-28)
      (- t_0 (/ (* l_m (/ (PI) F)) F))
      (if (<= l_m 2.5e+15) (fma (/ -1.0 (* F F)) (tan t_0) t_0) t_0)))))

Derivation

1. Split input into 3 regimes

2. if l < 1.14999999999999993e-28

   1. Initial program 74.3%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 79.5

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F} \]

   10. Applied rewrites 79.5%

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

   if 1.14999999999999993e-28 < l < 2.5e15

   1. Initial program 99.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{{F}^{2} \cdot \cos \left(\ell \cdot \mathsf{PI}\left(\right)\right)} + \ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) + \ell \cdot \mathsf{PI}\left(\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{{F}^{2}} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) + \ell \cdot \mathsf{PI}\left(\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{{F}^{2}}, \color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}, \ell \cdot \mathsf{PI}\left(\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{{F}^{2}}, \tan \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \ell\right)}, \ell \cdot \mathsf{PI}\left(\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\ell}\right), \ell \cdot \mathsf{PI}\left(\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\ell}\right), \ell \cdot \mathsf{PI}\left(\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \ell \cdot \mathsf{PI}\left(\right)\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \ell \cdot \mathsf{PI}\left(\right)\right) \]

      11. lift-tan.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \ell \cdot \mathsf{PI}\left(\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \mathsf{PI}\left(\right) \cdot \ell\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \mathsf{PI}\left(\right) \cdot \ell\right) \]

      14. lift-PI.f64 99.6

          \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \mathsf{PI}\left(\right) \cdot \ell\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{F \cdot F}, \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right), \mathsf{PI}\left(\right) \cdot \ell\right)} \]

   if 2.5e15 < l

   1. Initial program 61.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 99.6

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2000000000:\\ \;\;\;\;t\_0 - \frac{l\_m \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (* l_s (if (<= l_m 2000000000.0) (- t_0 (/ (* l_m (/ (PI) F)) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 2e9

   1. Initial program 75.7%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{{F}^{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\color{blue}{F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.0%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 80.1

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\ell \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F} \]

   10. Applied rewrites 80.1%

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

   if 2e9 < l

   1. Initial program 62.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 98.2

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2000000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 2000000000.0) (* (- (PI) (/ (PI) (* F F))) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 2e9

   1. Initial program 75.7%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \cdot \color{blue}{\ell} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \cdot \color{blue}{\ell} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \cdot \ell \]

      4. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \cdot \ell \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \cdot \ell \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \cdot \ell \]

      7. pow2 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot \ell \]

      8. lift-*.f64 70.5

         \[\leadsto \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot \ell \]

   5. Applied rewrites 70.5%

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

   if 2e9 < l

   1. Initial program 62.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

      3. lift-PI.f64 98.2

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.4% accurate, 22.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\mathsf{PI}\left(\right) \cdot l\_m\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m) :precision binary64 (* l_s (* (PI) l_m)))

Derivation

1. Initial program 71.9%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
2. Add Preprocessing

3. Taylor expanded in F around inf

   \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\ell} \]

   3. lift-PI.f64 70.3

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell \]

5. Applied rewrites 70.3%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))