VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 99.3%

Time: 4.0s

Alternatives: 9

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: 76.4% 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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 9 \cdot 10^{-13}:\\ \;\;\;\;t\_0 - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{elif}\;l\_m \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 9e-13)
      (- t_0 (/ (/ t_0 F) F))
      (if (<= l_m 2.6e+15) (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if l < 9e-13

   1. Initial program 81.7%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 88.3

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

   4. Applied rewrites 88.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 88.3%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.0

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

   9. Applied rewrites 83.0%

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

   if 9e-13 < l < 2.6e15

   1. Initial program 87.5%

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

   if 2.6e15 < l

   1. Initial program 65.4%

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

   3. Taylor expanded in F around 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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 2: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) -5e-237)
      (/ (* (- (PI)) l_m) (* F F))
      t_0))))

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)))) < -5.0000000000000002e-237

   1. Initial program 79.2%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 25.8

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

   5. Applied rewrites 25.8%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 25.4

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

   8. Applied rewrites 25.4%

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

   if -5.0000000000000002e-237 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 77.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 75.0

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

   5. Applied rewrites 75.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot \ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;t\_0 - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{elif}\;l\_m \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{\tan \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \mathsf{PI}\left(\right)\right)\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 5.2e-9)
      (- t_0 (/ (/ t_0 F) F))
      (if (<= l_m 2.6e+15)
        (- t_0 (/ (tan (fma (PI) l_m (PI))) (* F F)))
        t_0)))))

Derivation

1. Split input into 3 regimes

2. if l < 5.2000000000000002e-9

   1. Initial program 81.8%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 88.3

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

   4. Applied rewrites 88.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 88.4%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.1

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

   9. Applied rewrites 83.1%

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

   if 5.2000000000000002e-9 < l < 2.6e15

   1. Initial program 85.7%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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. 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. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. tan-+PI-rev N/A

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

      9. lower-tan.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-PI.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if 2.6e15 < l

   1. Initial program 65.4%

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

   3. Taylor expanded in F around 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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 2.6e+15) (- t_0 (/ (/ (tan t_0) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 2.6e15

   1. Initial program 82.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 88.3%

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

   if 2.6e15 < l

   1. Initial program 65.4%

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

   3. Taylor expanded in F around 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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 5: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1600:\\ \;\;\;\;t\_0 - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 1600.0) (- t_0 (/ (/ t_0 F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 1600

   1. Initial program 82.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 88.5%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.3

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

   9. Applied rewrites 83.3%

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

   if 1600 < l

   1. Initial program 66.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

Alternative 6: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1600:\\ \;\;\;\;t\_0 - \frac{l\_m \cdot \frac{\mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 1600.0) (- t_0 (/ (* l_m (/ (PI) F)) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 1600

   1. Initial program 82.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 88.5%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   8. 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 83.3

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

   9. Applied rewrites 83.3%

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

   if 1600 < l

   1. Initial program 66.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

Alternative 7: 93.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1600:\\ \;\;\;\;t\_0 - \frac{t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 1600.0) (- t_0 (/ t_0 (* F F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 1600

   1. Initial program 82.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 82.8

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in l around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if 1600 < l

   1. Initial program 66.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

Alternative 8: 92.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1600:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 1600.0) (* (- (PI) (/ (PI) (* F F))) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 1600

   1. Initial program 82.0%

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

   3. Taylor expanded in 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 76.8

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

   5. Applied rewrites 76.8%

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

   if 1600 < l

   1. Initial program 66.5%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

Alternative 9: 74.2% 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 78.0%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
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 73.1

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

5. Applied rewrites 73.1%

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

Reproduce

herbie shell --seed 2025027 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))