VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 99.0%

Time: 7.5s

Alternatives: 8

Speedup: 3.7×

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: 76.1% 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: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 500000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan t\_1}{F}, \frac{-1}{F}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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)) (t_1 (* l_m (PI))))
   (* l_s (if (<= t_0 500000000.0) (fma (/ (tan t_1) F) (/ -1.0 F) t_1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 85.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. un-div-inv N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-rgt-identity N/A

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

      12. times-frac N/A

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

      13. distribute-neg-frac2 N/A

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

      14. inv-pow N/A

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

   4. Applied rewrites 91.2%

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

   if 5e8 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5000:\\ \;\;\;\;t\_0 - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot l\_m, l\_m, {\mathsf{PI}\left(\right)}^{-1}\right)}{l\_m} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 5000.0)
      (-
       t_0
       (/
        (/ 1.0 F)
        (*
         (/ (fma (* (* -0.3333333333333333 (PI)) l_m) l_m (pow (PI) -1.0)) l_m)
         F)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e3

   1. Initial program 85.0%

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

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

      3. lift-/.f64 N/A

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

      4. inv-pow N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. lift-tan.f64 N/A

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

      8. tan-quot N/A

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

      9. clear-num N/A

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

      10. inv-pow N/A

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

      11. div-inv N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 N/A

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

      17. inv-pow N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 N/A

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\mathsf{PI}\left(\right)}}{\ell}} \cdot \left(-F\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{\mathsf{PI}\left(\right)}}{\ell}} \cdot \left(-F\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) \cdot {\ell}^{2}} + \frac{1}{\mathsf{PI}\left(\right)}}{\ell} \cdot \left(-F\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)} + \frac{1}{\mathsf{PI}\left(\right)}}{\ell} \cdot \left(-F\right)} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\color{blue}{\left(\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) \cdot \ell\right) \cdot \ell} + \frac{1}{\mathsf{PI}\left(\right)}}{\ell} \cdot \left(-F\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}}{\ell} \cdot \left(-F\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right) - \frac{-1}{6} \cdot \mathsf{PI}\left(\right)\right) \cdot \ell}, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot \left(-F\right)} \]

      7. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right)} \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot \left(-F\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot \left(-F\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\color{blue}{\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot \left(-F\right)} \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\left(\color{blue}{\frac{-1}{3}} \cdot \mathsf{PI}\left(\right)\right) \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot \left(-F\right)} \]

      11. lower-PI.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\left(\frac{-1}{3} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell} \cdot \left(-F\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\left(\frac{-1}{3} \cdot \mathsf{PI}\left(\right)\right) \cdot \ell, \ell, \color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{\ell} \cdot \left(-F\right)} \]

      13. lower-PI.f64 93.3

          \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot \ell, \ell, \frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}\right)}{\ell} \cdot \left(-F\right)} \]

   7. Applied rewrites 93.3%

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{-1}{F}}{\color{blue}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot \ell, \ell, \frac{1}{\mathsf{PI}\left(\right)}\right)}{\ell}} \cdot \left(-F\right)} \]

   if 5e3 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell \leq 5000:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot \ell, \ell, {\mathsf{PI}\left(\right)}^{-1}\right)}{\ell} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 500000000:\\ \;\;\;\;t\_0 - \frac{\frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 500000000.0) (- t_0 (/ (/ (tan (* l_m (PI))) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 85.0%

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

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 91.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if 5e8 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 4: 98.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 500000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{F} \cdot l\_m, \frac{-1}{F}, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 500000000.0)
      (fma (* (/ (PI) F) l_m) (/ -1.0 F) (* l_m (PI)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 85.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. un-div-inv N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. *-rgt-identity N/A

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

      12. times-frac N/A

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

      13. distribute-neg-frac2 N/A

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

      14. inv-pow N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if 5e8 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 5: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 500000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{-\mathsf{PI}\left(\right)}{F} \cdot \frac{l\_m}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 500000000.0)
      (fma (PI) l_m (* (/ (- (PI)) F) (/ l_m F)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 85.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-/.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if 5e8 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 6: 92.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 500000000:\\ \;\;\;\;t\_0 - \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 500000000.0) (- t_0 (* (/ (PI) (* F F)) l_m)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 85.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if 5e8 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 7: 92.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 500000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 500000000.0) (* (- (PI) (/ (PI) (* F F))) l_m) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 85.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if 5e8 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 8: 74.3% 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 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. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 75.5

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

5. Applied rewrites 75.5%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))