VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.9% → 99.0%

Time: 8.1s

Alternatives: 15

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: 75.9% 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

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2.5 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{l\_m}{\frac{F}{\mathsf{PI}\left(\right)}}, \frac{-1}{F}, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{\tan t\_0}{\left(-F\right) \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))) (t_1 (* (PI) l_m)))
   (*
    l_s
    (if (<= t_1 2.5e-64)
      (fma (/ l_m (/ F (PI))) (/ -1.0 F) t_0)
      (if (<= t_1 2e+15) (fma (PI) l_m (/ (tan t_0) (* (- F) F))) t_1)))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < 2.50000000000000017e-64

   1. Initial program 88.1%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 92.9%

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

   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 86.1

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

   7. Applied rewrites 86.1%

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

   9. Applied rewrites 86.2%

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

   if 2.50000000000000017e-64 < (*.f64 (PI.f64) l) < 2e15

   1. Initial program 97.7%

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

      1. lift--.f64 N/A

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin t\_0}{F}}{\cos \left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot l\_m\right)}, \frac{-1}{F}, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot l\_m\right)\\ \end{array} \end{array} \end{array} \]

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 2e+15)
      (fma
       (/ (/ (sin t_0) F) (cos (* (cbrt (pow (PI) 3.0)) l_m)))
       (/ -1.0 F)
       (* l_m (PI)))
      (*
       (pow (PI) 0.8333333333333334)
       (* (pow (PI) 0.16666666666666666) l_m))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e15

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. tan-quot N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 93.2

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

   6. Applied rewrites 93.2%

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

      1. rem-cbrt-cube N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cbrt.f64 93.0

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

   8. Applied rewrites 93.0%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.0%

      \[\leadsto \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot \ell \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

      \[\leadsto {\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \ell\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin t\_0}{F}}{\cos t\_0}, \frac{-1}{F}, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot l\_m\right)\\ \end{array} \end{array} \end{array} \]

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 2e+15)
      (fma (/ (/ (sin t_0) F) (cos t_0)) (/ -1.0 F) (* l_m (PI)))
      (*
       (pow (PI) 0.8333333333333334)
       (* (pow (PI) 0.16666666666666666) l_m))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e15

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. tan-quot N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 93.2

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

   6. Applied rewrites 93.2%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.0%

      \[\leadsto \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot \ell \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

      \[\leadsto {\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \ell\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - {\left(F \cdot F\right)}^{-1} \cdot \tan t\_0 \leq -1 \cdot 10^{-273}:\\ \;\;\;\;\left(-\mathsf{PI}\left(\right)\right) \cdot \frac{l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (pow (* F F) -1.0) (tan t_0))) -1e-273)
      (* (- (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)))) < -1e-273

   1. Initial program 86.3%

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

   3. 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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 21.3%

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

   10. Applied rewrites 18.2%

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

   if -1e-273 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 87.3%

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

   3. 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 66.1

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

   5. Applied rewrites 66.1%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - {\left(F \cdot F\right)}^{-1} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -1 \cdot 10^{-273}:\\ \;\;\;\;\left(-\mathsf{PI}\left(\right)\right) \cdot \frac{\ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot l\_m \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan t\_0}{F}, \frac{-1}{F}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;{\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot l\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))))
   (*
    l_s
    (if (<= (* (PI) l_m) 2e+15)
      (fma (/ (tan t_0) F) (/ -1.0 F) t_0)
      (*
       (pow (PI) 0.8333333333333334)
       (* (pow (PI) 0.16666666666666666) l_m))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e15

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.0%

      \[\leadsto \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot \ell \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

      \[\leadsto {\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \ell\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot l\_m \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan t\_0}{F}, \frac{-1}{F}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;{\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot l\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))))
   (*
    l_s
    (if (<= (* (PI) l_m) 2e+15)
      (fma (/ (tan t_0) F) (/ -1.0 F) t_0)
      (*
       (pow (PI) 0.16666666666666666)
       (* (pow (PI) 0.8333333333333334) l_m))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e15

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.0%

      \[\leadsto \sqrt[3]{{\mathsf{PI}\left(\right)}^{3}} \cdot \ell \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto {\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot \ell\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan t\_1}{F}, \frac{-1}{F}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 2e+15) (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) < 2e15

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 8: 99.2% accurate, 0.9× speedup

Math

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

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 2e+15) (- t_0 (/ (/ (tan (* l_m (PI))) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e15

   1. Initial program 88.7%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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. *-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 93.1

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

      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 93.1

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

   4. Applied rewrites 93.1%

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

   if 2e15 < (*.f64 (PI.f64) l)

   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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 9: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 50000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(1 - {\left(F \cdot F\right)}^{-1}\right)\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 50000.0) (* (* (PI) (- 1.0 (pow (* F F) -1.0))) l_m) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.7%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      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 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 81.7%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 78.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell \leq 50000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(1 - {\left(F \cdot F\right)}^{-1}\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(\frac{l\_m}{\frac{F}{\mathsf{PI}\left(\right)}}, \frac{-1}{F}, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 50000.0)
      (fma (/ l_m (/ F (PI))) (/ -1.0 F) (* l_m (PI)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

   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 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 86.2%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 78.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 11: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1}{F}, \frac{-1}{F}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 50000.0) (fma (/ t_1 F) (/ -1.0 F) t_1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

   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 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 86.2%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 78.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 12: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \frac{l\_m}{F}, \frac{-1}{F}, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 50000.0)
      (fma (* (PI) (/ l_m F)) (/ -1.0 F) (* l_m (PI)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 93.2%

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

   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 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 86.2%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 78.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 13: 98.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{\mathsf{PI}\left(\right)}{F} \cdot \frac{l\_m}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 50000.0) (fma (PI) l_m (* (/ (PI) F) (/ l_m (- F)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.7%

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

      1. lift--.f64 N/A

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

      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 89.0

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

   4. Applied rewrites 89.0%

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

   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 86.1

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

   7. Applied rewrites 86.1%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 78.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell \leq 50000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \frac{\mathsf{PI}\left(\right)}{F} \cdot \frac{\ell}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 50000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 50000.0) (* (- (PI) (/ (PI) (* F F))) l_m) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e4

   1. Initial program 88.7%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      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 81.7

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

   5. Applied rewrites 81.7%

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

   if 5e4 < (*.f64 (PI.f64) l)

   1. Initial program 78.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 15: 73.5% 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 86.8%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
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 71.9

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

5. Applied rewrites 71.9%

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

Reproduce

herbie shell --seed 2024319 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))