VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 99.2%

Time: 6.5s

Alternatives: 8

Speedup: 4.4×

Specification

Math

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

FPCore

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := {\mathsf{PI}\left(\right)}^{-1} + \mathsf{PI}\left(\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;t\_1 + \frac{\frac{\frac{\sin t\_1}{\cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \mathsf{PI}\left(\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, l\_m, \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - {\mathsf{PI}\left(\right)}^{-2}\right) \cdot l\_m}{t\_0}\right)}{2}\\ \end{array} \end{array} \end{array} \]

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 (+ (pow (PI) -1.0) (PI))) (t_1 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 5.6e+14)
      (+ t_1 (/ (/ (/ (sin t_1) (cos (fma (PI) l_m (PI)))) F) F))
      (/
       (fma t_0 l_m (/ (* (- (* (PI) (PI)) (pow (PI) -2.0)) l_m) t_0))
       2.0)))))

Derivation

1. Split input into 2 regimes

2. if l < 5.6e14

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. 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 90.3

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

      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 90.3

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

   4. Applied rewrites 90.3%

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

      1. *-lft-identity N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-+PI-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. tan-+PI-rev N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. associate-*r/ N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. cos-+PI-rev N/A

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

      18. lower-cos.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-PI.f64 N/A

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

   6. Applied rewrites 90.1%

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

   if 5.6e14 < l

   1. Initial program 71.2%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.8%

      \[\leadsto \frac{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{-1} + \mathsf{PI}\left(\right), \ell, \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - {\mathsf{PI}\left(\right)}^{-2}\right) \cdot \ell}{{\mathsf{PI}\left(\right)}^{-1} + \mathsf{PI}\left(\right)}\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell + \frac{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{PI}\left(\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{-1} + \mathsf{PI}\left(\right), \ell, \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - {\mathsf{PI}\left(\right)}^{-2}\right) \cdot \ell}{{\mathsf{PI}\left(\right)}^{-1} + \mathsf{PI}\left(\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;t\_0 + \frac{\frac{\frac{\sin t\_0}{\cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \mathsf{PI}\left(\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= l_m 5.6e+14)
      (+ t_0 (/ (/ (/ (sin t_0) (cos (fma (PI) l_m (PI)))) F) F))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 5.6e14

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. 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 90.3

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

      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 90.3

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

   4. Applied rewrites 90.3%

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

      1. *-lft-identity N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-+PI-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. tan-+PI-rev N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. associate-*r/ N/A

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

      12. frac-2neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. cos-+PI-rev N/A

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

      18. lower-cos.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-PI.f64 N/A

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

   6. Applied rewrites 90.1%

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

   if 5.6e14 < l

   1. Initial program 71.2%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell + \frac{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \mathsf{PI}\left(\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - {F}^{-1} \cdot \frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 5.6e+14)
      (- t_0 (* (pow F -1.0) (/ (tan (* l_m (PI))) F)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 5.6e14

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 90.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   if 5.6e14 < l

   1. Initial program 71.2%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.6%

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

Alternative 4: 84.0% accurate, 0.8× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) -4e-221)
      (/ (* (- (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)))) < -4.00000000000000007e-221

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 83.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 30.4%

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

   12. Applied rewrites 25.0%

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

   if -4.00000000000000007e-221 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 83.9%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 77.3%

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

4. Final simplification 54.0%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - \frac{\frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= l_m 5.6e+14) (- t_0 (/ (/ (tan (* l_m (PI))) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 5.6e14

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. 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 90.3

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

      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 90.3

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

   4. Applied rewrites 90.3%

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

   if 5.6e14 < l

   1. Initial program 71.2%

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

   3. Taylor expanded in F around inf

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

   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 6: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 30000000000000:\\ \;\;\;\;t\_0 - \frac{l\_m}{F} \cdot \frac{\mathsf{PI}\left(\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 30000000000000.0) (- t_0 (* (/ l_m F) (/ (PI) F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 3e13

   1. Initial program 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 90.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 90.2

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

   4. Applied rewrites 90.2%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 84.2%

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

   if 3e13 < l

   1. Initial program 71.7%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 98.0%

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

Alternative 7: 92.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 30000000000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 30000000000000.0)
    (* (- (PI) (/ (PI) (* F F))) l_m)
    (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 3e13

   1. Initial program 84.2%

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

   3. Taylor expanded in 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

   5. Applied rewrites 78.1%

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

   if 3e13 < l

   1. Initial program 71.7%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 98.0%

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

Alternative 8: 74.1% accurate, 22.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.3%

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

3. Taylor expanded in F around inf

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

5. Applied rewrites 73.5%

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

Reproduce

herbie shell --seed 2025019 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))