VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.2%

Time: 4.5s

Alternatives: 6

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: 76.9% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 780000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{\frac{\tan t\_0}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= l_m 780000000000.0)
      (fma (PI) l_m (/ (/ (tan t_0) F) (- F)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 7.8e11

   1. Initial program 80.3%

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

   3. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 80.3

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

   5. Applied rewrites 80.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. associate-*r/ N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. *-commutative N/A

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

      9. quot-tan N/A

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

      10. mul-1-neg N/A

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

      11. quot-tan N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 88.5%

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

   if 7.8e11 < l

   1. Initial program 72.1%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 780000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \frac{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq -4 \cdot 10^{-272}:\\ \;\;\;\;\frac{\frac{\left(-l\_m\right) \cdot \mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) -4e-272)
      (/ (/ (* (- l_m) (PI)) 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)))) < -3.99999999999999972e-272

   1. Initial program 76.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.8

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

   8. Applied rewrites 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mul-1-neg N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   10. Applied rewrites 32.7%

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

   if -3.99999999999999972e-272 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 80.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 68.1

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

   5. Applied rewrites 68.1%

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

Alternative 3: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq -4 \cdot 10^{-272}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot l\_m}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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-272)
      (/ (* (- (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)))) < -3.99999999999999972e-272

   1. Initial program 76.6%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 70.0

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

   5. Applied rewrites 70.0%

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

   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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.8

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

   8. Applied rewrites 24.8%

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

   if -3.99999999999999972e-272 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 80.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 68.1

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

   5. Applied rewrites 68.1%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -4 \cdot 10^{-272}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot \ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 7000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{\frac{\left(-l\_m\right) \cdot \mathsf{PI}\left(\right)}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 7000000.0)
    (fma (PI) l_m (/ (/ (* (- l_m) (PI)) F) F))
    (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 7e6

   1. Initial program 80.2%

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

   3. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. associate-*r/ N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. *-commutative N/A

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

      9. quot-tan N/A

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

      10. mul-1-neg N/A

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

      11. quot-tan N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 88.5%

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

   10. Taylor expanded in l around 0

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lift-PI.f64 84.6

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

   12. Applied rewrites 84.6%

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

   if 7e6 < l

   1. Initial program 72.6%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 5: 92.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 7000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 7000000.0) (* (- (PI) (/ (PI) (* F F))) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 7e6

   1. Initial program 80.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if 7e6 < l

   1. Initial program 72.6%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 6: 73.6% accurate, 22.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 68.0

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

5. Applied rewrites 68.0%

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

Reproduce

herbie shell --seed 2025051 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))