VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 98.6%

Time: 3.8s

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 320:\\ \;\;\;\;t\_0 - \frac{{F}^{-1} \cdot \sin t\_0}{F \cdot \cos t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 320.0)
      (- t_0 (/ (* (pow F -1.0) (sin t_0)) (* F (cos t_0))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 320

   1. Initial program 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 320 < l

   1. Initial program 67.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 2: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0 \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= (- t_0 (* (/ 1.0 (* F F)) (tan t_0))) -1e-247)
      (* (- 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)))) < -1e-247

   1. Initial program 80.4%

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

   3. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. quot-tan N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-tan.f64 26.5

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 24.9

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

   8. Applied rewrites 24.9%

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

   if -1e-247 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 83.2%

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

   3. Taylor expanded in F around inf

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

      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 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 320:\\ \;\;\;\;t\_0 - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (* l_s (if (<= l_m 320.0) (- t_0 (/ (/ (tan t_0) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 320

   1. Initial program 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-pow.f64 N/A

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

      12. inv-pow N/A

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

      13. associate-/r* N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-*.f64 N/A

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

   6. Applied rewrites 91.8%

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

   if 320 < l

   1. Initial program 67.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 4: 98.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 50:\\ \;\;\;\;t\_0 - \frac{\frac{\mathsf{PI}\left(\right)}{F} \cdot l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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

Derivation

1. Split input into 2 regimes

2. if l < 50

   1. Initial program 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-pow.f64 N/A

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

      12. inv-pow N/A

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

      13. associate-/r* N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-*.f64 N/A

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in l around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 84.9

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

   9. Applied rewrites 84.9%

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

   if 50 < l

   1. Initial program 67.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 5: 92.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 50:\\ \;\;\;\;t\_0 - \frac{t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (* l_s (if (<= l_m 50.0) (- t_0 (/ t_0 (* F F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 50

   1. Initial program 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-tan.f64 N/A

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

      9. quot-tan N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-pow.f64 N/A

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

      12. inv-pow N/A

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

      13. associate-/r* N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-*.f64 N/A

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 84.9

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

   9. Applied rewrites 84.9%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. pow2 N/A

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

       5. lower-/.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 79.9

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

   11. Applied rewrites 79.9%

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

   if 50 < l

   1. Initial program 67.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 6: 92.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 50:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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

Derivation

1. Split input into 2 regimes

2. if l < 50

   1. Initial program 86.5%

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

   3. Taylor expanded in 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 79.6

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

   5. Applied rewrites 79.6%

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

   if 50 < l

   1. Initial program 67.4%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 7: 73.8% 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 81.7%

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

3. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 75.4

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

5. Applied rewrites 75.4%

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

Alternative 8: 3.1% accurate, 135.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot 0 \end{array} \]

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 0.0))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * 0.0;
}

Fortran

l\_m =     private
l\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_s, f, l_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_s
    real(8), intent (in) :: f
    real(8), intent (in) :: l_m
    code = l_s * 0.0d0
end function

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * 0.0;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * 0.0

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * 0.0)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * 0.0;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. tan-+PI-rev N/A

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

   4. lower-tan.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 59.9

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

4. Applied rewrites 59.9%

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

5. Taylor expanded in l around 0

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

7. Applied rewrites 2.7%

   \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0} \]
8. Step-by-step derivation

   1. associate-/r* 2.7

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{F} \cdot F} \cdot 0 \]

   2. unpow-1 2.7

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   5. tan-quot N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot \color{blue}{F}} \cdot 0 \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   9. lift-*.f64 2.7

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   10. times-frac 2.7

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{F \cdot F}} \cdot 0 \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot \color{blue}{0} \]

   12. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

   14. lift-/.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot 0 \]

9. Applied rewrites 3.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))