VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.0% → 99.4%

Time: 7.7s

Alternatives: 9

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: 76.0% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;t\_0 + \frac{-1}{F} \cdot \frac{\tan t\_0}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 1.55e+15) (+ t_0 (* (/ -1.0 F) (/ (tan t_0) F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 1.55e15

   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. 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. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 90.0

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

   4. Applied rewrites 90.0%

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

   if 1.55e15 < l

   1. Initial program 57.5%

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

   3. Taylor expanded in 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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell + \frac{-1}{F} \cdot \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := t\_0 - \frac{1}{F \cdot F} \cdot \tan t\_0\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+234} \lor \neg \left(t\_1 \leq -5 \cdot 10^{-255}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)) (t_1 (- t_0 (* (/ 1.0 (* F F)) (tan t_0)))))
   (*
    l_s
    (if (or (<= t_1 -2e+234) (not (<= t_1 -5e-255)))
      t_0
      (* (- l_m) (/ (PI) (* F F)))))))

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)))) < -2.00000000000000004e234 or -4.9999999999999996e-255 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 64.5%

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

   3. Taylor expanded in 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 68.6

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

   5. Applied rewrites 68.6%

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

   if -2.00000000000000004e234 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -4.9999999999999996e-255

   1. Initial program 94.8%

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

   3. Taylor expanded in F around 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 21.1

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

   5. Applied rewrites 21.1%

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

   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. lower-neg.f64 N/A

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

      3. associate-/l* 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. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 18.8

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

   8. Applied rewrites 18.8%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -2 \cdot 10^{+234} \lor \neg \left(\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -5 \cdot 10^{-255}\right):\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{\frac{1}{F} \cdot \tan t\_0}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 1.55e+15) (- t_0 (/ (* (/ 1.0 F) (tan t_0)) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 1.55e15

   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. 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. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 90.0

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

   4. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-tan.f64 N/A

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

      18. lift-neg.f64 90.0

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

   6. Applied rewrites 90.0%

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

   if 1.55e15 < l

   1. Initial program 57.5%

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

   3. Taylor expanded in 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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 52000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{-1}{F} \cdot \frac{t\_0}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 52000.0) (fma (PI) l_m (* (/ -1.0 F) (/ t_0 F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 52000

   1. Initial program 78.4%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.4

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

   7. Applied rewrites 83.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 83.4%

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

       1. lift-neg.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. distribute-neg-frac N/A

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

       4. metadata-eval N/A

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

       5. lower-/.f64 83.4

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

   11. Applied rewrites 83.4%

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

   if 52000 < l

   1. Initial program 58.1%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 52000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \frac{-1}{F} \cdot \frac{\mathsf{PI}\left(\right) \cdot \ell}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 52000:\\ \;\;\;\;t\_0 - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 52000.0) (- t_0 (/ (/ t_0 F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 52000

   1. Initial program 78.4%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.4

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

   7. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. frac-times N/A

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

      12. sqr-neg-rev N/A

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

      13. pow2 N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 73.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-rgt-identity N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 83.4%

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

   if 52000 < l

   1. Initial program 58.1%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 52000:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{\frac{\mathsf{PI}\left(\right) \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 52000:\\ \;\;\;\;t\_0 - \frac{t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 52000.0) (- t_0 (/ t_0 (* F F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 52000

   1. Initial program 78.4%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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. pow2 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-tan.f64 N/A

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

      19. lower-neg.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 83.4

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

   7. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. frac-times N/A

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

      12. sqr-neg-rev N/A

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

      13. pow2 N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 73.2%

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

   11. Applied rewrites 73.2%

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

   if 52000 < l

   1. Initial program 58.1%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 52000:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell - \frac{\mathsf{PI}\left(\right) \cdot \ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 52000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 52000.0) (* (- (PI) (/ (PI) (* F F))) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 52000

   1. Initial program 78.4%

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

   3. Taylor expanded in 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 71.9

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

   5. Applied rewrites 71.9%

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

   if 52000 < l

   1. Initial program 58.1%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 8: 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 72.9%

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

3. Taylor expanded in F around inf

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 72.0

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

5. Applied rewrites 72.0%

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

Alternative 9: 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 72.9%

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

3. Taylor expanded in F around 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 21.2

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

5. Applied rewrites 21.2%

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

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. tan-+PI N/A

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

   6. tan-+PI-rev N/A

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

   7. lower-tan.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. associate-+l+ N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. count-2-rev N/A

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

   15. +-commutative N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. lower-+.f64 N/A

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

   20. lift-PI.f64 9.0

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

7. Applied rewrites 9.0%

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

8. Taylor expanded in l around 0

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

   1. associate-/r* N/A

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

   2. +-commutative N/A

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

   3. cos-+PI N/A

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

   4. count-2-rev N/A

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

   5. cos-+PI N/A

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

   6. cos-PI N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

10. Applied rewrites 3.1%

    \[\leadsto 0 \]

11. Final simplification 3.1%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025045 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))