VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 98.8%

Time: 8.5s

Alternatives: 8

Speedup: 3.7×

Specification

Math

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

FPCore

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

Initial Program: 76.6% 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: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 100000000:\\ \;\;\;\;t\_0 - \frac{\frac{\tan t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))))
   (* l_s (if (<= t_0 100000000.0) (- t_0 (/ (/ (tan t_0) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e8

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 92.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 1e8 < (*.f64 (PI.f64) l)

   1. Initial program 56.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) \leq 100000000:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right) - \frac{\frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \tan t\_0 \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{-236}:\\ \;\;\;\;\frac{l\_m}{F} \cdot \frac{-\mathsf{PI}\left(\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (*
    l_s
    (if (<= (- t_0 (* (tan t_0) (/ 1.0 (* F F)))) -1e-236)
      (* (/ l_m F) (/ (- (PI)) 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-236

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 84.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 65.7

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 39.1%

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

   if -1e-236 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 77.6%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) - \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{-236}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\mathsf{PI}\left(\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - \tan t\_0 \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (*
    l_s
    (if (<= (- t_0 (* (tan t_0) (/ 1.0 (* F F)))) -1e-236)
      (* (/ (- (PI)) (* F F)) l_m)
      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-236

   1. Initial program 75.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 84.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 65.7

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 30.3%

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

   if -1e-236 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 77.6%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) - \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{F \cdot F} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{\frac{-\mathsf{PI}\left(\right)}{F} \cdot l\_m}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (*
    l_s
    (if (<= t_0 100000000.0)
      (fma (PI) l_m (/ (* (/ (- (PI)) F) l_m) F))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e8

   1. Initial program 82.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-/.f64 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 86.3%

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

   if 1e8 < (*.f64 (PI.f64) l)

   1. Initial program 56.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \frac{\frac{-\mathsf{PI}\left(\right)}{F} \cdot \ell}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 100000000:\\ \;\;\;\;t\_0 - \frac{\frac{t\_0}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))))
   (* l_s (if (<= t_0 100000000.0) (- t_0 (/ (/ t_0 F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e8

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 92.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if 1e8 < (*.f64 (PI.f64) l)

   1. Initial program 56.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) \leq 100000000:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right) - \frac{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{l\_m}{F \cdot F} \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (*
    l_s
    (if (<= t_0 100000000.0)
      (fma (PI) l_m (* (/ l_m (* F F)) (- (PI))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e8

   1. Initial program 82.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in l around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-/.f64 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 77.1%

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

   if 1e8 < (*.f64 (PI.f64) l)

   1. Initial program 56.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), \ell, \frac{\ell}{F \cdot F} \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 100000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (* l_s (if (<= t_0 100000000.0) (* (- (PI) (/ (PI) (* F F))) l_m) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e8

   1. Initial program 82.2%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if 1e8 < (*.f64 (PI.f64) l)

   1. Initial program 56.0%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) \leq 100000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 22.5× speedup

Math

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

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 (* l_m (PI))))

Derivation

1. Initial program 76.7%

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

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 66.6

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

5. Applied rewrites 66.6%

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

6. Final simplification 66.6%

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

Reproduce

herbie shell --seed 2024267 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))