VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 99.4%

Time: 7.8s

Alternatives: 5

Speedup: 4.4×

Specification

Math

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

FPCore

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;t\_0 + \frac{\frac{\frac{\sin \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\mathsf{fma}\left(l\_m, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (*
    l_s
    (if (<= l_m 3.8e+14)
      (+ t_0 (/ (/ (/ (sin (* l_m (PI))) (cos (fma l_m (PI) (PI)))) F) F))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 3.8e14

   1. Initial program 80.9%

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

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 87.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. tan-+PI-rev N/A

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

      7. tan-quot N/A

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

      8. frac-2neg N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. cos-+PI-rev N/A

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

      18. lower-cos.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-PI.f64 N/A

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

      24. lower-fma.f64 87.4

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

   6. Applied rewrites 87.4%

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

   if 3.8e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell + \frac{\frac{\frac{\sin \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\mathsf{fma}\left(\ell, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)}}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0 - \frac{\frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (* l_s (if (<= l_m 3.6e+15) (- t_0 (/ (/ (tan (* l_m (PI))) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 3.6e15

   1. Initial program 80.9%

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

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 87.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

   if 3.6e15 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 3: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - \frac{\mathsf{PI}\left(\right) \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 3.8e+14) (- t_0 (/ (* (PI) (/ l_m F)) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if l < 3.8e14

   1. Initial program 80.9%

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

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 87.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 87.6

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{F}}}{F} \]
   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. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 82.9

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

   7. Applied rewrites 82.9%

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

   9. Applied rewrites 82.9%

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

   if 3.8e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 4: 92.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot l\_m\\ \end{array} \end{array} \]

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 3.8e+14) (* (- (PI) (/ (PI) (* F F))) l_m) (* (PI) l_m))))

Derivation

1. Split input into 2 regimes

2. if l < 3.8e14

   1. Initial program 80.9%

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

   3. Taylor expanded in 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 76.3

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

   5. Applied rewrites 76.3%

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

   if 3.8e14 < l

   1. Initial program 63.6%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 5: 73.0% 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 76.8%

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

3. Taylor expanded in F around 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 75.6

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

5. Applied rewrites 75.6%

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

Reproduce

herbie shell --seed 2024359 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))