VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.9% → 99.2%

Time: 17.3s

Alternatives: 9

Speedup: 3.7×

• could not determine a ground truth

  1. F = 2.4654765846250347e-57
  2. l = 639137522874327.5

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: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* l_m (PI))))
   (* l_s (if (<= t_0 1e+15) (+ (/ (* (tan t_0) (/ -1.0 F)) F) t_0) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e15

   1. Initial program 80.3%

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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 89.9

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

   4. Applied rewrites 89.9%

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

   if 1e15 < (*.f64 (PI.f64) l)

   1. Initial program 53.0%

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

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

   5. Applied rewrites 99.7%

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

4. Final simplification 92.3%

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

Alternative 2: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ t_1 := t\_0 - \tan t\_0 \cdot \frac{1}{F \cdot F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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))) (t_1 (- t_0 (* (tan t_0) (/ 1.0 (* F F))))))
   (*
    l_s
    (if (<= t_1 -1e+132)
      t_0
      (if (<= t_1 -1e-246) (* (/ (- (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)))) < -9.99999999999999991e131 or -9.99999999999999956e-247 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 66.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   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 71.4

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

   5. Applied rewrites 71.4%

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

   if -9.99999999999999991e131 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -9.99999999999999956e-247

   1. Initial program 97.8%

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

      1. lift-PI.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. add-cbrt-cube N/A

         \[\leadsto \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      3. lower-cbrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      4. rem-cube-cbrt N/A

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}^{3}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      5. add-cbrt-cube N/A

         \[\leadsto \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      6. lift-PI.f64 N/A

         \[\leadsto \sqrt[3]{{\color{blue}{\mathsf{PI}\left(\right)}}^{3}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

      7. lower-pow.f64 97.4

         \[\leadsto \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}} \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]

   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 94.5

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

   7. Applied rewrites 94.5%

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

   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 27.6%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) - \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{+132}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \mathbf{elif}\;\ell \cdot \mathsf{PI}\left(\right) - \tan \left(\ell \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{F \cdot F} \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{F \cdot F} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

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

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 1e+15) (- t_0 (/ (/ (tan t_0) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e15

   1. Initial program 80.3%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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 89.9

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

      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 89.9

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

   4. Applied rewrites 89.9%

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

   if 1e15 < (*.f64 (PI.f64) l)

   1. Initial program 53.0%

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

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

   5. Applied rewrites 99.7%

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

4. Final simplification 92.3%

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

Alternative 4: 98.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{\mathsf{PI}\left(\right)}{\frac{-F}{l\_m} \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 5.0) (fma (PI) l_m (/ (PI) (* (/ (- F) l_m) F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5

   1. Initial program 80.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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 80.1

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

   4. Applied rewrites 80.1%

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

   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. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 85.0%

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

   if 5 < (*.f64 (PI.f64) l)

   1. Initial program 55.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 88.4%

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

Alternative 5: 98.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right), l\_m, \frac{-\mathsf{PI}\left(\right)}{F} \cdot \frac{l\_m}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 5.0) (fma (PI) l_m (* (/ (- (PI)) F) (/ l_m F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5

   1. Initial program 80.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   2. Add Preprocessing
   3. 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 80.1

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

   4. Applied rewrites 80.1%

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

   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. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if 5 < (*.f64 (PI.f64) l)

   1. Initial program 55.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 88.4%

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

Alternative 6: 92.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;t\_0 - \frac{t\_0}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))))
   (* l_s (if (<= t_0 5.0) (- t_0 (/ t_0 (* F F))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5

   1. Initial program 80.0%

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

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 89.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 84.9

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

   7. Applied rewrites 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 75.2

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

   9. Applied rewrites 75.2%

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

   if 5 < (*.f64 (PI.f64) l)

   1. Initial program 55.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 81.2%

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

Alternative 7: 91.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;t\_0 - \frac{\mathsf{PI}\left(\right)}{F \cdot F} \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5

   1. Initial program 80.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if 5 < (*.f64 (PI.f64) l)

   1. Initial program 55.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 81.2%

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

Alternative 8: 91.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5

   1. Initial program 80.0%

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

   3. Taylor expanded in 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 75.1

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

   5. Applied rewrites 75.1%

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

   if 5 < (*.f64 (PI.f64) l)

   1. Initial program 55.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 81.2%

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

Alternative 9: 73.2% 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 73.6%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
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 71.5

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

5. Applied rewrites 71.5%

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

6. Final simplification 71.5%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))