VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.2% → 99.1%

Time: 8.0s

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: 76.2% 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: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 20000000000000:\\ \;\;\;\;t\_0 - {F}^{-1} \cdot \frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (PI) l_m)))
   (*
    l_s
    (if (<= t_0 20000000000000.0)
      (- t_0 (* (pow F -1.0) (/ (tan (* l_m (PI))) F)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e13

   1. Initial program 79.0%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 88.1

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 88.1

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

   4. Applied rewrites 88.1%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 60.5%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      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 2: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ t_1 := t\_0 - {\left(F \cdot F\right)}^{-1} \cdot \tan t\_0\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+162} \lor \neg \left(t\_1 \leq -2 \cdot 10^{-265}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-l\_m\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)) (t_1 (- t_0 (* (pow (* F F) -1.0) (tan t_0)))))
   (*
    l_s
    (if (or (<= t_1 -1e+162) (not (<= t_1 -2e-265)))
      t_0
      (* (- l_m) (/ (PI) (* F F)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -9.9999999999999994e161 or -1.99999999999999997e-265 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 67.3%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 70.9

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

   5. Applied rewrites 70.9%

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

   if -9.9999999999999994e161 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.99999999999999997e-265

   1. Initial program 96.8%

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

      1. lift-*.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. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 96.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

   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. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 95.5

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 24.3%

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

   12. Applied rewrites 24.5%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{PI}\left(\right) \cdot \ell - {\left(F \cdot F\right)}^{-1} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -1 \cdot 10^{+162} \lor \neg \left(\mathsf{PI}\left(\right) \cdot \ell - {\left(F \cdot F\right)}^{-1} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \leq -2 \cdot 10^{-265}\right):\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\mathsf{PI}\left(\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 20000000000000:\\ \;\;\;\;t\_0 - \frac{\frac{\tan \left(l\_m \cdot \mathsf{PI}\left(\right)\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 20000000000000.0)
      (- t_0 (/ (/ (tan (* l_m (PI))) F) F))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e13

   1. Initial program 79.0%

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

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

      4. *-lft-identity N/A

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

      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 88.1

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

      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 88.1

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

   4. Applied rewrites 88.1%

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 60.5%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 40000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot l\_m}{F}, {F}^{-1}, l\_m \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (PI) l_m)))
   (*
    l_s
    (if (<= t_0 40000000.0)
      (fma (/ (* (- (PI)) l_m) F) (pow F -1.0) (* l_m (PI)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e7

   1. Initial program 79.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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. 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. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. inv-pow N/A

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

      13. inv-pow N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in l around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-PI.f64 85.4

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

   7. Applied rewrites 85.4%

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

   if 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 96.0

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

   5. Applied rewrites 96.0%

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

Alternative 5: 98.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 40000000:\\ \;\;\;\;t\_0 - \frac{\frac{l\_m \cdot \mathsf{PI}\left(\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 40000000.0) (- t_0 (/ (/ (* l_m (PI)) F) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e7

   1. Initial program 79.0%

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

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

      4. *-lft-identity N/A

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

      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 88.2

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

      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 88.2

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

   4. Applied rewrites 88.2%

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

   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 85.4

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

   7. Applied rewrites 85.4%

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

   9. Applied rewrites 85.4%

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

   if 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 96.0

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

   5. Applied rewrites 96.0%

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

Alternative 6: 98.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 40000000:\\ \;\;\;\;t\_0 - \frac{\frac{\mathsf{PI}\left(\right)}{F} \cdot l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (PI) l_m)))
   (* l_s (if (<= t_0 40000000.0) (- t_0 (/ (* (/ (PI) F) l_m) F)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e7

   1. Initial program 79.0%

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

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

      4. *-lft-identity N/A

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

      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 88.2

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

      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 88.2

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

   4. Applied rewrites 88.2%

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

   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 85.4

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

   7. Applied rewrites 85.4%

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

   if 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 96.0

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

   5. Applied rewrites 96.0%

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

Alternative 7: 92.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot l\_m\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 40000000:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{F \cdot F}\right) \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (PI) l_m)))
   (* l_s (if (<= t_0 40000000.0) (* (- (PI) (/ (PI) (* F F))) l_m) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e7

   1. Initial program 79.0%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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 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 4e7 < (*.f64 (PI.f64) l)

   1. Initial program 61.1%

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

   3. Taylor expanded in F around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-PI.f64 96.0

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

   5. Applied rewrites 96.0%

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

Alternative 8: 73.9% 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 75.3%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
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 72.3

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

5. Applied rewrites 72.3%

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

Reproduce

herbie shell --seed 2024329 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))