VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 82.7%

Time: 9.9s

Alternatives: 7

Speedup: 1.0×

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: 76.9% 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: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := l\_m \cdot \mathsf{PI}\left(\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;t\_0 - \frac{l\_m}{F} \cdot \frac{\mathsf{PI}\left(\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{\tan t\_0}{F \cdot F}\\ \end{array} \end{array} \end{array} \]

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 (* l_m (PI))))
   (*
    l_s
    (if (<= t_0 1e-13)
      (- t_0 (* (/ l_m F) (/ (PI) F)))
      (- t_0 (/ (tan t_0) (* F F)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e-13

   1. Initial program 77.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{PI}\left(\right) \cdot \ell - \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 83.9%

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

   if 1e-13 < (*.f64 (PI.f64) l)

   1. Initial program 67.9%

      \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
   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. 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. lower-/.f64 67.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.9

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

   4. Applied rewrites 67.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \mathsf{PI}\left(\right) \leq 10^{-13}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right) - \frac{\ell}{F} \cdot \frac{\mathsf{PI}\left(\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \mathsf{PI}\left(\right) - \frac{\tan \left(\ell \cdot \mathsf{PI}\left(\right)\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.6× speedup

Math

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

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 (- t_0 (/ (pow F -1.0) (* (/ 1.0 (tan t_0)) F))))))

Derivation

1. Initial program 75.1%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
2. Add Preprocessing
3. 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. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. frac-times N/A

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

4. Applied rewrites 81.6%

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

5. Final simplification 81.6%

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

Alternative 3: 82.7% accurate, 1.0× speedup

Math

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

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 (- t_0 (/ (/ (tan t_0) F) F)))))

Derivation

1. Initial program 75.1%

   \[\mathsf{PI}\left(\right) \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \]
2. Add Preprocessing
3. 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 81.6

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

   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 81.6

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

4. Applied rewrites 81.6%

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

5. Final simplification 81.6%

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

Alternative 4: 76.0% accurate, 2.9× speedup

Math

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

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 (* l_m (PI))))
   (*
    l_s
    (if (<= t_0 5e-10)
      (- t_0 (* (/ l_m F) (/ (PI) F)))
      (- t_0 (* (- l_m) (/ (PI) (* F F))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5.00000000000000031e-10

   1. Initial program 78.3%

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

   3. Taylor expanded in 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 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 84.1%

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

   if 5.00000000000000031e-10 < (*.f64 (PI.f64) l)

   1. Initial program 66.5%

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

   3. Taylor expanded in 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 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 51.0%

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

4. Final simplification 75.1%

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

Alternative 5: 70.5% accurate, 3.1× speedup

Math

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

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 (* l_m (PI))))
   (*
    l_s
    (if (<= t_0 5e-10)
      (- t_0 (/ t_0 (* F F)))
      (- t_0 (* (- l_m) (/ (PI) (* F F))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5.00000000000000031e-10

   1. Initial program 78.3%

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

   3. Taylor expanded in 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 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.7%

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

   if 5.00000000000000031e-10 < (*.f64 (PI.f64) l)

   1. Initial program 66.5%

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

   3. Taylor expanded in 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 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 51.0%

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

4. Final simplification 68.9%

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

Alternative 6: 69.7% accurate, 4.5× speedup

Math

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

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

Derivation

1. Initial program 75.1%

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

3. Taylor expanded in 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 67.4

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

5. Applied rewrites 67.4%

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

7. Applied rewrites 67.7%

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

8. Final simplification 67.7%

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

Alternative 7: 69.7% accurate, 4.5× speedup

Math

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

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)) (* (/ l_m (* F F)) (PI)))))

Derivation

1. Initial program 75.1%

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

3. Taylor expanded in 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 67.4

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

5. Applied rewrites 67.4%

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

7. Applied rewrites 49.7%

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

9. Applied rewrites 67.7%

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

10. Final simplification 67.7%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* (PI) l) (* (/ 1.0 (* F F)) (tan (* (PI) l)))))