Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 9.9s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_2) (/ (sin t_1) t_1))))

Initial Program: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_2) (/ (sin t_1) t_1))))

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ t_2 := tau \cdot t\_1\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin t\_2}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* tau t_1)))
   (* (/ (sin t_1) t_1) (/ (sin t_2) t_2))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\color{blue}{\frac{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

   6. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]

6. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\frac{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\frac{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\frac{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

8. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]
9. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ t_2 := tau \cdot t\_1\\ \frac{\sin t\_1 \cdot \sin t\_2}{t\_1 \cdot t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* tau t_1)))
   (/ (* (sin t_1) (sin t_2)) (* t_1 t_2))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\color{blue}{\frac{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

   6. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x}} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

6. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\color{blue}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{x \cdot \mathsf{PI}\left(\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right)}} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \color{blue}{\left(\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)\right) \cdot tau}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\color{blue}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)} \cdot \left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)\right) \cdot tau} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\color{blue}{\left(x \cdot \left(-\mathsf{PI}\left(\right)\right)\right)} \cdot \left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)\right) \cdot tau} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(x \cdot \left(-\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)}\right) \cdot tau} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(x \cdot \left(-\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(x \cdot \left(-\mathsf{PI}\left(\right)\right)\right)}\right) \cdot tau} \]

   8. swap-sqr N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(-\mathsf{PI}\left(\right)\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)} \cdot tau} \]

   9. lift-neg.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)} \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right) \cdot tau} \]

   10. lift-neg.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(x \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot tau} \]

   11. sqr-neg N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot tau} \]

   12. swap-sqr N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot tau} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot tau} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot tau} \]

   15. associate-*r* N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   16. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   17. lower-*.f32 97.7

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

10. Applied rewrites 97.7%

    \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]
11. Add Preprocessing

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ \frac{\sin t\_1 \cdot \sin \left(tau \cdot t\_1\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)))
   (/ (* (sin t_1) (sin (* tau t_1))) (* (* tau (* x x)) (* (PI) (PI))))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\color{blue}{\frac{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

   6. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x}} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

6. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\color{blue}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{x \cdot \mathsf{PI}\left(\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \left(\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{tau \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(tau \cdot {x}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}}} \]

    2. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(tau \cdot {x}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}}} \]

    3. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(tau \cdot {x}^{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{2}} \]

    4. unpow2 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} \]

    5. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} \]

    6. unpow2 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

    7. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

    8. lower-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)} \]

    9. lower-PI.f32 96.8

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

11. Applied rewrites 96.8%

    \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{\left(tau \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}} \]
12. Add Preprocessing

Alternative 4: 70.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ \frac{1}{t\_1} \cdot \left(\frac{1}{tau} \cdot \sin \left(tau \cdot t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (* (/ 1.0 t_1) (* (/ 1.0 tau) (sin (* tau t_1))))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\left(\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right)} \]

5. Taylor expanded in x around 0

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

   1. lower-/.f32 69.9

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{1}{tau}} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right) \]

7. Applied rewrites 69.9%

   \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{1}{tau}} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right) \]
8. Add Preprocessing

Alternative 5: 70.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{PI}\left(\right) \cdot tau} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)}{x \cdot x} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (/ x (* (PI) tau)) (/ (sin (* (* (PI) x) tau)) (* x x))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. frac-2neg N/A

      \[\leadsto \frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\color{blue}{\frac{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

   6. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\mathsf{neg}\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}}{tau \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}{tau \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {x}^{2}\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{\left(tau \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}}} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot {\mathsf{PI}\left(\right)}^{2}} \cdot \frac{\sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}{{x}^{2}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot {\mathsf{PI}\left(\right)}^{2}} \cdot \frac{\sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}{{x}^{2}}} \]

7. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)}{x \cdot x}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{x}{tau \cdot \mathsf{PI}\left(\right)} \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)}}{x \cdot x} \]
9. Step-by-step derivation

10. Applied rewrites 69.9%

    \[\leadsto \frac{x}{\mathsf{PI}\left(\right) \cdot tau} \cdot \frac{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)}}{x \cdot x} \]
11. Add Preprocessing

Alternative 6: 64.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ \frac{1}{\frac{1}{\frac{\sin t\_1}{t\_1}}} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (/ 1.0 (/ 1.0 (/ (sin t_1) t_1)))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\left(\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right)} \]

5. Taylor expanded in tau around 0

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

   1. lower-sin.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   4. lower-PI.f32 63.1

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \]

7. Applied rewrites 63.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\mathsf{PI}\left(\right) \cdot x} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

9. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]
10. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

    2. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}}}} \]

    3. lower-/.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}}}} \]

    4. lift-/.f32 N/A

       \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}{x}}} \]

    5. associate-/l/ N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{x \cdot \mathsf{PI}\left(\right)}}}} \]

    6. lift-*.f32 N/A

       \[\leadsto \frac{1}{\frac{1}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}}}} \]

11. Applied rewrites 63.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}}}} \]
12. Add Preprocessing

Alternative 7: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ \frac{1}{\frac{t\_1}{\sin t\_1}} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (/ 1.0 (/ t_1 (sin t_1)))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\left(\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right)} \]

5. Taylor expanded in tau around 0

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

   1. lower-sin.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   4. lower-PI.f32 63.1

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \]

7. Applied rewrites 63.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

9. Applied rewrites 63.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
10. Add Preprocessing

Alternative 8: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (/ 1.0 (* (/ x (sin (* (PI) x))) (PI))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\left(\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right)} \]

5. Taylor expanded in tau around 0

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

   1. lower-sin.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   4. lower-PI.f32 63.1

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \]

7. Applied rewrites 63.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\mathsf{PI}\left(\right) \cdot x} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

9. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]
10. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

    2. lift-/.f32 N/A

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}}} \]

    3. associate-/r/ N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \mathsf{PI}\left(\right)}} \]

    4. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \mathsf{PI}\left(\right)}} \]

    5. lower-/.f32 63.1

       \[\leadsto \frac{1}{\color{blue}{\frac{x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \mathsf{PI}\left(\right)} \]

11. Applied rewrites 63.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \mathsf{PI}\left(\right)}} \]
12. Add Preprocessing

Alternative 9: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x} \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (/ (/ (sin (* (PI) x)) (PI)) x))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\left(\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right)} \]

5. Taylor expanded in tau around 0

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

   1. lower-sin.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   4. lower-PI.f32 63.1

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \]

7. Applied rewrites 63.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\mathsf{PI}\left(\right) \cdot x} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}} \]

9. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right)}}{x}} \]
10. Add Preprocessing

Alternative 10: 64.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (/ (sin t_1) t_1)))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \cdot \left(\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\left(\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \]

4. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)\right)} \]

5. Taylor expanded in tau around 0

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

   1. lower-sin.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   4. lower-PI.f32 63.1

      \[\leadsto \frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \]

7. Applied rewrites 63.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot x}} \]

   3. lift-/.f32 N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \color{blue}{\frac{1}{\mathsf{PI}\left(\right) \cdot x}} \]

   4. lift-*.f32 N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \]

   6. lift-*.f32 N/A

      \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \frac{1}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \]

   7. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   8. lower-/.f32 63.1

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

   11. lift-*.f32 63.1

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

9. Applied rewrites 63.1%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]
10. Add Preprocessing

Alternative 11: 63.2% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

float code(float x, float tau) {
	return 1.0f;
}

Fortran

real(4) function code(x, tau)
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0
end function

Julia

function code(x, tau)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0);
end

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 62.4%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 62.4%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024325 
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0)) (and (<= 1.0 tau) (<= tau 5.0)))
  (* (/ (sin (* (* x (PI)) tau)) (* (* x (PI)) tau)) (/ (sin (* x (PI))) (* x (PI)))))