Lanczos kernel

Percentage Accurate: 97.9% → 97.7%

Time: 9.7s

Alternatives: 9

Speedup: N/A×

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: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

6. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. div-inv N/A

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

   12. lift-/.f32 N/A

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

8. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-/.f32 N/A

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

10. Applied rewrites 98.0%

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

11. Final simplification 98.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

6. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. div-inv N/A

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

   12. lift-/.f32 N/A

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

8. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. frac-2neg N/A

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

10. Applied rewrites 97.9%

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

11. Final simplification 97.9%

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

Alternative 3: 71.1% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f32 N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-*l* N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 70.8

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

7. Applied rewrites 70.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 70.7

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

9. Applied rewrites 70.7%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. frac-times N/A

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

    5. lift-*.f32 N/A

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

    6. associate-*r* N/A

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

    7. lift-sqrt.f32 N/A

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

    8. lift-sqrt.f32 N/A

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

    9. rem-square-sqrt N/A

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

    10. lift-PI.f32 N/A

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

11. Applied rewrites 71.1%

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

12. Final simplification 71.1%

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

Alternative 4: 71.1% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

6. Applied rewrites 97.9%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 71.1

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

9. Applied rewrites 71.1%

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

10. Final simplification 71.1%

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

Alternative 5: 71.0% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f32 N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-*l* N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 70.8

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

7. Applied rewrites 70.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 70.7

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

9. Applied rewrites 70.7%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. frac-times N/A

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

11. Applied rewrites 71.0%

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

12. Final simplification 71.0%

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

Alternative 6: 71.0% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

6. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. div-inv N/A

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

   12. lift-/.f32 N/A

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in x around 0

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

    1. lower-/.f32 70.9

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

11. Applied rewrites 70.9%

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

12. Final simplification 70.9%

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

Alternative 7: 64.5% 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 97.8%

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

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

5. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 64.4

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

7. Applied rewrites 64.4%

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

Alternative 8: 63.7% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. frac-2neg N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

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

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-*.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-PI.f32 63.6

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

7. Applied rewrites 63.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 63.6%

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

10. Final simplification 63.6%

    \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{\frac{\left(-x\right) \cdot \mathsf{PI}\left(\right)}{x}}} \]
11. Add Preprocessing

Alternative 9: 63.7% 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 97.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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)))))