Lanczos kernel

Percentage Accurate: 98.0% → 97.8%

Time: 10.1s

Alternatives: 10

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: 98.0% 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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Final simplification 97.8%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 97.7%

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

5. Final simplification 97.7%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.0

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

4. Applied rewrites 97.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 97.7

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.7

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

6. Applied rewrites 97.7%

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

7. Final simplification 97.7%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

6. Applied rewrites 97.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(\left(tau \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(0.16666666666666666 \cdot x\right) - \frac{\frac{1}{\mathsf{PI}\left(\right)}}{x}\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (- (sin (* x (PI))))
  (- (* (* (* tau tau) (PI)) (* 0.16666666666666666 x)) (/ (/ 1.0 (PI)) x))))

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 97.6%

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

5. Taylor expanded in tau around 0

   \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left({tau}^{2} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) - \frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left({tau}^{2} \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} \cdot {tau}^{2}\right) \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot {tau}^{2}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{6} \cdot {tau}^{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} \cdot \left({tau}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot x + \left(\mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \left({tau}^{2} \cdot \mathsf{PI}\left(\right)\right), x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left({tau}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{6}}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left({tau}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{6}}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot {tau}^{2}\right)} \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot tau\right)}\right) \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot tau\right)} \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(tau \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right) \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(tau \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right) \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   15. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(tau \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot tau\right) \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{x \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   16. associate-/l/ N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{6}, x, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{\mathsf{PI}\left(\right)}}{x}}\right)\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

   17. distribute-frac-neg N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{1}{6}, x, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{\mathsf{PI}\left(\right)}\right)}{x}}\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

7. Applied rewrites 27.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot 0.16666666666666666, x, \frac{\frac{-1}{\mathsf{PI}\left(\right)}}{x}\right)} \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 78.2%

   \[\leadsto \left(\left(0.16666666666666666 \cdot x\right) \cdot \left(\left(tau \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{\frac{1}{\mathsf{PI}\left(\right)}}{x}}\right) \cdot \left(-\sin \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \]

10. Final simplification 78.2%

    \[\leadsto \left(-\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(\left(tau \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(0.16666666666666666 \cdot x\right) - \frac{\frac{1}{\mathsf{PI}\left(\right)}}{x}\right) \]
11. Add Preprocessing

Alternative 6: 70.9% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 70.4%

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

6. Final simplification 70.4%

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

Alternative 7: 70.7% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.0

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 70.1%

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

8. Final simplification 70.1%

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

Alternative 8: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(-0.16666666666666666 \cdot \mathsf{fma}\left(tau, tau, 1\right)\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (pow (* x (PI)) 2.0) (* -0.16666666666666666 (fma tau tau 1.0))) 1.0))

Derivation

1. Initial program 97.7%

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

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

6. Applied rewrites 97.7%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {x}^{2} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right)} + 1 \]

   3. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{6} \cdot {tau}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot {x}^{2} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   4. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{6} \cdot {tau}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {x}^{2}\right)} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {tau}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   6. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\frac{-1}{6} \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   7. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left({tau}^{2} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{-1}{6}} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   8. associate-*l* N/A

      \[\leadsto \left(\color{blue}{{tau}^{2} \cdot \left(\left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{-1}{6}\right)} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   9. *-commutative N/A

      \[\leadsto \left({tau}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}\right) + 1 \]

   10. associate-*l* N/A

       \[\leadsto \left({tau}^{2} \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \color{blue}{\frac{-1}{6} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {x}^{2}\right)}\right) + 1 \]

   11. *-commutative N/A

       \[\leadsto \left({tau}^{2} \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{-1}{6} \cdot \color{blue}{\left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) + 1 \]

   12. distribute-lft1-in N/A

       \[\leadsto \color{blue}{\left({tau}^{2} + 1\right) \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1 \]

9. Applied rewrites 62.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 42.2%

    \[\leadsto \left(\mathsf{fma}\left(tau, tau, 1\right) \cdot -0.16666666666666666\right) \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2} + \color{blue}{1} \]

12. Final simplification 42.1%

    \[\leadsto {\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \left(-0.16666666666666666 \cdot \mathsf{fma}\left(tau, tau, 1\right)\right) + 1 \]
13. Add Preprocessing

Alternative 9: 64.3% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in tau around 0

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
4. 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 63.7

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

5. Applied rewrites 63.7%

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

6. Final simplification 63.7%

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

Alternative 10: 63.5% 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 97.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.9%

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

Reproduce

herbie shell --seed 2024271 
(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)))))