Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 10.4s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* t_1 tau)))
   (* (/ (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. Final simplification 97.7%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

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

6. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. frac-times N/A

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

   8. frac-2neg N/A

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

8. Applied rewrites 97.5%

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

9. Final simplification 97.5%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)))
   (/ (* (sin (* t_1 tau)) (sin t_1)) (* (* (* tau x) (PI)) 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. 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. frac-2neg N/A

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

   11. lift-*.f32 N/A

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

6. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. lift-*.f32 N/A

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

   4. frac-times N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. frac-2neg N/A

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

8. Applied rewrites 97.1%

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

9. Final simplification 97.1%

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

Alternative 4: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

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. frac-2neg N/A

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

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

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

4. Applied rewrites 97.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 96.9

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

6. Applied rewrites 96.9%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 97.0

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

8. Applied rewrites 96.8%

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

9. Final simplification 96.8%

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

Alternative 5: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

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. frac-2neg N/A

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

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

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

4. Applied rewrites 97.0%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 96.9

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

6. Applied rewrites 96.6%

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

7. Final simplification 96.6%

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

Alternative 6: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

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. frac-2neg N/A

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

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

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

4. Applied rewrites 97.0%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. lower-/.f32 N/A

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

6. Applied rewrites 96.5%

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

7. Final simplification 96.5%

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

Alternative 7: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ \frac{1 + \left({t\_1}^{2} \cdot -0.16666666666666666\right) \cdot \left(tau \cdot tau\right)}{t\_1} \cdot \sin t\_1 \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)))
   (*
    (/ (+ 1.0 (* (* (pow t_1 2.0) -0.16666666666666666) (* tau tau))) t_1)
    (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. 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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f32 N/A

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

7. Applied rewrites 26.9%

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

9. Applied rewrites 76.0%

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

10. Final simplification 76.0%

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

Alternative 8: 70.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* (PI) x) tau))) (/ (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. 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. frac-2neg N/A

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

   11. lift-*.f32 N/A

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 68.5

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

9. Applied rewrites 68.5%

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

10. Final simplification 68.5%

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

Alternative 9: 64.4% accurate, 2.2× speedup

Math

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

FPCore

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

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

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 60.8%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 60.8%

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

10. Applied rewrites 61.8%

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

11. Final simplification 61.8%

    \[\leadsto 1 + {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2} \cdot -0.16666666666666666 \]
12. 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 60.8%

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

Reproduce

herbie shell --seed 2024270 
(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)))))