Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 9.3s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

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: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 97.5%

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

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

6. Applied rewrites 97.5%

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

8. Applied rewrites 97.3%

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

9. Final simplification 97.3%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)))
   (* (/ (sin t_1) (* tau x)) (/ (sin (* t_1 tau)) (* t_1 (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. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. unswap-sqr N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. times-frac N/A

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

   9. times-frac N/A

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

   10. lower-*.f32 N/A

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

5. Applied rewrites 96.9%

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

6. Final simplification 96.9%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)))
   (/ (* (sin (* (* tau (PI)) x)) (sin t_1)) (* (* t_1 tau) 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 97.5%

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

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

6. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-/l/ N/A

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

   5. associate-*l/ N/A

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

   6. frac-times N/A

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

   7. frac-2neg N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

8. Applied rewrites 96.9%

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

9. Final simplification 96.9%

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

Alternative 5: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

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

   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 97.5%

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

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

   3. associate-/l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

6. Applied rewrites 97.5%

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

8. Applied rewrites 96.7%

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

9. Final simplification 96.7%

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

Alternative 6: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

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

   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 97.5%

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-PI.f32 96.6

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

9. Applied rewrites 96.6%

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

10. Final simplification 96.6%

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

Alternative 7: 70.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* t_1 tau))) (* (/ (/ (sin t_2) t_2) 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 97.5%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 68.5

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

7. Applied rewrites 68.5%

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

8. Final simplification 68.5%

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

Alternative 8: 70.9% accurate, 1.6× speedup

Math

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

FPCore

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

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

   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 97.5%

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 68.4

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

9. Applied rewrites 68.4%

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

10. Final simplification 68.4%

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

Alternative 9: 70.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (/ (* (/ 1.0 tau) (sin (* t_1 tau))) 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. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f32 68.3

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

7. Applied rewrites 68.3%

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

8. Final simplification 68.3%

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

Alternative 10: 70.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (* (/ 1.0 (* (* (PI) x) tau)) (sin (* (* tau (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 97.5%

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 68.2

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

9. Applied rewrites 68.2%

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

10. Final simplification 68.2%

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

Alternative 11: 64.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (/ (* (sin t_1) tau) (* 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. 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 97.5%

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

5. Taylor expanded in tau around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 61.2

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

7. Applied rewrites 61.2%

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

8. Final simplification 61.2%

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

Alternative 12: 64.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (* (- (sin t_1)) (/ -1.0 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 97.5%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 61.2

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

7. Applied rewrites 61.2%

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

8. Final simplification 61.2%

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

Alternative 13: 64.5% accurate, 10.8× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (+ 1.0 (* (* -0.16666666666666666 (* (PI) (PI))) (* x 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. 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 60.3%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 59.8%

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

10. Applied rewrites 61.1%

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

11. Final simplification 61.1%

    \[\leadsto 1 + \left(-0.16666666666666666 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(x \cdot x\right) \]
12. Add Preprocessing

Alternative 14: 63.6% 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 60.4%

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

Reproduce

herbie shell --seed 2024295 
(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)))))