Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 6.2s

Alternatives: 8

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

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

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))))

Derivation

1. Initial program 98.3%

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

Alternative 2: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\\ \mathbf{if}\;x \leq 0.03400000184774399:\\ \;\;\;\;\frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(x, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot x\right)}{\left(\left(-x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x (PI)) tau)))
   (if (<= x 0.03400000184774399)
     (*
      (/ (sin t_1) t_1)
      (fma (* (* (PI) (PI)) -0.16666666666666666) (* x x) 1.0))
     (/
      (* (sin (fma x (PI) (PI))) (sin (* (* tau (PI)) x)))
      (* (* (- x) (PI)) (* (* (PI) x) tau))))))

Derivation

1. Split input into 2 regimes

2. if x < 0.0340000018

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.4%

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

   if 0.0340000018 < x

   1. Initial program 96.9%

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

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f32 N/A

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

      7. frac-2neg N/A

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

      8. frac-times N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. remove-double-neg N/A

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

   4. Applied rewrites 93.0%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f32 N/A

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

   6. Applied rewrites 97.0%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. frac-2neg N/A

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

      4. lift-*.f32 N/A

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

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

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

      6. lift-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f32 N/A

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

   8. Applied rewrites 97.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-neg.f32 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. sin-neg-rev N/A

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

      10. sin-+PI N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-fma.f32 N/A

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

      14. lift-sin.f32 N/A

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

      15. lower-*.f32 94.0

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

   10. Applied rewrites 93.6%

       \[\leadsto \frac{\color{blue}{\sin \left(\mathsf{fma}\left(x, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot x\right)}}{\left(\left(-x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. lift-/.f32 N/A

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

   7. frac-2neg N/A

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

   8. frac-times N/A

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

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

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

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

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. remove-double-neg N/A

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

4. Applied rewrites 77.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. lift-*.f32 N/A

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

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

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f32 N/A

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

8. Applied rewrites 98.2%

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

9. Final simplification 98.2%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. lift-/.f32 N/A

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

   7. frac-2neg N/A

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

   8. frac-times N/A

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

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

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

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

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. remove-double-neg N/A

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

4. Applied rewrites 77.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. lift-*.f32 N/A

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

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

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f32 N/A

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

8. Applied rewrites 98.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 97.5

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.5

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

10. Applied rewrites 97.5%

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

11. Final simplification 97.5%

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

Alternative 5: 84.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x (PI)) tau)))
   (*
    (/ (sin t_1) t_1)
    (fma (* (* (PI) (PI)) -0.16666666666666666) (* x x) 1.0))))

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 86.5%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, x \cdot x, 1\right)} \]
6. Add Preprocessing

Alternative 6: 79.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)))
   (*
    (sin t_1)
    (/
     (fma (* -0.16666666666666666 (* tau tau)) (* (* (PI) (PI)) (* x x)) 1.0)
     t_1))))

Derivation

1. Initial program 98.3%

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

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

   3. lift-/.f32 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 79.6%

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

Alternative 7: 78.3% accurate, 7.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

5. Applied rewrites 78.9%

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

Alternative 8: 63.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025019 
(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)))))