Lanczos kernel

Percentage Accurate: 97.9% → 98.2%

Time: 6.3s

Alternatives: 12

Speedup: 0.5×

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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ t_3 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \leq 0.9850000143051147:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(x \cdot tau, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(\left(-x\right) \cdot x\right) \cdot \left(t\_3 \cdot tau\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), t\_3 \cdot -0.16666666666666666, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(0.027777777777777776 \cdot tau, tau, \mathsf{fma}\left(tau \cdot tau, \left(tau \cdot tau\right) \cdot 0.008333333333333333, 0.008333333333333333\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)) (t_3 (* (PI) (PI))))
   (if (<= (* (/ (sin t_2) t_2) (/ (sin t_1) t_1)) 0.9850000143051147)
     (/
      (* (sin (fma (* x tau) (PI) (PI))) (sin (* (PI) x)))
      (* (* (- x) x) (* t_3 tau)))
     (fma
      (fma
       (fma tau tau 1.0)
       (* t_3 -0.16666666666666666)
       (*
        (*
         (pow (PI) 4.0)
         (fma
          (* 0.027777777777777776 tau)
          tau
          (fma
           (* tau tau)
           (* (* tau tau) 0.008333333333333333)
           0.008333333333333333)))
        (* x x)))
      (* x x)
      1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (/.f32 (sin.f32 (*.f32 (*.f32 x (PI.f32)) tau)) (*.f32 (*.f32 x (PI.f32)) tau)) (/.f32 (sin.f32 (*.f32 x (PI.f32))) (*.f32 x (PI.f32)))) < 0.985000014

   1. Initial program 96.2%

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f32 N/A

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

   5. Applied rewrites 95.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. frac-times N/A

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

      10. lower-/.f32 N/A

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

   7. Applied rewrites 94.6%

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

   if 0.985000014 < (*.f32 (/.f32 (sin.f32 (*.f32 (*.f32 x (PI.f32)) tau)) (*.f32 (*.f32 x (PI.f32)) tau)) (/.f32 (sin.f32 (*.f32 x (PI.f32))) (*.f32 x (PI.f32))))

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(0.027777777777777776 \cdot tau, tau, \mathsf{fma}\left({tau}^{4}, 0.008333333333333333, 0.008333333333333333\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right)} \]
   6. Step-by-step derivation

      1. lift-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, {tau}^{4} \cdot \frac{1}{120} + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, {tau}^{4} \cdot \frac{1}{120} + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, {tau}^{\left(2 \cdot 2\right)} \cdot \frac{1}{120} + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      4. pow-pow N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, {\left({tau}^{2}\right)}^{2} \cdot \frac{1}{120} + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, {\left(tau \cdot tau\right)}^{2} \cdot \frac{1}{120} + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, \left(\left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right)\right) \cdot \frac{1}{120} + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, \left(tau \cdot tau\right) \cdot \left(\left(tau \cdot tau\right) \cdot \frac{1}{120}\right) + \frac{1}{120}\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, \mathsf{fma}\left(tau \cdot tau, \left(tau \cdot tau\right) \cdot \frac{1}{120}, \frac{1}{120}\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, \mathsf{fma}\left(tau \cdot tau, \left(tau \cdot tau\right) \cdot \frac{1}{120}, \frac{1}{120}\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{6}, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(\frac{1}{36} \cdot tau, tau, \mathsf{fma}\left(tau \cdot tau, \left(tau \cdot tau\right) \cdot \frac{1}{120}, \frac{1}{120}\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

      11. lower-*.f32 99.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(0.027777777777777776 \cdot tau, tau, \mathsf{fma}\left(tau \cdot tau, \left(tau \cdot tau\right) \cdot 0.008333333333333333, 0.008333333333333333\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]

   7. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(0.027777777777777776 \cdot tau, tau, \mathsf{fma}\left(tau \cdot tau, \left(tau \cdot tau\right) \cdot 0.008333333333333333, 0.008333333333333333\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sin.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

7. Applied rewrites 98.0%

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

8. Final simplification 98.0%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.6%

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

7. Final simplification 97.6%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.1%

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

7. Final simplification 97.1%

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

Alternative 5: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

3. 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. *-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.7%

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

8. Final simplification 96.7%

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

Alternative 6: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

3. 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. *-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in x around inf

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

   1. lower-sin.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 96.7

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

10. Applied rewrites 96.7%

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

11. Final simplification 96.7%

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

Alternative 7: 85.1% 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 97.9%

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

3. 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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 82.8

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

5. Applied rewrites 82.8%

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

Alternative 8: 85.1% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sin.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

7. Applied rewrites 98.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 82.8

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

10. Applied rewrites 82.8%

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

11. Final simplification 82.8%

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

Alternative 9: 79.1% accurate, 4.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

5. Applied rewrites 78.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 77.6

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

8. Applied rewrites 77.6%

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

Alternative 10: 78.4% 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 97.9%

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

3. 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

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 77.3%

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

Alternative 11: 78.4% accurate, 7.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

5. Applied rewrites 82.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, \left({\mathsf{PI}\left(\right)}^{4} \cdot \mathsf{fma}\left(0.027777777777777776 \cdot tau, tau, \mathsf{fma}\left({tau}^{4}, 0.008333333333333333, 0.008333333333333333\right)\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 77.3%

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

Alternative 12: 63.1% 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 97.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.8%

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

Reproduce

herbie shell --seed 2025026 
(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)))))