Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.6s

Alternatives: 19

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

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 98.3%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return sinf(t_1) * (sinf((x * ((float) M_PI))) / ((x * ((float) M_PI)) * t_1));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(sin(t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(Float32(x * Float32(pi)) * t_1)))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = sin(t_1) * (sin((x * single(pi))) / ((x * single(pi)) * t_1));
end

Derivation

1. Initial program 98.3%

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

   1. associate-*r/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.1%

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

   1. associate-/l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sin-lowering-sin.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. sin-lowering-sin.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

6. Applied egg-rr 98.1%

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

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

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

   2. associate-*l* N/A

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

   3. sqr-neg N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. unswap-sqr N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. *-lowering-*.f32 N/A

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

8. Applied egg-rr 98.1%

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

9. Final simplification 98.1%

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

Alternative 3: 90.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{tau}, -0.16666666666666666 \cdot \frac{\pi}{tau}\right), \frac{1}{\pi \cdot tau}\right)}{x} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (sin (* x (* PI tau)))
  (/
   (fma
    (* x x)
    (fma
     (* x x)
     (/ (* 0.008333333333333333 (* PI (* PI PI))) tau)
     (* -0.16666666666666666 (/ PI tau)))
    (/ 1.0 (* PI tau)))
   x)))

C

float code(float x, float tau) {
	return sinf((x * (((float) M_PI) * tau))) * (fmaf((x * x), fmaf((x * x), ((0.008333333333333333f * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) / tau), (-0.16666666666666666f * (((float) M_PI) / tau))), (1.0f / (((float) M_PI) * tau))) / x);
}

Julia

function code(x, tau)
	return Float32(sin(Float32(x * Float32(Float32(pi) * tau))) * Float32(fma(Float32(x * x), fma(Float32(x * x), Float32(Float32(Float32(0.008333333333333333) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) / tau), Float32(Float32(-0.16666666666666666) * Float32(Float32(pi) / tau))), Float32(Float32(1.0) / Float32(Float32(pi) * tau))) / x))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. sin-lowering-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. associate-/r* N/A

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

   14. associate-/l/ N/A

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

   15. /-lowering-/.f32 N/A

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

5. Simplified 97.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]

8. Simplified 90.8%

   \[\leadsto \sin \left(x \cdot \left(tau \cdot \pi\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{tau}, -0.16666666666666666 \cdot \frac{\pi}{tau}\right), \frac{1}{tau \cdot \pi}\right)}{x}} \]

9. Final simplification 90.8%

   \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{tau}, -0.16666666666666666 \cdot \frac{\pi}{tau}\right), \frac{1}{\pi \cdot tau}\right)}{x} \]
10. Add Preprocessing

Alternative 4: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{tau}, \frac{\pi \cdot -0.16666666666666666}{tau}\right), \frac{1}{\pi \cdot tau}\right)}{x} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (sin (* (* x PI) tau))
  (/
   (fma
    (* x x)
    (fma
     (* x x)
     (/ (* 0.008333333333333333 (* PI (* PI PI))) tau)
     (/ (* PI -0.16666666666666666) tau))
    (/ 1.0 (* PI tau)))
   x)))

C

float code(float x, float tau) {
	return sinf(((x * ((float) M_PI)) * tau)) * (fmaf((x * x), fmaf((x * x), ((0.008333333333333333f * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) / tau), ((((float) M_PI) * -0.16666666666666666f) / tau)), (1.0f / (((float) M_PI) * tau))) / x);
}

Julia

function code(x, tau)
	return Float32(sin(Float32(Float32(x * Float32(pi)) * tau)) * Float32(fma(Float32(x * x), fma(Float32(x * x), Float32(Float32(Float32(0.008333333333333333) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) / tau), Float32(Float32(Float32(pi) * Float32(-0.16666666666666666)) / tau)), Float32(Float32(1.0) / Float32(Float32(pi) * tau))) / x))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*r/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.1%

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

   1. associate-/l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sin-lowering-sin.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. sin-lowering-sin.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in x around 0

   \[\leadsto \sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]

9. Simplified 90.4%

   \[\leadsto \sin \left(tau \cdot \left(x \cdot \pi\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{tau}, \frac{\pi \cdot -0.16666666666666666}{tau}\right), \frac{1}{tau \cdot \pi}\right)}{x}} \]

10. Final simplification 90.4%

    \[\leadsto \sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{tau}, \frac{\pi \cdot -0.16666666666666666}{tau}\right), \frac{1}{\pi \cdot tau}\right)}{x} \]
11. Add Preprocessing

Alternative 5: 85.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * fmaf((((float) M_PI) * ((float) M_PI)), ((x * x) * -0.16666666666666666f), 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(x * x) * Float32(-0.16666666666666666)), Float32(1.0)))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

Alternative 6: 85.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return sinf(t_1) * (fmaf(((float) M_PI), (((float) M_PI) * ((x * x) * -0.16666666666666666f)), 1.0f) / t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(sin(t_1) * Float32(fma(Float32(pi), Float32(Float32(pi) * Float32(Float32(x * x) * Float32(-0.16666666666666666))), Float32(1.0)) / t_1))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sin-lowering-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. PI-lowering-PI.f32 N/A

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

   9. /-lowering-/.f32 N/A

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

7. Applied egg-rr 84.3%

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

8. Final simplification 84.3%

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

Alternative 7: 84.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, t\_1, t\_1 \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \left(tau \cdot tau\right) \cdot 0.027777777777777776\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* PI PI) (* PI PI))))
   (fma
    (* x x)
    (fma
     (* x x)
     (fma
      0.008333333333333333
      t_1
      (*
       t_1
       (fma
        0.008333333333333333
        (* (* tau tau) (* tau tau))
        (* (* tau tau) 0.027777777777777776))))
     (* (* PI PI) (fma -0.16666666666666666 (* tau tau) -0.16666666666666666)))
    1.0)))

C

float code(float x, float tau) {
	float t_1 = (((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI));
	return fmaf((x * x), fmaf((x * x), fmaf(0.008333333333333333f, t_1, (t_1 * fmaf(0.008333333333333333f, ((tau * tau) * (tau * tau)), ((tau * tau) * 0.027777777777777776f)))), ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi)))
	return fma(Float32(x * x), fma(Float32(x * x), fma(Float32(0.008333333333333333), t_1, Float32(t_1 * fma(Float32(0.008333333333333333), Float32(Float32(tau * tau) * Float32(tau * tau)), Float32(Float32(tau * tau) * Float32(0.027777777777777776))))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*r/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.1%

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

   1. associate-/l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sin-lowering-sin.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. sin-lowering-sin.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

6. Applied egg-rr 98.1%

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

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

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

   2. associate-*l* N/A

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

   3. sqr-neg N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. unswap-sqr N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. *-lowering-*.f32 N/A

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

8. Applied egg-rr 98.1%

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

9. Taylor expanded in x around 0

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

11. Simplified 83.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.008333333333333333, \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \left(tau \cdot tau\right) \cdot 0.027777777777777776\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]
12. Add Preprocessing

Alternative 8: 83.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi \cdot \pi, \left(x \cdot x\right) \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right), 0.008333333333333333 \cdot \left(\left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right)\right), \left(tau \cdot tau\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (fma (* PI PI) (* (* x x) -0.16666666666666666) 1.0)
  (fma
   (* x x)
   (fma
    (* x (* x (* (* PI PI) (* PI PI))))
    (* 0.008333333333333333 (* (* tau tau) (* tau tau)))
    (* (* tau tau) (* (* PI PI) -0.16666666666666666)))
   1.0)))

C

float code(float x, float tau) {
	return fmaf((((float) M_PI) * ((float) M_PI)), ((x * x) * -0.16666666666666666f), 1.0f) * fmaf((x * x), fmaf((x * (x * ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))))), (0.008333333333333333f * ((tau * tau) * (tau * tau))), ((tau * tau) * ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return Float32(fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(x * x) * Float32(-0.16666666666666666)), Float32(1.0)) * fma(Float32(x * x), fma(Float32(x * Float32(x * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(0.008333333333333333) * Float32(Float32(tau * tau) * Float32(tau * tau))), Float32(Float32(tau * tau) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666)))), Float32(1.0)))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 83.6

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

7. Applied egg-rr 83.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{120} \cdot \left({tau}^{4} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \left(x \cdot x\right) \cdot \frac{-1}{6}, 1\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

10. Simplified 83.2%

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

11. Final simplification 83.2%

    \[\leadsto \mathsf{fma}\left(\pi \cdot \pi, \left(x \cdot x\right) \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right), 0.008333333333333333 \cdot \left(\left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right)\right), \left(tau \cdot tau\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right)\right), 1\right) \]
12. Add Preprocessing

Alternative 9: 83.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi \cdot \pi, \left(x \cdot x\right) \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(tau \cdot tau, \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), x \cdot \left(x \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right)\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (fma (* PI PI) (* (* x x) -0.16666666666666666) 1.0)
  (fma
   (* tau tau)
   (fma
    (* tau tau)
    (* (* (* PI PI) (* PI PI)) (* 0.008333333333333333 (* (* x x) (* x x))))
    (* x (* x (* (* PI PI) -0.16666666666666666))))
   1.0)))

C

float code(float x, float tau) {
	return fmaf((((float) M_PI) * ((float) M_PI)), ((x * x) * -0.16666666666666666f), 1.0f) * fmaf((tau * tau), fmaf((tau * tau), (((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * (0.008333333333333333f * ((x * x) * (x * x)))), (x * (x * ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f)))), 1.0f);
}

Julia

function code(x, tau)
	return Float32(fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(x * x) * Float32(-0.16666666666666666)), Float32(1.0)) * fma(Float32(tau * tau), fma(Float32(tau * tau), Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(Float32(0.008333333333333333) * Float32(Float32(x * x) * Float32(x * x)))), Float32(x * Float32(x * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))))), Float32(1.0)))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 83.6

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

7. Applied egg-rr 83.6%

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

8. Taylor expanded in tau around 0

   \[\leadsto \color{blue}{\left(1 + {tau}^{2} \cdot \left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{120} \cdot \left({tau}^{2} \cdot \left({x}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \left(x \cdot x\right) \cdot \frac{-1}{6}, 1\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

10. Simplified 83.2%

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

11. Final simplification 83.2%

    \[\leadsto \mathsf{fma}\left(\pi \cdot \pi, \left(x \cdot x\right) \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(tau \cdot tau, \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(0.008333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), x \cdot \left(x \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right)\right)\right), 1\right) \]
12. Add Preprocessing

Alternative 10: 83.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \left(tau \cdot tau\right) \cdot 0.027777777777777776\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (fma
  (* x x)
  (fma
   x
   (*
    x
    (*
     (* (* PI PI) (* PI PI))
     (fma
      0.008333333333333333
      (* (* tau tau) (* tau tau))
      (* (* tau tau) 0.027777777777777776))))
   (* (* PI PI) (fma -0.16666666666666666 (* tau tau) -0.16666666666666666)))
  1.0))

C

float code(float x, float tau) {
	return fmaf((x * x), fmaf(x, (x * (((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * fmaf(0.008333333333333333f, ((tau * tau) * (tau * tau)), ((tau * tau) * 0.027777777777777776f)))), ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(x, Float32(x * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * fma(Float32(0.008333333333333333), Float32(Float32(tau * tau) * Float32(tau * tau)), Float32(Float32(tau * tau) * Float32(0.027777777777777776))))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 83.6

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

7. Applied egg-rr 83.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

10. Simplified 82.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \left(tau \cdot tau\right) \cdot 0.027777777777777776\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]
11. Add Preprocessing

Alternative 11: 79.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf(-0.16666666666666666f, fmaf((tau * (((float) M_PI) * (tau * (((float) M_PI) * (x * x))))), fmaf((x * x), ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f), 1.0f), (x * (x * (((float) M_PI) * ((float) M_PI))))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), fma(Float32(tau * Float32(Float32(pi) * Float32(tau * Float32(Float32(pi) * Float32(x * x))))), fma(Float32(x * x), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666)), Float32(1.0)), Float32(x * Float32(x * Float32(Float32(pi) * Float32(pi))))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 83.6

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

7. Applied egg-rr 83.6%

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

8. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

10. Simplified 78.4%

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

11. Final simplification 78.4%

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

Alternative 12: 79.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf(-0.16666666666666666f, fmaf(x, (x * (((float) M_PI) * ((float) M_PI))), ((tau * (tau * (((float) M_PI) * (x * (x * ((float) M_PI)))))) * fmaf((x * x), (((float) M_PI) * (((float) M_PI) * -0.16666666666666666f)), 1.0f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), fma(x, Float32(x * Float32(Float32(pi) * Float32(pi))), Float32(Float32(tau * Float32(tau * Float32(Float32(pi) * Float32(x * Float32(x * Float32(pi)))))) * fma(Float32(x * x), Float32(Float32(pi) * Float32(Float32(pi) * Float32(-0.16666666666666666))), Float32(1.0)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

8. Simplified 78.4%

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

Alternative 13: 79.2% accurate, 4.4× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf((((float) M_PI) * ((float) M_PI)), ((x * x) * -0.16666666666666666f), 1.0f) * fmaf((tau * tau), (x * (x * ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return Float32(fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(x * x) * Float32(-0.16666666666666666)), Float32(1.0)) * fma(Float32(tau * tau), Float32(x * Float32(x * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666)))), Float32(1.0)))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 83.6

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

7. Applied egg-rr 83.6%

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

8. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

10. Simplified 78.2%

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

11. Final simplification 78.2%

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

Alternative 14: 78.4% accurate, 7.8× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf(x, (x * ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(x, Float32(x * Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*r/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.1%

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

   1. associate-/l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sin-lowering-sin.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. sin-lowering-sin.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

9. Simplified 77.1%

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

Alternative 15: 64.1% accurate, 11.7× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf((((float) M_PI) * ((x * x) * -0.16666666666666666f)), ((float) M_PI), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(Float32(pi) * Float32(Float32(x * x) * Float32(-0.16666666666666666))), Float32(pi), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 N/A

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

   16. PI-lowering-PI.f32 64.7

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

8. Simplified 64.7%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. PI-lowering-PI.f32 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. PI-lowering-PI.f32 64.7

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

10. Applied egg-rr 64.7%

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

Alternative 16: 64.1% accurate, 11.7× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf((x * (((float) M_PI) * ((float) M_PI))), (x * -0.16666666666666666f), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * Float32(Float32(pi) * Float32(pi))), Float32(x * Float32(-0.16666666666666666)), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 N/A

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

   16. PI-lowering-PI.f32 64.7

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

8. Simplified 64.7%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. PI-lowering-PI.f32 N/A

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

   12. PI-lowering-PI.f32 N/A

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

   13. *-lowering-*.f32 64.7

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

10. Applied egg-rr 64.7%

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

Alternative 17: 64.1% accurate, 11.7× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return fmaf((x * x), (((float) M_PI) * (((float) M_PI) * -0.16666666666666666f)), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), Float32(Float32(pi) * Float32(Float32(pi) * Float32(-0.16666666666666666))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. PI-lowering-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 N/A

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

   16. PI-lowering-PI.f32 64.7

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

8. Simplified 64.7%

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

Alternative 18: 63.2% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

float code(float x, float tau) {
	return 1.0f;
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

5. Simplified 63.9%

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

Alternative 19: 6.3% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 0.0)

C

float code(float x, float tau) {
	return 0.0f;
}

Fortran

real(4) function code(x, tau)
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 0.0e0
end function

Julia

function code(x, tau)
	return Float32(0.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.3%

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

   1. frac-times N/A

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

   2. sin-mult N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f32 N/A

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

4. Applied egg-rr 64.8%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot \left(\pi \cdot tau - \pi\right)\right) - \cos \left(x \cdot \mathsf{fma}\left(\pi, tau, \pi\right)\right)}{\left(x \cdot \left(tau \cdot \left(\pi \cdot \left(x \cdot \pi\right)\right)\right)\right) \cdot 2}} \]

5. Taylor expanded in tau around 0

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

   1. div-sub N/A

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

   2. cos-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

   5. metadata-eval 6.3

      \[\leadsto \color{blue}{0} \]

7. Simplified 6.3%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))