Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.5s

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 98.0%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

4. Applied egg-rr 97.4%

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

   1. associate-*r/ N/A

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

   2. associate-/r/ N/A

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

   3. /-rgt-identity N/A

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

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

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. div-inv N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

4. Applied egg-rr 97.1%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.0%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 97.1

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

7. Applied egg-rr 97.1%

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

8. Final simplification 97.1%

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

Alternative 5: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.0%

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

6. Final simplification 97.0%

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

Alternative 6: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

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

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

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

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

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

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

7. Simplified 96.7%

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

8. Final simplification 96.7%

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

Alternative 7: 90.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.0%

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

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 91.4%

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

9. Final simplification 91.4%

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

Alternative 8: 84.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

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

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

4. Applied egg-rr 97.4%

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

   1. associate-*r/ N/A

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

   2. associate-/r/ N/A

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

   3. /-rgt-identity N/A

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

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

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in tau around 0

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

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

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

9. Simplified 86.4%

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

10. Final simplification 86.4%

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

Alternative 9: 84.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative N/A

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

   2. frac-2neg N/A

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

   3. frac-times N/A

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

   4. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\left(-\sin \left(x \cdot \pi\right)\right) \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\pi \cdot \left(x \cdot \left(tau \cdot \left(0 - x \cdot \pi\right)\right)\right)}} \]

5. Taylor expanded in x around 0

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

7. Simplified 86.1%

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

Alternative 10: 84.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

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

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

   12. PI-lowering-PI.f32 85.4

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

7. Simplified 85.4%

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

8. Taylor expanded in x around 0

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

10. Simplified 85.4%

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

11. Final simplification 85.4%

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

Alternative 11: 79.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

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

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

   12. PI-lowering-PI.f32 85.4

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

7. Simplified 85.4%

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

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

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

10. Simplified 80.3%

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

Alternative 12: 79.1% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

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

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   12. PI-lowering-PI.f32 79.7

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

7. Applied egg-rr 79.7%

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

8. Final simplification 79.7%

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

Alternative 13: 79.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   11. PI-lowering-PI.f32 79.7

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

7. Applied egg-rr 79.7%

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

8. Final simplification 79.7%

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

Alternative 14: 79.1% accurate, 7.8× speedup

Math

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

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(Float32(x * 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 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

Alternative 15: 70.2% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

6. Taylor expanded in tau around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

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

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

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

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

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

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-lowering-*.f32 70.1

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

8. Simplified 70.1%

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

9. Final simplification 70.1%

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

Alternative 16: 64.9% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

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. accelerator-lowering-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. PI-lowering-PI.f32 65.3

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

8. Simplified 65.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f32 65.3

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

10. Applied egg-rr 65.3%

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

11. Final simplification 65.3%

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

Alternative 17: 64.9% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

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. accelerator-lowering-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. PI-lowering-PI.f32 65.3

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

8. Simplified 65.3%

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

9. Final simplification 65.3%

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

Alternative 18: 64.9% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 79.7

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

5. Simplified 79.7%

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

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. accelerator-lowering-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. PI-lowering-PI.f32 65.3

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

8. Simplified 65.3%

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

9. Taylor expanded in x around 0

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

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

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

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

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

    5. unpow2 N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 65.3

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

11. Simplified 65.3%

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

Alternative 19: 64.0% 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 98.0%

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

Reproduce

herbie shell --seed 2024196 
(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))))