Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 15.0s

Alternatives: 20

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: 98.0% 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: 98.0% 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 98.1%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. frac-times N/A

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

   7. frac-2neg N/A

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

   8. *-lft-identity N/A

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

   9. remove-double-neg N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

6. Applied rewrites 97.7%

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

7. Final simplification 97.7%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

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(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(Float32(x * Float32(pi))) * sin(t_1)) / Float32(x * Float32(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 98.1%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.2%

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

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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sin.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sin.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

7. Applied rewrites 97.6%

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

8. Final simplification 97.6%

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

Alternative 4: 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 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-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. lower-/.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. Applied rewrites 97.1%

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

6. Final simplification 97.1%

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

Alternative 5: 90.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. frac-times N/A

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

   7. frac-2neg N/A

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

   8. *-lft-identity N/A

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

   9. remove-double-neg N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.9%

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

6. Applied rewrites 97.7%

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

8. Applied rewrites 97.6%

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

9. Taylor expanded in x around 0

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

    1. lower-/.f32 N/A

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

11. Applied rewrites 92.8%

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

12. Final simplification 92.8%

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

Alternative 6: 85.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 88.5

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

5. Applied rewrites 88.5%

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

Alternative 7: 85.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 87.5%

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

7. Applied rewrites 87.5%

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

Alternative 8: 85.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 87.5%

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

7. Applied rewrites 87.5%

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

Alternative 9: 80.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 87.5%

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

7. Applied rewrites 87.4%

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

8. Taylor expanded in tau around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, \frac{1}{36}, \frac{1}{120}\right) \cdot \left(\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(x \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(tau, tau \cdot \frac{-1}{6}, \frac{-1}{6}\right), 1\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 82.9%

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

Alternative 10: 79.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 87.5%

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

7. Applied rewrites 87.4%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 81.9%

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

11. Final simplification 81.9%

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

Alternative 11: 79.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

7. Applied rewrites 81.9%

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

8. Final simplification 81.9%

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

Alternative 12: 79.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
6. 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. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 81.9%

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

Alternative 13: 79.3% 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 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

Alternative 14: 70.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

6. Taylor expanded in tau around inf

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

8. Applied rewrites 72.1%

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

Alternative 15: 70.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

6. Taylor expanded in tau around inf

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

8. Applied rewrites 72.1%

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

9. Final simplification 72.1%

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

Alternative 16: 70.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 66.5%

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

9. Taylor expanded in tau around inf

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

11. Applied rewrites 72.1%

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

Alternative 17: 65.1% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 66.5%

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

10. Applied rewrites 66.5%

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

Alternative 18: 65.1% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 66.5%

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

10. Applied rewrites 66.5%

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

Alternative 19: 65.1% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 81.9

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

5. Applied rewrites 81.9%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 66.5%

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

10. Applied rewrites 66.5%

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

11. Final simplification 66.5%

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

Alternative 20: 64.1% accurate, 258.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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 98.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 65.6%

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

Reproduce

herbie shell --seed 2024219 
(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))))