Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.3s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Final simplification 97.8%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg2 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l/ N/A

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

6. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-neg.f32 N/A

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

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

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. sqr-neg N/A

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

   15. associate-*l* N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 97.6

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

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

   3. lower-*.f32 N/A

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 85.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 85.5

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

5. Applied rewrites 85.5%

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

6. Final simplification 85.5%

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

Alternative 5: 85.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 85.5

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

5. Applied rewrites 85.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 84.9

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

7. Applied rewrites 84.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 85.5

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 85.5

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

9. Applied rewrites 85.5%

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

10. Final simplification 85.5%

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

Alternative 6: 85.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 85.5

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

5. Applied rewrites 85.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

7. Applied rewrites 85.4%

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

8. Final simplification 85.4%

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

Alternative 7: 79.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 N/A

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

   18. lower-PI.f32 80.1

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

5. Applied rewrites 80.1%

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

6. Final simplification 80.1%

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

Alternative 8: 79.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 85.5

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

5. Applied rewrites 85.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 N/A

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

   18. lower-PI.f32 80.0

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

8. Applied rewrites 80.0%

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

9. Final simplification 80.0%

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

Alternative 9: 79.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 79.4%

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

Alternative 10: 64.9% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 85.5

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

5. Applied rewrites 85.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 65.2%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. lower-fma.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f32 N/A

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

    9. lower-PI.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f32 65.2

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

11. Applied rewrites 65.2%

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

12. Final simplification 65.2%

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

Alternative 11: 64.9% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 85.5

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

5. Applied rewrites 85.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 65.2%

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

10. Applied rewrites 65.2%

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

11. Final simplification 65.2%

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

Alternative 12: 63.9% 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 97.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2024235 
(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))))