Lanczos kernel

Percentage Accurate: 97.9% → 98.0%

Time: 7.0s

Alternatives: 24

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: 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 97.9%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

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(x * Float32(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 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 97.8%

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

Alternative 4: 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 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 97.7%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* PI x) tau)))
   (* (/ (sin (* PI x)) (* (* PI x) t_1)) (sin 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) * t_1)) * sinf(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(Float32(pi) * x) * t_1)) * sin(t_1))
end

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-sin.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

9. Applied rewrites 97.6%

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

Alternative 6: 97.4% 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 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 97.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sin.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f32 N/A

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

9. Applied rewrites 97.4%

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

Alternative 7: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-sin.f32 N/A

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

   7. lower-*.f32 N/A

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

8. Applied rewrites 96.8%

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

Alternative 8: 91.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * fmaf((fmaf(((((float) M_PI) * ((float) M_PI)) / (x * x)), -0.16666666666666666f, ((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI)) * 0.008333333333333333f)) * (x * x)), (x * x), 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(fma(Float32(Float32(Float32(pi) * Float32(pi)) / Float32(x * x)), Float32(-0.16666666666666666), Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)) * Float32(0.008333333333333333))) * Float32(x * x)), Float32(x * x), Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 91.2%

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

Alternative 9: 91.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \frac{-1}{6} + \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right), x \cdot x, 1\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \frac{-1}{6} + \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right), x \cdot x, 1\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \frac{-1}{6} + \left(\frac{1}{120} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right), x \cdot x, 1\right) \]

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. unpow-prod-down N/A

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

   13. pow-prod-up N/A

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

   14. metadata-eval N/A

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

   15. associate-*l* N/A

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

6. Applied rewrites 91.2%

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

Alternative 10: 91.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * fmaf((fmaf(((x * x) * 0.008333333333333333f), (((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI)), (-0.16666666666666666f * (((float) M_PI) * ((float) M_PI)))) * x), x, 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(fma(Float32(Float32(x * x) * Float32(0.008333333333333333)), Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)), Float32(Float32(-0.16666666666666666) * Float32(Float32(pi) * Float32(pi)))) * x), x, Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

6. Applied rewrites 91.2%

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

Alternative 11: 90.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 90.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

9. Applied rewrites 90.5%

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

Alternative 12: 84.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 84.9

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

8. Applied rewrites 84.9%

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

Alternative 13: 84.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 98.0

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*l* N/A

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

   9. pow2 N/A

      \[\leadsto \sin \left(\left(\pi \cdot x\right) \cdot tau\right) \cdot \frac{\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot \left(\pi \cdot \pi\right) + 1}{\left(\pi \cdot x\right) \cdot tau} \]

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

10. Applied rewrites 84.7%

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

Alternative 14: 84.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 90.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 84.3

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

9. Applied rewrites 84.3%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-sin.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. associate-/r* N/A

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

    8. *-commutative N/A

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

    9. *-commutative N/A

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

    10. lower-/.f32 N/A

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

11. Applied rewrites 84.4%

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

Alternative 15: 84.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 90.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 84.3

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

9. Applied rewrites 84.3%

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

    1. lift-*.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-PI.f32 84.6

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

11. Applied rewrites 84.6%

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

Alternative 16: 84.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 90.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 84.3

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

9. Applied rewrites 84.3%

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

    1. lift-PI.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lift-PI.f32 84.3

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

11. Applied rewrites 84.3%

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

Alternative 17: 78.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 90.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 84.3

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

9. Applied rewrites 84.3%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

12. Applied rewrites 78.7%

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

Alternative 18: 78.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.3%

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

6. Applied rewrites 78.3%

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

Alternative 19: 78.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.3%

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

6. Applied rewrites 78.3%

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

Alternative 20: 69.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.3%

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

5. Taylor expanded in tau around inf

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

   1. pow-prod-down N/A

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

   2. pow2 N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-PI.f32 69.3

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

7. Applied rewrites 69.3%

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

Alternative 21: 64.1% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 90.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-PI.f32 84.3

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

9. Applied rewrites 84.3%

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

10. Taylor expanded in x around 0

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

    1. lift-PI.f32 64.1

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

12. Applied rewrites 64.1%

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

Alternative 22: 64.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.3%

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

5. Taylor expanded in tau around 0

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

   1. pow2 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 64.1

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

7. Applied rewrites 64.1%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

      \[\leadsto \left(\frac{-1}{6} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(x \cdot x\right) + 1 \]

   4. pow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

9. Applied rewrites 64.1%

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

Alternative 23: 64.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.3%

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

5. Taylor expanded in tau around 0

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

   1. pow2 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 64.1

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

7. Applied rewrites 64.1%

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

8. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 64.1

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

10. Applied rewrites 64.1%

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

Alternative 24: 63.2% accurate, 94.3× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 63.2%

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

Reproduce

herbie shell --seed 2025131 
(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))))