Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 5.8s

Alternatives: 25

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

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 97.9

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

5. Applied rewrites 97.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.9

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

5. Applied rewrites 97.9%

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

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 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin(((tau * x) * single(pi))) * (sin((single(pi) * x)) / (((single(pi) * x) * (single(pi) * x)) * 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. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}} \]
3. 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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

4. Applied rewrites 97.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. unswap-sqr N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 97.1

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

6. Applied rewrites 97.1%

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

Alternative 5: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin(((tau * x) * single(pi))) * (sin((single(pi) * x)) / ((single(pi) * single(pi)) * ((x * x) * 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. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}} \]
3. 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. *-commutative N/A

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

   3. lower-*.f32 N/A

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

4. Applied rewrites 97.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. lower-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 97.0

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

6. Applied rewrites 97.0%

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

Alternative 6: 91.0% accurate, 1.0× speedup

Math

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

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 91.0%

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

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 91.0%

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

Alternative 7: 91.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
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 91.0%

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

6. Applied rewrites 91.0%

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

Alternative 8: 91.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (*
    (/ (sin t_1) t_1)
    (fma
     (fma
      (* PI PI)
      -0.16666666666666666
      (* (* (* (* 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)), -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(fma(Float32(Float32(pi) * Float32(pi)), Float32(-0.16666666666666666), Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)) * Float32(Float32(Float32(0.008333333333333333) * 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 91.0%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]
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. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. unpow-prod-down N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

6. Applied rewrites 91.0%

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

Alternative 9: 91.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (*
    (/ (sin t_1) t_1)
    (fma
     (*
      (fma
       (* (* x x) (* (* (* PI PI) PI) PI))
       0.008333333333333333
       (* (* PI PI) -0.16666666666666666))
      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) * (((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI))), 0.008333333333333333f, ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f)) * 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(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi))), Float32(0.008333333333333333), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))) * 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 91.0%

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

6. Applied rewrites 91.0%

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

Alternative 10: 91.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (*
    (/ (sin t_1) t_1)
    (fma
     (fma
      (* 0.008333333333333333 (* x x))
      (* (* PI PI) (* PI PI))
      (* (* PI PI) -0.16666666666666666))
     (* 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((0.008333333333333333f * (x * x)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f)), (x * x), 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * fma(fma(Float32(Float32(0.008333333333333333) * Float32(x * x)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), 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. 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 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 91.0%

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

Alternative 11: 90.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.6

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

4. Applied rewrites 84.6%

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

9. Applied rewrites 90.3%

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

Alternative 12: 84.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (fma (* (* PI PI) -0.16666666666666666) (* 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(((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f), (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(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666)), 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

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.6

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

4. Applied rewrites 84.6%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. inv-pow N/A

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

   3. pow-plus N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \left(\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2} + {\left({x}^{2}\right)}^{0}\right) \]

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 84.6

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

7. Applied rewrites 84.6%

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

Alternative 13: 84.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(Float32(tau * x) * Float32(pi))) / tau) * Float32(fma(Float32(Float32(x * x) * Float32(pi)), Float32(-0.16666666666666666), Float32(Float32(1.0) / Float32(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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.6

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

4. Applied rewrites 84.6%

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-PI.f32 84.0

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

9. Applied rewrites 84.0%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-PI.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lift-PI.f32 84.0

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

11. Applied rewrites 84.0%

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

Alternative 14: 84.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(x * Float32(Float32(pi) * tau))) / tau) * Float32(fma(Float32(Float32(x * x) * Float32(pi)), Float32(-0.16666666666666666), Float32(Float32(1.0) / Float32(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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.6

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

4. Applied rewrites 84.6%

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-PI.f32 84.0

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

9. Applied rewrites 84.0%

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

    1. lift-*.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lift-PI.f32 84.0

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

11. Applied rewrites 84.0%

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

Alternative 15: 83.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(Float32(Float32(pi) * x) * tau)) / tau) * Float32(fma(Float32(-0.16666666666666666), Float32(pi), Float32(Float32(1.0) / Float32(Float32(x * x) * Float32(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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.6

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

4. Applied rewrites 84.6%

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-PI.f32 84.0

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

9. Applied rewrites 84.0%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower-fma.f32 N/A

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

    4. lift-PI.f32 N/A

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

    5. lower-/.f32 N/A

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

    6. pow2 N/A

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

    7. lift-*.f32 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-PI.f32 83.9

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

12. Applied rewrites 83.9%

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

Alternative 16: 79.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(fma(Float32(Float32(-0.16666666666666666) * Float32(tau * tau)), Float32(Float32(x * x) * Float32(Float32(pi) * Float32(pi))), Float32(1.0)) * fma(fma(Float32(Float32(0.008333333333333333) * Float32(x * x)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), 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 91.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. pow2 N/A

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

   13. lift-*.f32 79.0

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

7. Applied rewrites 79.0%

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

Alternative 17: 79.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(fma(Float32(Float32(Float32(x * x) * Float32(Float32(pi) * Float32(pi))) * Float32(-0.16666666666666666)), Float32(tau * tau), Float32(1.0)) * fma(fma(Float32(Float32(0.008333333333333333) * Float32(x * x)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), 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 91.0%

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

5. Taylor expanded in tau around 0

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

7. Applied rewrites 84.5%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 79.0

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

10. Applied rewrites 79.0%

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

Alternative 18: 78.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.6

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

4. Applied rewrites 84.6%

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

6. Applied rewrites 84.3%

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

8. Applied rewrites 84.1%

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

9. Taylor expanded in tau around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

11. Applied rewrites 78.3%

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

Alternative 19: 78.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * fma(Float32(pi), Float32(pi), 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 78.0%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. lift-PI.f32 N/A

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

   20. lift-PI.f32 78.0

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

6. Applied rewrites 78.0%

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

Alternative 20: 78.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(Float32(fma(tau, tau, Float32(1.0)) * Float32(Float32(pi) * Float32(pi))) * Float32(-0.16666666666666666)) * 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 78.0%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

6. Applied rewrites 78.0%

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

Alternative 21: 78.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(x * x) * Float32(-0.16666666666666666)), 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 78.0%

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

6. Applied rewrites 78.0%

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

Alternative 22: 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 78.0%

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

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 23: 64.3% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666)) * 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 78.0%

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

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 64.3

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

7. Applied rewrites 64.3%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right), x \cdot x, 1\right) \]
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. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

9. Applied rewrites 64.3%

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

Alternative 24: 64.3% 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 78.0%

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

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 64.3

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

7. Applied rewrites 64.3%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right), x \cdot x, 1\right) \]
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 64.3%

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

Alternative 25: 63.3% 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 63.3%

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

Reproduce

herbie shell --seed 2025127 
(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))))