Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 4.8s

Alternatives: 16

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

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

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

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

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

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

FPCore

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

C

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

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * x) * Float32(pi))
	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 = (tau * x) * single(pi);
	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-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 97.6%

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.6%

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

3. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

5. Applied rewrites 97.8%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = (single(pi) * tau) * x;
	tmp = sin(t_1) * (sin((single(pi) * x)) / ((t_1 * 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-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

7. Applied rewrites 97.4%

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

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (*
    (/ (sin t_1) t_1)
    (+ 1.0 (* -0.16666666666666666 (* (pow x 2.0) (pow PI 2.0)))))))

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (1.0f + (-0.16666666666666666f * (powf(x, 2.0f) * powf(((float) M_PI), 2.0f))));
}

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(Float32(-0.16666666666666666) * Float32((x ^ Float32(2.0)) * (Float32(pi) ^ Float32(2.0))))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (single(1.0) + (single(-0.16666666666666666) * ((x ^ single(2.0)) * (single(pi) ^ single(2.0)))));
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

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 85.3%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right) \]

4. Applied rewrites 85.3%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right)} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \left(tau \cdot x\right) \cdot \pi\\ \frac{\sin t\_1}{t\_1} \cdot \left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* tau x) PI)))
   (*
    (/ (sin t_1) t_1)
    (+ 1.0 (* -0.16666666666666666 (* (pow x 2.0) (pow PI 2.0)))))))

C

float code(float x, float tau) {
	float t_1 = (tau * x) * ((float) M_PI);
	return (sinf(t_1) / t_1) * (1.0f + (-0.16666666666666666f * (powf(x, 2.0f) * powf(((float) M_PI), 2.0f))));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * x) * Float32(pi))
	return Float32(Float32(sin(t_1) / t_1) * Float32(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32((x ^ Float32(2.0)) * (Float32(pi) ^ Float32(2.0))))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * x) * single(pi);
	tmp = (sin(t_1) / t_1) * (single(1.0) + (single(-0.16666666666666666) * ((x ^ single(2.0)) * (single(pi) ^ single(2.0)))));
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-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 85.3%

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot x\right) \cdot \pi} \cdot \left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right) \]

8. Applied rewrites 85.3%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot x\right) \cdot \pi} \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)\right)} \]
9. Add Preprocessing

Alternative 6: 78.8% accurate, 1.1× speedup

Math

\[1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{2}, -0.16666666666666666 \cdot {\pi}^{2}\right) \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+
  1.0
  (*
   (pow x 2.0)
   (fma
    -0.16666666666666666
    (* (pow tau 2.0) (pow PI 2.0))
    (* -0.16666666666666666 (pow PI 2.0))))))

C

float code(float x, float tau) {
	return 1.0f + (powf(x, 2.0f) * fmaf(-0.16666666666666666f, (powf(tau, 2.0f) * powf(((float) M_PI), 2.0f)), (-0.16666666666666666f * powf(((float) M_PI), 2.0f))));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * fma(Float32(-0.16666666666666666), Float32((tau ^ Float32(2.0)) * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-0.16666666666666666) * (Float32(pi) ^ Float32(2.0))))))
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

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-PI.f32 78.8%

       \[\leadsto 1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{2}, -0.16666666666666666 \cdot {\pi}^{2}\right) \]

4. Applied rewrites 78.8%

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{2}, -0.16666666666666666 \cdot {\pi}^{2}\right)} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (((float) M_PI) / ((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(Float32(pi) / Float32(pi)))
end

MATLAB

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

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 97.6%

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.6%

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

3. Applied rewrites 97.6%

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

4. Taylor expanded in x around 0

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

   1. lower-PI.f32 71.0%

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

6. Applied rewrites 71.0%

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

Alternative 8: 64.7% accurate, 2.2× speedup

Math

\[1 \cdot \frac{\left(1 + \frac{\left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{x \cdot \pi}\right) \cdot \left(x \cdot \pi\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * (((single(1.0) + (((x * ((x * x) * single(-0.16666666666666666))) * ((single(pi) * single(pi)) * single(pi))) / (x * single(pi)))) * (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. Taylor expanded in x around 0

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. lift-*.f32 N/A

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

   5. sum-to-mult N/A

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

   6. lower-unsound-*.f32 N/A

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

9. Applied rewrites 64.7%

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

Alternative 9: 64.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. div-flip N/A

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

   5. lower-unsound-/.f32 N/A

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

9. Applied rewrites 64.7%

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

Alternative 10: 64.7% accurate, 3.1× speedup

Math

\[1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * ((x * (single(pi) + (single(-0.16666666666666666) * (((x * x) * single(pi)) * (single(pi) * 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. Taylor expanded in x around 0

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f32 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left({x}^{2} \cdot \pi\right) \cdot {\color{blue}{\pi}}^{2}\right)\right)}{x \cdot \pi} \]

   9. lift-pow.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot {\pi}^{2}\right)\right)}{x \cdot \pi} \]

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right)}{x \cdot \pi} \]

9. Applied rewrites 64.7%

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

Alternative 11: 64.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left({x}^{2} \cdot {\pi}^{3}, \color{blue}{-0.16666666666666666}, \pi\right)}{x \cdot \pi} \]

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left(\left({\pi}^{3} \cdot x\right) \cdot x, -0.16666666666666666, \pi\right)}{x \cdot \pi} \]

   13. lift-pow.f32 N/A

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

   14. unpow3 N/A

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

   15. unpow2 N/A

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

   16. lift-pow.f32 N/A

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

   17. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left(\left(\left({\pi}^{2} \cdot \pi\right) \cdot x\right) \cdot x, -0.16666666666666666, \pi\right)}{x \cdot \pi} \]

   18. lift-pow.f32 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f32 64.7%

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

9. Applied rewrites 64.7%

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

Alternative 12: 64.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. lift-pow.f32 N/A

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

   7. unpow3 N/A

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

   8. unpow2 N/A

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

   9. lift-pow.f32 N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left(\left({x}^{2} \cdot -0.16666666666666666\right) \cdot {\pi}^{2}, \pi, \pi\right)}{x \cdot \pi} \]

   15. lift-pow.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot {\pi}^{2}, \pi, \pi\right)}{x \cdot \pi} \]

   18. lift-pow.f32 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f32 64.7%

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

9. Applied rewrites 64.7%

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

Alternative 13: 64.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, {\pi}^{3} \cdot \color{blue}{-0.16666666666666666}, \pi\right)}{x \cdot \pi} \]

   12. lift-pow.f32 N/A

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

   13. unpow3 N/A

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

   14. unpow2 N/A

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

   15. lift-pow.f32 N/A

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

   16. lower-*.f32 64.7%

       \[\leadsto 1 \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, \left({\pi}^{2} \cdot \pi\right) \cdot -0.16666666666666666, \pi\right)}{x \cdot \pi} \]

   17. lift-pow.f32 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f32 64.7%

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

9. Applied rewrites 64.7%

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

Alternative 14: 64.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

9. Applied rewrites 64.6%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. associate-/l/ N/A

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

    4. lift-*.f32 N/A

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

    5. *-commutative N/A

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

    6. lift-*.f32 N/A

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

11. Applied rewrites 64.6%

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

Alternative 15: 64.6% accurate, 3.4× speedup

Math

\[\frac{\frac{\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \pi, 9.869604110717773, \pi\right) \cdot x\right) \cdot 1}{\pi}}{x} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (/
  (/
   (*
    (* (fma (* (* (* x x) -0.16666666666666666) PI) 9.869604110717773 PI) x)
    1.0)
   PI)
  x))

C

float code(float x, float tau) {
	return (((fmaf((((x * x) * -0.16666666666666666f) * ((float) M_PI)), 9.869604110717773f, ((float) M_PI)) * x) * 1.0f) / ((float) M_PI)) / x;
}

Julia

function code(x, tau)
	return Float32(Float32(Float32(Float32(fma(Float32(Float32(Float32(x * x) * Float32(-0.16666666666666666)) * Float32(pi)), Float32(9.869604110717773), Float32(pi)) * x) * 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

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

9. Applied rewrites 64.6%

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

10. Evaluated real constant 64.6%

    \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \pi, 9.869604110717773, \pi\right) \cdot x\right) \cdot 1}{\pi}}{x} \]
11. Add Preprocessing

Alternative 16: 63.7% accurate, 7.1× speedup

Math

\[1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * ((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. Taylor expanded in x around 0

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.7%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.7%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]

8. Taylor expanded in x around 0

   \[\leadsto 1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]
9. Step-by-step derivation

   1. lower-PI.f32 63.7%

      \[\leadsto 1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]

10. Applied rewrites 63.7%

    \[\leadsto 1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025207 
(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))))