Lanczos kernel

Percentage Accurate: 97.9% → 98.0%

Time: 4.6s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (tau * ((float) M_PI)) * x;
	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 * Float32(pi)) * x)
	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 * single(pi)) * x;
	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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

Alternative 2: 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 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

7. Applied rewrites 97.7%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

7. Applied rewrites 97.4%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

7. Applied rewrites 97.4%

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

Alternative 6: 85.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 97.7%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.7%

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 97.7%

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.7%

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 85.0%

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

6. Applied rewrites 85.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f32 N/A

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

8. Applied rewrites 85.1%

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

Alternative 7: 85.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 97.7%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.7%

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 97.7%

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.7%

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 85.0%

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

6. Applied rewrites 85.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. mult-flip N/A

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

   5. lower-*.f32 N/A

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

8. Applied rewrites 84.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

10. Applied rewrites 85.1%

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

Alternative 8: 85.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 97.7%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.7%

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 97.7%

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.7%

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 85.0%

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

6. Applied rewrites 85.0%

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

8. Applied rewrites 85.2%

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

Alternative 9: 85.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 97.7%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.7%

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 97.7%

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.7%

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 85.0%

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

6. Applied rewrites 85.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. mult-flip N/A

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

   5. lower-*.f32 N/A

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

8. Applied rewrites 84.8%

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

10. Applied rewrites 85.2%

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

Alternative 10: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

3. Applied rewrites 97.5%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 79.2%

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

6. Applied rewrites 79.2%

   \[\leadsto \frac{\sin \left(\pi \cdot x\right)}{x} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot \left({x}^{2} \cdot \pi\right), \frac{1}{\pi}\right)} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-pow.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 79.2%

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

8. Applied rewrites 79.2%

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

Alternative 11: 79.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

3. Applied rewrites 97.5%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 79.2%

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

6. Applied rewrites 79.2%

   \[\leadsto \frac{\sin \left(\pi \cdot x\right)}{x} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot \left({x}^{2} \cdot \pi\right), \frac{1}{\pi}\right)} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 79.1%

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

8. Applied rewrites 79.1%

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

Alternative 12: 71.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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. frac-2neg N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

3. Applied rewrites 97.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 71.3%

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

6. Applied rewrites 71.3%

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

Alternative 13: 71.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.3%

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

7. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.9%

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

9. Applied rewrites 97.9%

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

10. Taylor expanded in x around 0

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

    1. lower-*.f32 N/A

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

    2. lower-PI.f32 71.4%

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

12. Applied rewrites 71.4%

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

Alternative 14: 71.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 71.4%

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

8. Applied rewrites 71.4%

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

Alternative 15: 71.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. remove-double-neg N/A

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

   2. lift-sin.f32 N/A

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

   3. sin-+PI-rev N/A

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

   4. sin-neg-rev N/A

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

   5. lower-sin.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 81.5%

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

3. Applied rewrites 81.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 60.1%

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

6. Applied rewrites 60.1%

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

8. Applied rewrites 71.3%

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

Alternative 16: 71.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

3. Applied rewrites 97.5%

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

4. Taylor expanded in x around 0

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

   1. lower-PI.f32 71.3%

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

6. Applied rewrites 71.3%

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

Alternative 17: 71.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin(((-x * tau) * single(pi))) * (single(-1.0) / (tau * (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

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

   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. frac-2neg N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

3. Applied rewrites 97.8%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 71.1%

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

6. Applied rewrites 71.1%

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

Alternative 18: 70.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

3. Applied rewrites 97.5%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 79.2%

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

6. Applied rewrites 79.2%

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

7. Taylor expanded in x around 0

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

   1. lower-PI.f32 70.2%

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

9. Applied rewrites 70.2%

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

Alternative 19: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * single(pi))) * (single(1.0) / (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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.0%

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

5. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 64.8%

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

10. Applied rewrites 64.8%

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

Alternative 20: 64.0% accurate, 94.3× speedup

Math

\[1 \]

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 64.0%

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

Alternative 21: 6.3% accurate, 94.3× speedup

Math

\[0 \]

FPCore

(FPCore (x tau) :precision binary32 0.0)

C

float code(float x, float tau) {
	return 0.0f;
}

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 = 0.0e0
end function

Julia

function code(x, tau)
	return Float32(0.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(0.0);
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. remove-double-neg N/A

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

   2. lift-sin.f32 N/A

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

   3. sin-+PI-rev N/A

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

   4. sin-neg-rev N/A

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

   5. lower-sin.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 81.5%

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

3. Applied rewrites 81.5%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 14.8%

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

6. Applied rewrites 14.8%

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

   1. lift-sin.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. sin-neg N/A

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

   4. lift-PI.f32 N/A

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

   5. sin-PI N/A

      \[\leadsto \frac{\mathsf{neg}\left(0\right)}{tau \cdot \left(x \cdot \pi\right)} \]

   6. metadata-eval 6.3%

      \[\leadsto \frac{0}{\color{blue}{tau} \cdot \left(x \cdot \pi\right)} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{0}{tau \cdot \color{blue}{\left(x \cdot \pi\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{0}{tau \cdot \left(x \cdot \color{blue}{\pi}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{0}{tau \cdot \left(\pi \cdot \color{blue}{x}\right)} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{0}{tau \cdot \left(\pi \cdot \color{blue}{x}\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{0}{\left(\pi \cdot x\right) \cdot \color{blue}{tau}} \]

   12. lower-*.f32 6.3%

       \[\leadsto \frac{0}{\left(\pi \cdot x\right) \cdot \color{blue}{tau}} \]

8. Applied rewrites 6.3%

   \[\leadsto \frac{0}{\color{blue}{\left(\pi \cdot x\right) \cdot tau}} \]

9. Taylor expanded in x around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 6.3%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025192 
(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))))