Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 15.0s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / t_1) * (sin((single(pi) * x)) / (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. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.1%

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

3. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Final simplification 97.4%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 97.8%

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

3. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (single(pi) * x);
	tmp = (sin((single(pi) * x)) / (single(pi) * x)) * (sin(t_1) / 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. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. *-commutative 97.3%

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

   3. *-commutative 97.3%

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

   4. associate-*r* 97.9%

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

   5. associate-*r/ 97.8%

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

   6. *-commutative 97.8%

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

   7. associate-/r* 97.7%

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

   8. associate-*l/ 97.7%

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

6. Applied egg-rr 97.8%

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

   1. associate-/l* 97.9%

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

   2. associate-/r* 97.6%

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

   3. rem-log-exp 78.0%

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

   4. frac-times 78.0%

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

   5. associate-*r* 78.0%

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

   6. *-commutative 78.0%

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

   7. *-commutative 78.0%

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

   8. associate-*r* 77.9%

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

   9. associate-*r/ 77.9%

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

8. Applied egg-rr 97.7%

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

   1. *-commutative 97.7%

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

   2. *-commutative 97.7%

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

   3. associate-*r* 96.9%

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

   4. *-commutative 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*r* 97.6%

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

   7. associate-/l/ 97.9%

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

   8. *-commutative 97.9%

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

10. Simplified 97.9%

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

11. Final simplification 97.9%

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

Alternative 5: 70.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (single(pi) * x);
	tmp = (sin(t_1) / t_1) * ((single(pi) * x) / (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. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in x around 0 97.3%

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

6. Taylor expanded in x around 0 70.9%

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

7. Taylor expanded in x around inf 71.1%

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

8. Final simplification 71.1%

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

Alternative 6: 70.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = ((sin((tau * (single(pi) * x))) / (single(pi) * x)) / tau) * ((single(pi) * x) / (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. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in x around 0 97.3%

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

6. Taylor expanded in x around 0 70.9%

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

   1. *-un-lft-identity 70.9%

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

   2. *-commutative 70.9%

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

   3. times-frac 70.9%

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

8. Applied egg-rr 70.9%

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

   1. associate-*l/ 71.0%

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

   2. *-lft-identity 71.0%

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

   3. *-commutative 71.0%

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

   4. *-commutative 71.0%

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

   5. associate-*r* 71.2%

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

   6. *-commutative 71.2%

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

10. Simplified 71.2%

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

11. Final simplification 71.2%

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

Alternative 7: 70.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = x * (sin(t_1) / (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. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-/l/ 97.3%

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

   4. associate-*l* 97.2%

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

   5. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in x around 0 71.1%

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

6. Final simplification 71.1%

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

Alternative 8: 70.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * (single(pi) * tau))) * (single(1.0) / (tau * (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. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in x around 0 70.9%

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

6. Final simplification 70.9%

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

Alternative 9: 70.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((single(pi) * (x * tau))) * (single(1.0) / tau)) / (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. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. *-commutative 97.3%

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

   3. *-commutative 97.3%

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

   4. associate-*r* 97.9%

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

   5. associate-*r/ 97.8%

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

   6. *-commutative 97.8%

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

   7. associate-/r* 97.7%

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

   8. associate-*l/ 97.7%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 70.9%

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

8. Final simplification 70.9%

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

Alternative 10: 70.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = ((sin((single(pi) * (x * tau))) * (single(1.0) / tau)) / 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. associate-*r/ 97.7%

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

   2. times-frac 97.4%

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

   3. associate-/l/ 97.3%

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

   4. associate-*l* 97.2%

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

   5. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. add-log-exp 77.9%

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

   2. *-commutative 77.9%

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

6. Applied egg-rr 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-*r* 77.8%

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

   3. *-commutative 77.8%

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

   4. associate-*r* 77.9%

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

   5. associate-*r/ 77.9%

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

   6. associate-*r* 78.0%

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

   7. *-commutative 78.0%

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

   8. *-commutative 78.0%

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

   9. associate-*r* 78.0%

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

   10. frac-times 78.0%

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

8. Applied egg-rr 97.5%

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

9. Taylor expanded in x around 0 70.9%

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

10. Final simplification 70.9%

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

Alternative 11: 70.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. *-commutative 97.3%

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

   3. frac-times 97.4%

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

   4. associate-/r* 97.4%

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

   5. clear-num 97.1%

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

   6. frac-times 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in x around 0 70.8%

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

8. Final simplification 70.8%

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

Alternative 12: 63.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in tau around 0 64.4%

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

6. Taylor expanded in x around 0 64.4%

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

   1. +-commutative 64.4%

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

   2. fma-define 64.4%

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

   3. *-commutative 64.4%

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

   4. unpow2 64.4%

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

   5. unpow2 64.4%

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

   6. swap-sqr 64.4%

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

   7. unpow2 64.4%

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

   8. *-commutative 64.4%

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

8. Simplified 64.4%

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

   1. unpow2 64.4%

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

   2. *-commutative 64.4%

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

   3. associate-*r* 64.4%

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

10. Applied egg-rr 64.4%

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

11. Final simplification 64.4%

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

Alternative 13: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + -0.16666666666666666 \cdot {\left(\pi \cdot x\right)}^{2} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in tau around 0 64.4%

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

6. Taylor expanded in x around 0 64.4%

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

   1. +-commutative 64.4%

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

   2. fma-define 64.4%

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

   3. *-commutative 64.4%

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

   4. unpow2 64.4%

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

   5. unpow2 64.4%

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

   6. swap-sqr 64.4%

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

   7. unpow2 64.4%

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

   8. *-commutative 64.4%

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

8. Simplified 64.4%

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

   1. fma-undefine 64.4%

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

   2. *-commutative 64.4%

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

10. Applied egg-rr 64.4%

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

11. Final simplification 64.4%

    \[\leadsto 1 + -0.16666666666666666 \cdot {\left(\pi \cdot x\right)}^{2} \]
12. Add Preprocessing

Alternative 14: 63.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((single(pi) * x)) / (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. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in tau around 0 64.4%

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

6. Final simplification 64.4%

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

Alternative 15: 62.9% accurate, 219.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. *-commutative 97.3%

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

   3. *-commutative 97.3%

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

   4. associate-*r* 97.9%

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

   5. associate-*r/ 97.8%

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

   6. *-commutative 97.8%

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

   7. associate-/r* 97.7%

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

   8. associate-*l/ 97.7%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 63.6%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 63.6%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(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))))