Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 13.8s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

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* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-/l* 97.1%

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

   5. *-commutative 97.1%

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

   6. associate-*l* 97.0%

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

   7. associate-/l/ 97.0%

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

   8. *-commutative 97.0%

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

3. Simplified 97.0%

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

   1. associate-*r* 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 97.0%

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

   4. *-un-lft-identity 97.0%

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

   5. times-frac 96.9%

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

   6. *-commutative 96.9%

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

   7. associate-/r* 96.9%

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

   8. *-commutative 96.9%

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

   9. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

   1. associate-/r/ 97.0%

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

   2. associate-/l/ 97.1%

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

   3. associate-*r* 97.4%

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

   4. frac-times 97.4%

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

   5. *-un-lft-identity 97.4%

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

7. Applied egg-rr 97.4%

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

8. Final simplification 97.4%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{tau \cdot {\left(\pi \cdot x\right)}^{2}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (sin (* tau (* PI x))) (/ (sin (* PI x)) (* tau (pow (* PI x) 2.0)))))

C

float code(float x, float tau) {
	return sinf((tau * (((float) M_PI) * x))) * (sinf((((float) M_PI) * x)) / (tau * powf((((float) M_PI) * x), 2.0f)));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(tau * Float32(Float32(pi) * x))) * Float32(sin(Float32(Float32(pi) * x)) / Float32(tau * (Float32(Float32(pi) * x) ^ Float32(2.0)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (single(pi) * x))) * (sin((single(pi) * x)) / (tau * ((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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.4%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around inf 97.0%

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

   1. associate-/l* 96.8%

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

   2. *-commutative 96.8%

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

   3. *-commutative 96.8%

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

   4. *-commutative 96.8%

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

   5. associate-*r* 96.9%

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

   6. *-commutative 96.9%

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

   7. associate-/r/ 97.0%

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

6. Simplified 97.2%

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

7. Final simplification 97.2%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot x\right) \cdot \left(\sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \frac{{\left(\pi \cdot x\right)}^{-2}}{tau}\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (sin (* PI x)) (* (sin (* tau (* PI x))) (/ (pow (* PI x) -2.0) tau))))

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((single(pi) * x)) * (sin((tau * (single(pi) * x))) * (((single(pi) * x) ^ single(-2.0)) / 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-/r/ 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-/l* 97.1%

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

   5. associate-*l* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*r* 97.2%

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

   8. *-commutative 97.2%

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

   9. associate-*r* 96.9%

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

   10. associate-*l* 96.9%

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

3. Simplified 96.9%

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

   1. add-log-exp 96.7%

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

   2. associate-/r/ 96.7%

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

   3. associate-*r* 96.3%

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

   4. *-commutative 96.3%

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

   5. associate-*r* 96.8%

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

   6. *-commutative 96.8%

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

   7. associate-*r* 96.9%

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

   8. associate-*r* 96.6%

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

   9. *-commutative 96.6%

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

   10. *-commutative 96.6%

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

   11. associate-*l* 96.6%

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

5. Applied egg-rr 96.8%

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

   1. *-un-lft-identity 96.8%

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

   2. log-prod 96.8%

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

   3. metadata-eval 96.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau}}\right) \]

   4. rem-log-exp 97.1%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau}} \]

   5. *-commutative 97.1%

      \[\leadsto 0 + \frac{\color{blue}{\sin \left(\pi \cdot x\right) \cdot \sin \left(x \cdot \left(\pi \cdot tau\right)\right)}}{{\left(\pi \cdot x\right)}^{2} \cdot tau} \]

   6. times-frac 97.0%

      \[\leadsto 0 + \color{blue}{\frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2}} \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau}} \]

7. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{0 + \left(\sin \left(x \cdot \pi\right) \cdot {\left(x \cdot \pi\right)}^{-2}\right) \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau}} \]
8. Step-by-step derivation

   1. +-lft-identity 96.9%

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

   2. associate-*l* 96.9%

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

   3. associate-*r/ 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*r/ 97.0%

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

   6. associate-*r* 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 5: 85.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / t_1) * (single(1.0) + (((single(pi) * x) ^ single(2.0)) * single(-0.16666666666666666)));
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* 98.0%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)}} \]

   2. *-commutative 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot x}\right)\right)} \]

5. Applied egg-rr 97.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)}} \]

6. Taylor expanded in x around 0 84.5%

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

   1. unpow2 84.5%

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

   2. unpow2 84.5%

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

   3. swap-sqr 84.5%

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

   4. unpow2 84.5%

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

8. Simplified 84.5%

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

9. Final simplification 84.5%

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

Alternative 6: 78.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma((x ^ Float32(2.0)), Float32(Float32(-0.16666666666666666) * Float32(Float32(Float32(1.0) + (tau ^ Float32(2.0))) * (Float32(pi) ^ Float32(2.0)))), 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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.4%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 76.8%

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

   1. +-commutative 76.8%

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

   2. fma-def 76.8%

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

   3. distribute-lft-out 76.8%

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

   4. distribute-lft1-in 76.8%

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

6. Simplified 76.8%

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

7. Final simplification 76.8%

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

Alternative 7: 71.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = 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. *-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* 98.0%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)}} \]

   2. *-commutative 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\pi \cdot x}\right)\right)} \]

5. Applied egg-rr 97.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)}} \]

6. Taylor expanded in x around 0 69.4%

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

7. Final simplification 69.4%

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

Alternative 8: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), Float32(x * Float32(x * (Float32(pi) ^ Float32(2.0)))), 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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.4%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 62.4%

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

5. Taylor expanded in x around 0 62.5%

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

   1. +-commutative 62.5%

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

   2. fma-def 62.5%

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

   3. *-commutative 62.5%

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

   4. unpow2 62.5%

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

   5. unpow2 62.5%

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

   6. swap-sqr 62.5%

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

   7. unpow2 62.5%

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

   8. *-commutative 62.5%

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

7. Simplified 62.5%

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

   1. *-commutative 62.5%

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

   2. pow2 62.5%

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

   3. associate-*r* 62.5%

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

   4. *-commutative 62.5%

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

   5. associate-*l* 62.5%

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

   6. pow2 62.5%

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

9. Applied egg-rr 62.5%

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

10. Final simplification 62.5%

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

Alternative 9: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), (Float32(Float32(pi) * x) ^ Float32(2.0)), 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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.4%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 62.4%

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

5. Taylor expanded in x around 0 62.5%

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

   1. +-commutative 62.5%

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

   2. fma-def 62.5%

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

   3. *-commutative 62.5%

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

   4. unpow2 62.5%

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

   5. unpow2 62.5%

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

   6. swap-sqr 62.5%

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

   7. unpow2 62.5%

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

   8. *-commutative 62.5%

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

7. Simplified 62.5%

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

8. Final simplification 62.5%

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

Alternative 10: 64.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (((single(pi) * x) ^ single(2.0)) * single(-0.16666666666666666));
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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.4%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 62.4%

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

5. Taylor expanded in x around 0 62.5%

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

   1. +-commutative 62.5%

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

   2. fma-def 62.5%

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

   3. *-commutative 62.5%

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

   4. unpow2 62.5%

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

   5. unpow2 62.5%

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

   6. swap-sqr 62.5%

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

   7. unpow2 62.5%

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

   8. *-commutative 62.5%

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

7. Simplified 62.5%

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

   1. fma-udef 62.5%

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

   2. *-commutative 62.5%

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

9. Applied egg-rr 62.5%

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

10. Final simplification 62.5%

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

Alternative 11: 63.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \frac{tau}{tau} \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (/ tau tau))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = tau / 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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.4%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l/ 96.9%

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

   3. associate-*r* 97.0%

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

   4. *-commutative 97.0%

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

   5. associate-*r* 96.8%

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

   6. *-commutative 96.8%

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

   7. associate-*r* 96.9%

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

   8. pow2 96.9%

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

   9. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

6. Taylor expanded in x around 0 61.6%

   \[\leadsto \frac{\color{blue}{tau}}{tau} \]

7. Final simplification 61.6%

   \[\leadsto \frac{tau}{tau} \]

Reproduce

herbie shell --seed 2023310 
(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))))