Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 15.0s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around inf 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*r* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-*r* 98.0%

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

   6. *-commutative 98.0%

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

4. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * single(pi))) * ((sin((x * (tau * single(pi)))) / tau) / ((x * single(pi)) ^ single(2.0)));
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

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. times-frac 97.0%

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

   2. *-commutative 97.0%

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

   3. associate-*r* 97.1%

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

   4. unpow2 97.1%

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

   5. unpow2 97.1%

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

   6. swap-sqr 97.2%

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

   7. unpow2 97.2%

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

   8. associate-*r/ 97.3%

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

   9. associate-*l/ 97.3%

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

   10. *-commutative 97.3%

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

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((single(pi) * (x * tau))) * (sin((x * single(pi))) / (tau * ((x * single(pi)) ^ single(2.0))));
end

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.6%

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

   6. associate-/l/ 97.5%

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

   7. associate-*r* 97.3%

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

   8. *-commutative 97.3%

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

   9. associate-*r* 97.0%

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

   10. associate-*l* 97.1%

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

3. Simplified 97.1%

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

   1. *-commutative 97.1%

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

   2. associate-*l* 97.0%

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

   3. associate-*l* 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*l* 97.5%

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

   6. *-commutative 97.5%

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

   7. add-sqr-sqrt 96.7%

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

   8. pow2 96.7%

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

   9. associate-*l* 96.5%

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

   10. *-commutative 96.5%

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

   11. *-commutative 96.5%

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

   12. associate-*r* 96.5%

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

   13. associate-*r* 96.5%

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

   14. sqrt-prod 96.5%

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

   15. sqrt-prod 96.5%

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

   16. add-sqr-sqrt 96.5%

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

5. Applied egg-rr 96.5%

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

6. Taylor expanded in x around inf 97.0%

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

   1. *-commutative 97.0%

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

   2. *-commutative 97.0%

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

   3. associate-*r* 97.1%

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

   4. associate-*r* 97.0%

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

   5. unpow2 97.0%

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

   6. associate-*r* 97.3%

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

   7. *-commutative 97.3%

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

   8. associate-*r* 97.3%

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

   9. *-commutative 97.3%

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

   10. associate-*r* 97.1%

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

   11. associate-*l/ 97.2%

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

   12. *-commutative 97.2%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (x * single(pi)))) / (tau * (((x * single(pi)) ^ single(2.0)) / sin((x * single(pi)))));
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. clear-num 97.2%

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

   2. frac-times 97.2%

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

   3. *-un-lft-identity 97.2%

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

   4. associate-*r* 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*r* 97.2%

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

   7. associate-*r* 97.2%

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

   8. pow2 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around inf 97.5%

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

7. Final simplification 97.5%

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

Alternative 5: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. add-log-exp 97.5%

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

5. Applied egg-rr 97.5%

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

6. Taylor expanded in x around 0 84.9%

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

   1. unpow2 84.9%

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

   2. unpow2 84.9%

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

   3. swap-sqr 84.9%

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

   4. unpow2 84.9%

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

8. Simplified 84.9%

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

   1. unpow2 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*r* 84.9%

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

10. Applied egg-rr 84.9%

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

11. Final simplification 84.9%

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

Alternative 6: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. add-log-exp 97.5%

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

5. Applied egg-rr 97.5%

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

6. Taylor expanded in x around 0 84.9%

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

   1. unpow2 84.9%

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

   2. unpow2 84.9%

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

   3. swap-sqr 84.9%

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

   4. unpow2 84.9%

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

8. Simplified 84.9%

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

9. Taylor expanded in x around inf 84.9%

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

10. Final simplification 84.9%

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

Alternative 7: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. add-log-exp 97.5%

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

5. Applied egg-rr 97.5%

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

6. Taylor expanded in x around 0 84.9%

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

   1. unpow2 84.9%

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

   2. unpow2 84.9%

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

   3. swap-sqr 84.9%

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

   4. unpow2 84.9%

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

8. Simplified 84.9%

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

9. Taylor expanded in x around inf 84.9%

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

    1. *-commutative 84.9%

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

    2. associate-*r* 84.3%

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

    3. *-commutative 84.3%

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

    4. associate-*r* 84.9%

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

11. Simplified 84.9%

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

12. Final simplification 84.9%

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

Alternative 8: 85.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))))
   (* (/ (sin t_1) t_1) (+ 1.0 (* (pow (* x PI) 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((x * ((float) M_PI)), 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(x * Float32(pi)) ^ 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) + (((x * single(pi)) ^ single(2.0)) * single(-0.16666666666666666)));
end

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. add-log-exp 97.5%

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

5. Applied egg-rr 97.5%

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

6. Taylor expanded in x around 0 84.9%

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

   1. unpow2 84.9%

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

   2. unpow2 84.9%

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

   3. swap-sqr 84.9%

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

   4. unpow2 84.9%

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

8. Simplified 84.9%

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

9. Final simplification 84.9%

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

Alternative 9: 84.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. associate-*r* 97.3%

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

   3. pow2 97.3%

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

   4. associate-*r* 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 84.2%

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

7. Final simplification 84.2%

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

Alternative 10: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot {1}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))))
   (* (/ (sin t_1) t_1) (pow 1.0 0.3333333333333333))))

C

float code(float x, float tau) {
	float t_1 = ((float) M_PI) * (x * tau);
	return (sinf(t_1) / t_1) * powf(1.0f, 0.3333333333333333f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(pi) * Float32(x * tau))
	return Float32(Float32(sin(t_1) / t_1) * (Float32(1.0) ^ Float32(0.3333333333333333)))
end

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / t_1) * (single(1.0) ^ single(0.3333333333333333));
end

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. add-cbrt-cube 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \color{blue}{\sqrt[3]{\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}} \]

   2. pow1/3 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 97.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

5. Applied egg-rr 97.9%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot \color{blue}{{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in x around 0 71.1%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot {\color{blue}{1}}^{0.3333333333333333} \]

7. Final simplification 71.1%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi \cdot \left(x \cdot tau\right)} \cdot {1}^{0.3333333333333333} \]

Alternative 11: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. associate-*r* 97.3%

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

   3. pow2 97.3%

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

   4. associate-*r* 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 70.8%

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

7. Final simplification 70.8%

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

Alternative 12: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in tau around 0 64.4%

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

5. Taylor expanded in x around 0 64.9%

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

   1. +-commutative 64.9%

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

   2. fma-def 64.9%

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

   3. unpow2 64.9%

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

   4. unpow2 64.9%

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

   5. swap-sqr 64.9%

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

   6. unpow2 64.9%

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

7. Simplified 64.9%

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

8. Final simplification 64.9%

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

Alternative 13: 64.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in tau around 0 64.4%

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

   1. add-exp-log 63.7%

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

6. Applied egg-rr 63.7%

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

7. Taylor expanded in x around 0 64.9%

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

   1. unpow2 64.9%

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

   2. unpow2 64.9%

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

   3. swap-sqr 64.9%

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

   4. unpow2 64.9%

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

9. Simplified 64.9%

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

10. Final simplification 64.9%

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

Alternative 14: 63.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \pi}{x \cdot \pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l/ 97.9%

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

   2. *-commutative 97.9%

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

   3. times-frac 97.7%

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

   4. associate-/l/ 97.6%

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

   5. associate-*l* 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in tau around 0 64.4%

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

5. Taylor expanded in x around 0 63.6%

   \[\leadsto \frac{\color{blue}{x \cdot \pi}}{x \cdot \pi} \]

6. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2023311 
(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))))