Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.7s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 98.1%

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

2. Final simplification 98.1%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 96.9%

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

   7. associate-*r* 97.0%

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

   8. swap-sqr 97.2%

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

   9. pow2 97.2%

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

   10. *-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*r* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. unpow2 97.1%

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

   7. unpow2 97.1%

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

   8. swap-sqr 97.1%

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

   9. unpow2 97.1%

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

   10. associate-*l/ 97.2%

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

   11. *-commutative 97.2%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 96.9%

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

   7. associate-*r* 97.0%

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

   8. swap-sqr 97.2%

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

   9. pow2 97.2%

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

   10. *-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

   1. div-inv 97.1%

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

   2. associate-/l* 97.1%

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

   3. pow-flip 97.1%

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

   4. metadata-eval 97.1%

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

7. Applied egg-rr 97.1%

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

   1. associate-/r/ 97.3%

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

   2. *-commutative 97.3%

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

   3. associate-*r* 97.6%

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

   4. *-commutative 97.6%

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

   5. associate-*r* 97.3%

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

   6. *-commutative 97.3%

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

   7. *-commutative 97.3%

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

   8. *-commutative 97.3%

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

   9. *-commutative 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 5: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around inf 96.9%

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

   1. times-frac 96.9%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. *-commutative 97.0%

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

   5. *-commutative 97.0%

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

   6. unpow2 97.0%

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

   7. unpow2 97.0%

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

   8. swap-sqr 97.3%

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

   9. unpow2 97.3%

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

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Alternative 6: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. clear-num 97.0%

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

   2. un-div-inv 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-/l* 96.9%

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

   6. associate-*r* 96.9%

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

   7. swap-sqr 97.3%

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

   8. pow2 97.3%

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

   9. *-commutative 97.3%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 7: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-*l/ 96.8%

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

   3. associate-/r* 97.0%

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

   4. associate-*l/ 97.1%

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

   5. *-commutative 97.1%

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

   6. associate-*r* 96.9%

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

   7. *-commutative 96.9%

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

   8. associate-*l* 97.0%

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

   9. associate-*r* 97.2%

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

   10. swap-sqr 97.4%

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

   11. pow2 97.4%

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

   12. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 8: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-/r* 96.8%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in x around 0 97.5%

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

7. Final simplification 97.5%

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

Alternative 9: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 96.9%

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

   7. associate-*r* 97.0%

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

   8. swap-sqr 97.2%

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

   9. pow2 97.2%

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

   10. *-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around -inf 97.5%

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

7. Final simplification 97.5%

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

Alternative 10: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in x around 0 86.1%

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

   1. associate-*r* 86.1%

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

   2. unpow2 86.1%

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

4. Simplified 86.1%

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

5. Final simplification 86.1%

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

Alternative 11: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 86.1%

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

   1. associate-*r* 86.1%

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

   2. unpow2 86.1%

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

6. Simplified 86.1%

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

7. Taylor expanded in x around 0 86.1%

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

   1. unpow2 86.1%

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

   2. *-commutative 86.1%

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

   3. unpow2 86.1%

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

   4. swap-sqr 86.1%

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

   5. unpow2 86.1%

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

   6. *-commutative 86.1%

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

9. Simplified 86.1%

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

10. Final simplification 86.1%

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

Alternative 12: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

   1. associate-/r* 97.7%

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

   2. div-inv 97.7%

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

   3. *-commutative 97.7%

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

3. Applied egg-rr 97.7%

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

4. Taylor expanded in x around 0 86.1%

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

   1. unpow2 86.1%

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

   2. *-commutative 86.1%

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

   3. unpow2 86.1%

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

   4. swap-sqr 86.1%

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

   5. unpow2 86.1%

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

   6. *-commutative 86.1%

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

6. Simplified 86.1%

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

7. Final simplification 86.1%

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

Alternative 13: 80.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. add-exp-log 92.5%

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

   2. associate-*r/ 92.5%

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

   3. associate-*l/ 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*r* 92.2%

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

   6. *-commutative 92.2%

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

   7. associate-*l* 92.4%

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

   8. associate-/r* 92.5%

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

   9. *-commutative 92.5%

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

5. Applied egg-rr 92.8%

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

6. Taylor expanded in x around 0 81.4%

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

   1. distribute-lft-out 81.4%

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

   2. distribute-rgt1-in 81.4%

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

   3. unpow2 81.4%

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

   4. unpow2 81.4%

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

8. Simplified 81.4%

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

9. Final simplification 81.4%

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

Alternative 14: 78.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-/r* 96.8%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in x around 0 79.3%

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

   1. fma-def 79.3%

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

   2. distribute-lft-out 79.3%

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

   3. distribute-rgt-out 79.3%

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

   4. unpow2 79.3%

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

8. Simplified 79.3%

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

9. Taylor expanded in tau around 0 79.8%

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

    1. +-commutative 79.8%

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

    2. distribute-lft-out 79.8%

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

    3. fma-def 79.8%

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

    4. *-commutative 79.8%

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

    5. distribute-lft1-in 79.8%

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

    6. unpow2 79.8%

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

    7. *-commutative 79.8%

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

    8. unpow2 79.8%

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

    9. unpow2 79.8%

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

    10. swap-sqr 79.8%

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

    11. unpow2 79.8%

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

11. Simplified 79.8%

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

12. Final simplification 79.8%

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

Alternative 15: 70.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 96.9%

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

   7. associate-*r* 97.0%

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

   8. swap-sqr 97.2%

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

   9. pow2 97.2%

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

   10. *-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 71.1%

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

7. Taylor expanded in x around -inf 71.4%

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

8. Final simplification 71.4%

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

Alternative 16: 70.6% 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 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 96.9%

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

   7. associate-*r* 97.0%

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

   8. swap-sqr 97.2%

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

   9. pow2 97.2%

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

   10. *-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 71.1%

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

7. Taylor expanded in x around inf 71.4%

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

   1. associate-*r* 71.2%

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

   2. *-commutative 71.2%

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

   3. *-commutative 71.2%

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

   4. *-commutative 71.2%

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

   5. *-commutative 71.2%

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

   6. associate-*r* 71.4%

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

   7. *-commutative 71.4%

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

9. Simplified 71.4%

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

10. Final simplification 71.4%

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

Alternative 17: 69.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.9%

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

   5. *-commutative 96.9%

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

   6. associate-*l* 96.9%

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

   7. associate-*r* 97.0%

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

   8. swap-sqr 97.2%

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

   9. pow2 97.2%

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

   10. *-commutative 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 71.1%

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

7. Taylor expanded in x around 0 70.3%

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

   1. unpow2 70.3%

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

   2. unpow2 70.3%

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

9. Simplified 70.3%

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

10. Final simplification 70.3%

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

Alternative 18: 64.2% 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 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in tau around 0 65.2%

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

   1. *-commutative 65.2%

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

6. Simplified 65.2%

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

   1. div-inv 65.3%

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

8. Applied egg-rr 65.3%

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

9. Taylor expanded in x around 0 65.1%

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

    1. unpow2 65.1%

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

    2. *-commutative 65.1%

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

11. Simplified 65.1%

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

12. Taylor expanded in x around 0 65.1%

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

    1. unpow2 65.1%

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

    2. unpow2 65.1%

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

    3. swap-sqr 65.1%

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

    4. unpow2 65.1%

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

14. Simplified 65.1%

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

15. Final simplification 65.1%

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

Alternative 19: 63.2% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around 0 64.3%

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

5. Final simplification 64.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023216 
(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))))