Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 13.0s

Alternatives: 14

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ t_2 := \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin t_2}{t_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)) (t_2 (expm1 (log1p (* x PI)))))
   (* (/ (sin t_1) t_1) (/ (sin t_2) t_2))))

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	float t_2 = expm1f(log1pf((x * ((float) M_PI))));
	return (sinf(t_1) / t_1) * (sinf(t_2) / t_2);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	t_2 = expm1(log1p(Float32(x * Float32(pi))))
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(t_2) / t_2))
end

Derivation

1. Initial program 97.8%

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

   1. expm1-log1p-u 97.8%

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

3. Applied egg-rr 97.8%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 97.8%

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

5. Applied egg-rr 97.8%

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

6. Final simplification 97.8%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

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(x * Float32(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 97.8%

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

2. Taylor expanded in x around inf 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*r* 97.2%

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

   3. *-commutative 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*r* 97.8%

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

   6. *-commutative 97.8%

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

4. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

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(sin(Float32(x * Float32(pi))) * Float32(Float32(sin(Float32(x * Float32(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 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

   1. clear-num 97.0%

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

   2. frac-times 97.1%

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

   3. *-un-lft-identity 97.1%

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

   4. associate-*r* 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*r* 97.2%

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

   7. associate-*r* 97.2%

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

   8. pow2 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*r* 97.1%

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

   3. unpow2 97.1%

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

   4. unpow2 97.1%

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

   5. swap-sqr 97.2%

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

   6. unpow2 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*l/ 97.1%

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

   9. *-commutative 97.1%

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

   10. associate-/r* 97.2%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

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(sin(Float32(x * Float32(pi))) * Float32(sin(Float32(Float32(x * Float32(pi)) * tau)) / Float32(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 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

   1. clear-num 97.0%

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

   2. frac-times 97.1%

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

   3. *-un-lft-identity 97.1%

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

   4. associate-*r* 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*r* 97.2%

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

   7. associate-*r* 97.2%

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

   8. pow2 97.2%

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

5. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.2%

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

   2. associate-/r/ 97.1%

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

7. Applied egg-rr 97.1%

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

8. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. unpow2 96.9%

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

   3. unpow2 96.9%

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

   4. swap-sqr 97.4%

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

   5. unpow2 97.4%

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

10. Simplified 97.4%

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

11. Final simplification 97.4%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

   1. clear-num 97.0%

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

   2. frac-times 97.1%

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

   3. *-un-lft-identity 97.1%

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

   4. associate-*r* 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*r* 97.2%

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

   7. associate-*r* 97.2%

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

   8. pow2 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around inf 97.7%

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

7. Final simplification 97.7%

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

Alternative 6: 85.1% 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 97.8%

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

   1. expm1-log1p-u 97.8%

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

3. Applied egg-rr 97.8%

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

4. Taylor expanded in x around 0 85.7%

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

   1. unpow2 85.7%

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

   2. unpow2 85.7%

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

   3. swap-sqr 85.7%

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

   4. unpow2 85.7%

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

6. Simplified 85.7%

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

7. Final simplification 85.7%

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

Alternative 7: 84.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 85.0%

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

   1. fma-def 85.0%

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

   2. associate-/r* 85.1%

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

6. Simplified 85.1%

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

7. Final simplification 85.1%

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

Alternative 8: 84.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 85.0%

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

5. Final simplification 85.0%

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

Alternative 9: 70.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 72.0%

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

5. Final simplification 72.0%

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

Alternative 10: 70.5% 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 97.8%

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

   1. expm1-log1p-u 97.8%

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

3. Applied egg-rr 97.8%

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

4. Taylor expanded in x around 0 72.3%

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

5. Final simplification 72.3%

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

Alternative 11: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 66.3%

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

5. Taylor expanded in x around 0 66.7%

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

6. Final simplification 66.7%

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

Alternative 12: 63.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 66.3%

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

5. Final simplification 66.3%

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

Alternative 13: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 66.3%

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

5. Taylor expanded in x around 0 66.7%

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

   1. +-commutative 66.7%

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

   2. fma-def 66.7%

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

   3. unpow2 66.7%

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

   4. unpow2 66.7%

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

   5. swap-sqr 66.7%

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

   6. unpow2 66.7%

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

7. Simplified 66.7%

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

8. Final simplification 66.7%

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

Alternative 14: 63.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*l* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in tau around 0 66.3%

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

   1. add-log-exp 66.0%

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

6. Applied egg-rr 66.0%

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

   1. rem-log-exp 66.3%

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

   2. associate-/r* 66.2%

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

8. Applied egg-rr 66.2%

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

9. Taylor expanded in x around 0 65.5%

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

10. Final simplification 65.5%

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

Reproduce

herbie shell --seed 2023336 
(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))))