Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.5s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

   1. clear-num 97.2%

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

   2. frac-times 97.3%

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

   3. *-un-lft-identity 97.3%

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

   4. associate-*r* 97.3%

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

   5. *-commutative 97.3%

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

   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* 97.4%

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

   8. pow2 97.4%

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

   9. *-commutative 97.4%

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

   10. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

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%

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

   2. unpow2 96.9%

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

   3. unpow2 96.9%

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

   4. swap-sqr 97.3%

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

   5. unpow2 97.3%

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

   6. *-commutative 97.3%

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

   7. *-commutative 97.3%

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

   8. *-commutative 97.3%

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

   9. associate-*r* 97.3%

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

   10. *-commutative 97.3%

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

   11. associate-*r/ 97.2%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

   1. clear-num 97.2%

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

   2. frac-times 97.3%

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

   3. *-un-lft-identity 97.3%

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

   4. associate-*r* 97.3%

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

   5. *-commutative 97.3%

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

   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* 97.4%

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

   8. pow2 97.4%

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

   9. *-commutative 97.4%

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

   10. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

   1. expm1-log1p-u 97.3%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot x\right)}^{2}\right)\right)}}{\sin \left(\pi \cdot x\right)} \cdot tau} \]

7. Applied egg-rr 97.3%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\pi \cdot x\right)}^{2}\right)\right)}}{\sin \left(\pi \cdot x\right)} \cdot tau} \]

8. Taylor expanded in x around inf 96.9%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 4: 84.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. associate-/r* 97.6%

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

   2. div-inv 97.7%

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

   3. *-commutative 97.7%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in x around 0 85.8%

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

   1. *-commutative 85.8%

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

   2. unpow2 85.8%

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

   3. unpow2 85.8%

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

   4. swap-sqr 85.8%

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

   5. unpow2 85.8%

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

   6. *-commutative 85.8%

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

8. Simplified 85.8%

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

9. Final simplification 85.8%

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

Alternative 5: 84.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around 0 85.2%

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

5. Final simplification 85.2%

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

Alternative 6: 70.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around 0 70.3%

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

5. Final simplification 70.3%

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

Alternative 7: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. associate-/r* 97.6%

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

   2. div-inv 97.7%

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

   3. *-commutative 97.7%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in x around 0 70.4%

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

7. Final simplification 70.4%

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

Alternative 8: 64.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 63.5%

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

5. Final simplification 63.5%

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

Alternative 9: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 63.5%

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

5. Taylor expanded in x around 0 63.7%

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

   1. +-commutative 63.7%

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

   2. fma-def 63.7%

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

   3. *-commutative 63.7%

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

   4. unpow2 63.7%

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

   5. unpow2 63.7%

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

   6. swap-sqr 63.7%

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

   7. unpow2 63.7%

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

   8. *-commutative 63.7%

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

7. Simplified 63.7%

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

8. Final simplification 63.7%

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

Alternative 10: 63.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0, 1\right) \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (fma x 0.0 1.0))

C

float code(float x, float tau) {
	return fmaf(x, 0.0f, 1.0f);
}

Julia

function code(x, tau)
	return fma(x, Float32(0.0), 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/ 97.7%

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

   2. *-commutative 97.7%

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

   3. times-frac 97.5%

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

   4. associate-/l/ 97.5%

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

   5. associate-*l* 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 63.5%

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

   1. *-commutative 63.5%

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

   2. expm1-log1p-u 63.5%

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

   3. expm1-udef 55.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\pi \cdot x\right)\right)} - 1}}{x \cdot \pi} \]

6. Applied egg-rr 55.2%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin \left(\pi \cdot x\right)\right)} - 1}}{x \cdot \pi} \]

7. Taylor expanded in x around 0 62.6%

   \[\leadsto \color{blue}{1 + x \cdot \left(-0.5 \cdot \pi + 0.5 \cdot \pi\right)} \]
8. Step-by-step derivation

   1. +-commutative 62.6%

      \[\leadsto \color{blue}{x \cdot \left(-0.5 \cdot \pi + 0.5 \cdot \pi\right) + 1} \]

   2. fma-def 62.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.5 \cdot \pi + 0.5 \cdot \pi, 1\right)} \]

   3. distribute-rgt-out 62.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\pi \cdot \left(-0.5 + 0.5\right)}, 1\right) \]

   4. metadata-eval 62.6%

      \[\leadsto \mathsf{fma}\left(x, \pi \cdot \color{blue}{0}, 1\right) \]

   5. mul0-rgt 62.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0}, 1\right) \]

9. Simplified 62.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0, 1\right)} \]

10. Final simplification 62.6%

    \[\leadsto \mathsf{fma}\left(x, 0, 1\right) \]

Reproduce

herbie shell --seed 2023320 
(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))))