Lanczos kernel

Percentage Accurate: 97.8% → 97.8%

Time: 12.9s

Alternatives: 10

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: 97.8% 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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. associate-*l* 97.5%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 2: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((x * (single(pi) * tau))) / tau) * (sin((x * single(pi))) / ((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. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.3%

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

   3. associate-*r* 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*l* 97.1%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in x around inf 96.7%

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

   1. times-frac 96.7%

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

   2. *-commutative 96.7%

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

   3. *-commutative 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*r* 96.9%

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

   6. *-commutative 96.9%

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

8. Simplified 96.9%

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

   1. *-un-lft-identity 96.9%

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

   2. pow-prod-down 97.3%

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

10. Applied egg-rr 97.3%

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

    1. *-lft-identity 97.3%

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

12. Simplified 97.3%

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

13. Final simplification 97.3%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

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(Float32(sin(Float32(x * Float32(Float32(pi) * tau))) / 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. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

   1. associate-/l* 97.8%

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

   2. associate-/r/ 97.8%

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

   3. associate-*r* 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-*l* 97.3%

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

   6. associate-*r* 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.7%

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

5. Applied egg-rr 97.7%

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

   1. associate-*l/ 97.8%

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

   2. associate-*r* 97.2%

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

   3. *-commutative 97.2%

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

   4. times-frac 97.2%

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

   5. associate-/r* 97.1%

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

   6. unpow2 97.1%

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

   7. pow-prod-down 96.8%

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

   8. clear-num 96.8%

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

   9. *-commutative 96.8%

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

   10. associate-*r* 96.8%

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

7. Applied egg-rr 97.4%

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

8. Final simplification 97.4%

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

Alternative 4: 84.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. associate-*l* 97.5%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 84.2%

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

   1. unpow2 62.0%

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

   2. unpow2 62.0%

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

   3. swap-sqr 62.0%

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

   4. unpow2 62.0%

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

6. Simplified 84.2%

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

7. Final simplification 84.2%

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

Alternative 5: 84.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(x * Float32(Float32(pi) * 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((x * (single(pi) * 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. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.3%

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

   3. associate-*r* 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*l* 97.1%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in x around inf 96.7%

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

   1. times-frac 96.7%

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

   2. *-commutative 96.7%

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

   3. *-commutative 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*r* 96.9%

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

   6. *-commutative 96.9%

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

8. Simplified 96.9%

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

9. Taylor expanded in x around 0 83.6%

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

10. Final simplification 83.6%

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

Alternative 6: 78.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * (((single(1.0) + (tau ^ single(2.0))) * (single(pi) ^ single(2.0))) * (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/ 97.8%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in x around 0 77.5%

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

   1. *-commutative 77.5%

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

   2. distribute-lft-out 77.5%

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

   3. associate-*l* 77.5%

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

   4. distribute-lft1-in 77.5%

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

6. Simplified 77.5%

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

7. Final simplification 77.5%

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

Alternative 7: 70.4% accurate, 2.0× speedup

Math

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

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(x * Float32(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. *-commutative 97.8%

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

   2. associate-*l* 97.5%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. expm1-log1p-u 97.9%

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

5. Applied egg-rr 97.9%

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

6. Taylor expanded in x around 0 68.8%

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

7. Final simplification 68.8%

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

Alternative 8: 63.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * (x * (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. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in tau around 0 61.8%

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

5. Taylor expanded in x around 0 62.0%

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

   1. unpow2 62.0%

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

   2. unpow2 62.0%

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

   3. swap-sqr 62.0%

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

   4. unpow2 62.0%

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

7. Simplified 62.0%

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

   1. pow-prod-down 62.0%

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

   2. *-commutative 62.0%

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

   3. unpow2 62.0%

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

   4. associate-*r* 62.0%

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

9. Applied egg-rr 62.0%

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

10. Final simplification 62.0%

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

Alternative 9: 63.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in tau around 0 61.8%

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

5. Taylor expanded in x around 0 62.0%

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

   1. unpow2 62.0%

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

   2. unpow2 62.0%

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

   3. swap-sqr 62.0%

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

   4. unpow2 62.0%

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

7. Simplified 62.0%

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

8. Final simplification 62.0%

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

Alternative 10: 62.9% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.8%

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

3. Simplified 97.8%

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

   1. associate-/l* 97.8%

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

   2. associate-/r/ 97.8%

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

   3. associate-*r* 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-*l* 97.3%

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

   6. associate-*r* 97.4%

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

   7. *-commutative 97.4%

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

   8. associate-*l* 97.7%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in x around 0 60.9%

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

7. Final simplification 60.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(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))))