Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.2s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-*l* 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \sin \left(x \cdot \pi\right) \cdot \frac{\sin t\_1}{\left(x \cdot \pi\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*r* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-/l* 97.0%

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

   6. unpow2 97.0%

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

   7. unpow2 97.0%

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

   8. swap-sqr 97.0%

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

   9. unpow2 97.0%

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

   10. associate-/r* 97.3%

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

7. Simplified 97.4%

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

   1. div-inv 97.2%

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

   2. associate-*r* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. unpow2 97.1%

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

   6. times-frac 97.2%

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

   7. associate-/r* 97.3%

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

   8. associate-*r* 97.6%

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

   9. *-commutative 97.6%

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

   10. *-commutative 97.6%

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

9. Applied egg-rr 97.6%

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

    1. associate-*r/ 97.7%

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

    2. *-rgt-identity 97.7%

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

    3. associate-/l/ 97.8%

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

    4. associate-*r* 97.4%

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

    5. *-commutative 97.4%

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

    6. associate-*r* 97.9%

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

    7. *-commutative 97.9%

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

11. Simplified 97.9%

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

12. Final simplification 97.9%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in x around inf 97.3%

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

6. Final simplification 97.3%

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

Alternative 5: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in x around inf 97.0%

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

6. Final simplification 97.0%

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

Alternative 6: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-*l* 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. log1p-expm1-u 97.9%

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in x around 0 80.1%

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

   1. associate-*r* 80.1%

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

   2. unpow2 80.1%

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

   3. unpow2 80.1%

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

   4. swap-sqr 80.1%

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

   5. *-commutative 80.1%

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

   6. *-commutative 80.1%

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

   7. unpow2 80.1%

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

   8. swap-sqr 80.1%

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

   9. *-commutative 80.1%

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

   10. *-commutative 80.1%

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

   11. unpow2 80.1%

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

   12. associate-*r* 80.1%

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

   13. *-commutative 80.1%

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

9. Simplified 80.1%

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

10. Final simplification 80.1%

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

Alternative 7: 80.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \left(1 + -0.16666666666666666 \cdot \left(t\_1 \cdot t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (*
    (/ (sin (* x PI)) (* x PI))
    (+ 1.0 (* -0.16666666666666666 (* 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))) * (1.0f + (-0.16666666666666666f * (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(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32(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))) * (single(1.0) + (single(-0.16666666666666666) * (t_1 * t_1)));
end

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-*l* 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. log1p-expm1-u 97.9%

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in x around 0 80.1%

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

   1. associate-*r* 80.1%

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

   2. unpow2 80.1%

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

   3. unpow2 80.1%

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

   4. swap-sqr 80.1%

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

   5. *-commutative 80.1%

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

   6. *-commutative 80.1%

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

   7. unpow2 80.1%

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

   8. swap-sqr 80.1%

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

   9. *-commutative 80.1%

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

   10. *-commutative 80.1%

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

   11. unpow2 80.1%

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

   12. associate-*r* 80.1%

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

   13. *-commutative 80.1%

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

9. Simplified 80.1%

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

    1. unpow2 80.1%

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

    2. associate-*r* 80.1%

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

    3. associate-*r* 80.1%

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

11. Applied egg-rr 80.1%

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

12. Final simplification 80.1%

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

Alternative 8: 71.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \sin t\_1 \cdot \frac{1}{t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-undefine 81.2%

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

4. Applied egg-rr 81.2%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \pi\right)} - 1\right)}}{x \cdot \pi} \]
5. Step-by-step derivation

   1. expm1-define 98.0%

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

6. Simplified 98.0%

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

   1. *-commutative 98.0%

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

   2. expm1-log1p-u 98.1%

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

   3. clear-num 98.0%

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

   4. *-commutative 98.0%

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

   5. associate-*r* 97.4%

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

   6. *-commutative 97.4%

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

   7. associate-*r* 97.9%

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

   8. clear-num 97.8%

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

   9. frac-times 97.8%

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

   10. metadata-eval 97.8%

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

8. Applied egg-rr 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-/r* 96.8%

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

   3. associate-/r* 97.0%

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

   4. associate-/l* 97.1%

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

   5. associate-*r* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*r/ 97.3%

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

10. Simplified 97.3%

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

11. Taylor expanded in x around 0 72.3%

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

    1. *-inverses 72.3%

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

    2. associate-/l/ 72.3%

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

    3. *-un-lft-identity 72.3%

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

    4. div-inv 72.3%

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

    5. clear-num 72.3%

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

    6. frac-times 72.4%

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

    7. *-inverses 72.4%

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

    8. *-un-lft-identity 72.4%

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

    9. *-inverses 72.4%

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

    10. associate-*r/ 72.3%

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

    11. times-frac 72.3%

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

    12. *-commutative 72.3%

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

    13. associate-*r* 72.3%

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

    14. *-commutative 72.3%

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

    15. *-un-lft-identity 72.3%

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

13. Applied egg-rr 72.6%

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

Alternative 9: 71.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-undefine 81.2%

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

4. Applied egg-rr 81.2%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \pi\right)} - 1\right)}}{x \cdot \pi} \]
5. Step-by-step derivation

   1. expm1-define 98.0%

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

6. Simplified 98.0%

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

   1. *-commutative 98.0%

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

   2. expm1-log1p-u 98.1%

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

   3. clear-num 98.0%

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

   4. *-commutative 98.0%

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

   5. associate-*r* 97.4%

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

   6. *-commutative 97.4%

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

   7. associate-*r* 97.9%

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

   8. clear-num 97.8%

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

   9. frac-times 97.8%

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

   10. metadata-eval 97.8%

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

8. Applied egg-rr 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-/r* 96.8%

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

   3. associate-/r* 97.0%

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

   4. associate-/l* 97.1%

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

   5. associate-*r* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*r/ 97.3%

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

10. Simplified 97.3%

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

11. Taylor expanded in x around 0 72.3%

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

12. Taylor expanded in x around inf 72.6%

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

13. Final simplification 72.6%

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

Alternative 10: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in tau around 0 66.2%

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

Alternative 11: 64.2% accurate, 219.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in x around 0 65.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024169 
(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))))