Lanczos kernel

Percentage Accurate: 97.8% → 97.8%

Time: 12.7s

Alternatives: 18

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

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

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 98.0%

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

2. Final simplification 98.0%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 97.8% 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 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

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. Final simplification 98.0%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.7%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.1%

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

   8. associate-*r* 97.0%

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

   9. pow2 97.0%

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

   10. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

6. Taylor expanded in tau 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. unpow2 96.7%

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

   2. unpow2 96.7%

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

   3. swap-sqr 97.3%

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

   4. unpow2 97.3%

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

   5. *-commutative 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*r* 97.0%

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

   8. associate-*l/ 97.0%

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

   9. *-commutative 97.0%

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

   10. associate-*r* 97.1%

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

   11. *-commutative 97.1%

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

8. Simplified 97.1%

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

9. Final simplification 97.1%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

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(Float32(x * 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 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around inf 96.7%

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

   1. times-frac 97.0%

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

   2. *-commutative 97.0%

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

   3. *-commutative 97.0%

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

   4. *-commutative 97.0%

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

   5. unpow2 97.0%

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

   6. unpow2 97.0%

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

   7. swap-sqr 97.3%

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

   8. unpow2 97.3%

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

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Alternative 6: 84.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

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. Taylor expanded in x around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

7. Final simplification 84.6%

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

Alternative 7: 84.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0 84.6%

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

   1. unpow2 84.6%

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

4. Simplified 84.6%

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

5. Final simplification 84.6%

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

Alternative 8: 84.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.7%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.1%

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

   8. associate-*r* 97.0%

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

   9. pow2 97.0%

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

   10. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 97.3%

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

   6. associate-*l* 97.0%

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

   7. *-commutative 97.0%

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

   8. *-commutative 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in x around 0 83.8%

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

   1. fma-def 83.8%

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

10. Simplified 83.8%

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

11. Final simplification 83.8%

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

Alternative 9: 84.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.7%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.1%

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

   8. associate-*r* 97.0%

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

   9. pow2 97.0%

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

   10. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 97.3%

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

   6. associate-*l* 97.0%

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

   7. *-commutative 97.0%

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

   8. *-commutative 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in x around 0 83.8%

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

9. Final simplification 83.8%

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

Alternative 10: 78.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 78.4%

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

   1. +-commutative 78.4%

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

   2. fma-def 78.4%

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

   3. unpow2 78.4%

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

   4. distribute-lft-out 78.4%

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

   5. distribute-lft1-in 78.4%

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

   6. unpow2 78.4%

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

6. Simplified 78.4%

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

7. Final simplification 78.4%

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

Alternative 11: 78.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.7%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.1%

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

   8. associate-*r* 97.0%

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

   9. pow2 97.0%

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

   10. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 97.3%

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

   6. associate-*l* 97.0%

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

   7. *-commutative 97.0%

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

   8. *-commutative 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in x around 0 78.4%

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

   1. +-commutative 78.4%

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

   2. unpow2 78.4%

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

   3. *-commutative 78.4%

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

   4. fma-def 78.4%

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

   5. associate-*r* 78.4%

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

   6. distribute-rgt-out 78.4%

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

   7. unpow2 78.4%

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

10. Simplified 78.4%

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

11. Final simplification 78.4%

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

Alternative 12: 70.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.7%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.1%

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

   8. associate-*r* 97.0%

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

   9. pow2 97.0%

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

   10. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 97.3%

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

   6. associate-*l* 97.0%

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

   7. *-commutative 97.0%

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

   8. *-commutative 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in x around 0 69.9%

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

9. Final simplification 69.9%

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

Alternative 13: 70.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return (sinf((x * (((float) M_PI) * tau))) / tau) * ((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(1.0) / x) / Float32(pi)))
end

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.7%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.1%

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

   8. associate-*r* 97.0%

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

   9. pow2 97.0%

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

   10. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. times-frac 97.3%

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

   6. associate-*l* 97.0%

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

   7. *-commutative 97.0%

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

   8. *-commutative 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in x around 0 69.9%

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

   1. associate-/r* 69.9%

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

10. Simplified 69.9%

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

11. Final simplification 69.9%

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

Alternative 14: 70.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

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. Step-by-step derivation

   1. expm1-log1p-u 97.6%

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

   2. expm1-udef 97.6%

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

   3. *-commutative 97.6%

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

   4. *-commutative 97.6%

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

5. Applied egg-rr 97.6%

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

6. Taylor expanded in x around 0 70.2%

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

7. Final simplification 70.2%

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

Alternative 15: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 64.4%

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

   1. *-commutative 64.4%

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

   2. *-commutative 64.4%

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

6. Simplified 64.4%

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

7. Taylor expanded in x around 0 64.5%

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

   1. unpow2 64.5%

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

   2. unpow2 64.5%

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

   3. swap-sqr 64.5%

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

   4. unpow2 64.5%

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

9. Simplified 64.5%

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

    1. *-commutative 64.5%

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

    2. unpow-prod-down 64.5%

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

    3. pow2 64.5%

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

11. Applied egg-rr 64.5%

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

    1. +-commutative 64.5%

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

    2. associate-*r* 64.5%

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

    3. fma-def 64.5%

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

13. Applied egg-rr 64.5%

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

14. Final simplification 64.5%

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

Alternative 16: 64.1% accurate, 2.9× speedup

Math

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

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(Float32(x * 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 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 64.4%

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

   1. *-commutative 64.4%

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

   2. *-commutative 64.4%

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

6. Simplified 64.4%

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

7. Taylor expanded in x around 0 64.5%

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

   1. unpow2 64.5%

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

   2. unpow2 64.5%

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

   3. swap-sqr 64.5%

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

   4. unpow2 64.5%

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

9. Simplified 64.5%

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

    1. *-commutative 64.5%

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

    2. unpow-prod-down 64.5%

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

    3. pow2 64.5%

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

11. Applied egg-rr 64.5%

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

12. Final simplification 64.5%

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

Alternative 17: 64.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 64.4%

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

   1. *-commutative 64.4%

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

   2. *-commutative 64.4%

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

6. Simplified 64.4%

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

7. Taylor expanded in x around 0 64.5%

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

   1. unpow2 64.5%

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

   2. unpow2 64.5%

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

   3. swap-sqr 64.5%

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

   4. unpow2 64.5%

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

9. Simplified 64.5%

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

10. Final simplification 64.5%

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

Alternative 18: 63.1% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. times-frac 97.5%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 63.6%

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

5. Final simplification 63.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(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))))