Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.6s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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* 97.3%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 2: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l/ 97.1%

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

   3. associate-/l* 96.8%

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

   4. associate-*r* 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.5%

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

   7. associate-/l* 96.7%

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

   8. *-commutative 96.7%

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

   9. associate-*r* 96.7%

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

   10. swap-sqr 96.9%

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

   11. pow2 96.9%

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

   12. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

   1. associate-/r/ 96.9%

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

   2. frac-times 97.0%

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

   3. associate-*r* 97.5%

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

   4. *-commutative 97.5%

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

   5. *-commutative 97.5%

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

   6. *-commutative 97.5%

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

   7. pow-prod-down 96.9%

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

   8. div-inv 96.8%

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

   9. *-commutative 96.8%

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

   10. *-commutative 96.8%

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

   11. associate-*r* 96.7%

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

7. Applied egg-rr 96.9%

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

8. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. *-commutative 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r* 96.8%

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

   5. unpow2 96.8%

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

   6. unpow2 96.8%

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

   7. swap-sqr 97.0%

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

   8. unpow2 97.0%

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

   9. *-lft-identity 97.0%

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

   10. associate-*l/ 96.9%

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

10. Simplified 97.3%

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

11. Final simplification 97.3%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((x * single(pi))) * (sin((tau * (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. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l/ 97.1%

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

   3. associate-/l* 96.8%

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

   4. associate-*r* 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.5%

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

   7. associate-/l* 96.7%

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

   8. *-commutative 96.7%

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

   9. associate-*r* 96.7%

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

   10. swap-sqr 96.9%

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

   11. pow2 96.9%

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

   12. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

   1. expm1-log1p-u 96.8%

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

7. Applied egg-rr 96.8%

   \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{tau}{\frac{\sin \left(\pi \cdot x\right)}{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}}^{2}}}} \]
8. Step-by-step derivation

   1. associate-/r/ 96.9%

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

   2. div-inv 96.9%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau} \cdot \color{blue}{\left(\sin \left(\pi \cdot x\right) \cdot \frac{1}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}^{2}}\right)} \]

   3. associate-*r* 97.0%

      \[\leadsto \color{blue}{\left(\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau} \cdot \sin \left(\pi \cdot x\right)\right) \cdot \frac{1}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}^{2}}} \]

   4. expm1-log1p-u 97.0%

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

   5. pow-flip 97.0%

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

   6. metadata-eval 97.0%

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

9. Applied egg-rr 97.0%

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

10. Taylor expanded in x around -inf 97.5%

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

11. Final simplification 97.5%

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

Alternative 4: 85.7% accurate, 1.2× speedup

Math

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

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(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * Float32(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32(x * Float32(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. Step-by-step derivation

   1. associate-*l* 97.3%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 85.6%

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

   1. associate-*r* 85.6%

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

   2. unpow2 85.6%

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

6. Simplified 85.6%

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

7. Taylor expanded in x around 0 85.6%

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

   1. unpow2 85.6%

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

   2. *-commutative 85.6%

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

   3. unpow2 85.6%

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

   4. swap-sqr 85.6%

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

   5. unpow2 85.6%

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

   6. *-commutative 85.6%

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

9. Simplified 85.6%

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

    1. unpow-prod-down 85.6%

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

    2. exp-to-pow 85.6%

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

    3. pow2 85.6%

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

    4. associate-*r* 85.6%

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

    5. *-commutative 85.6%

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

    6. exp-to-pow 85.6%

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

11. Applied egg-rr 85.6%

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

12. Final simplification 85.6%

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

Alternative 5: 85.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (single(1.0) + (single(-0.16666666666666666) * ((x * single(pi)) ^ single(2.0)))) * (sin((tau * (x * single(pi)))) / (x * (single(pi) * tau)));
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* 97.3%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 85.6%

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

   1. associate-*r* 85.6%

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

   2. unpow2 85.6%

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

6. Simplified 85.6%

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

7. Taylor expanded in x around 0 85.6%

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

   1. unpow2 85.6%

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

   2. *-commutative 85.6%

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

   3. unpow2 85.6%

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

   4. swap-sqr 85.6%

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

   5. unpow2 85.6%

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

   6. *-commutative 85.6%

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

9. Simplified 85.6%

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

10. Taylor expanded in x around -inf 85.1%

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

11. Final simplification 85.1%

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

Alternative 6: 85.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (* (/ (sin t_1) t_1) (+ 1.0 (* -0.16666666666666666 (pow (* x 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 * powf((x * ((float) M_PI)), 2.0f)));
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * Float32(Float32(1.0) + Float32(Float32(-0.16666666666666666) * (Float32(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 * 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* 97.3%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 85.6%

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

   1. associate-*r* 85.6%

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

   2. unpow2 85.6%

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

6. Simplified 85.6%

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

7. Taylor expanded in x around 0 85.6%

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

   1. unpow2 85.6%

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

   2. *-commutative 85.6%

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

   3. unpow2 85.6%

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

   4. swap-sqr 85.6%

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

   5. unpow2 85.6%

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

   6. *-commutative 85.6%

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

9. Simplified 85.6%

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

10. Final simplification 85.6%

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

Alternative 7: 85.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(sin(Float32(Float32(pi) * Float32(x * tau))) / Float32(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((single(pi) * (x * 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. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l/ 97.1%

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

   3. associate-/l* 96.8%

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

   4. associate-*r* 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.5%

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

   7. associate-/l* 96.7%

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

   8. *-commutative 96.7%

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

   9. associate-*r* 96.7%

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

   10. swap-sqr 96.9%

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

   11. pow2 96.9%

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

   12. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

   1. expm1-log1p-u 96.8%

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

7. Applied egg-rr 96.8%

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

8. Taylor expanded in x around 0 84.9%

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

9. Final simplification 84.9%

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

Alternative 8: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around 0 79.2%

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

   1. distribute-lft-out 79.2%

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

   2. *-commutative 79.2%

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

   3. unpow2 79.2%

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

   4. unpow2 79.2%

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

6. Simplified 79.2%

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

7. Final simplification 79.2%

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

Alternative 9: 79.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * Float32((Float32(pi) ^ Float32(2.0)) * Float32(Float32(1.0) + 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. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around 0 79.2%

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

   1. +-commutative 79.2%

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

   2. fma-def 79.2%

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

   3. distribute-lft-out 79.2%

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

   4. distribute-rgt1-in 79.2%

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

   5. unpow2 79.2%

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

   6. unpow2 79.2%

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

6. Simplified 79.2%

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

7. Final simplification 79.2%

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

Alternative 10: 79.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(x * x), Float32((Float32(pi) ^ Float32(2.0)) * Float32(Float32(-0.16666666666666666) + Float32(Float32(-0.16666666666666666) * 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. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

   1. *-commutative 96.9%

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

   2. associate-*l/ 97.1%

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

   3. associate-/l* 96.8%

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

   4. associate-*r* 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 96.5%

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

   7. associate-/l* 96.7%

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

   8. *-commutative 96.7%

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

   9. associate-*r* 96.7%

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

   10. swap-sqr 96.9%

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

   11. pow2 96.9%

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

   12. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

6. Taylor expanded in x around 0 79.2%

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

   1. +-commutative 79.2%

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

   2. *-commutative 79.2%

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

   3. fma-def 79.2%

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

   4. unpow2 79.2%

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

   5. associate-*r* 79.2%

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

   6. distribute-rgt-out 79.2%

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

   7. unpow2 79.2%

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

8. Simplified 79.2%

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

9. Final simplification 79.2%

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

Alternative 11: 71.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (x * single(pi));
	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. associate-/r* 97.6%

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

   2. div-inv 97.6%

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

   3. *-commutative 97.6%

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

3. Applied egg-rr 97.6%

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

4. Taylor expanded in x around 0 71.8%

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

5. Final simplification 71.8%

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

Alternative 12: 65.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in tau around 0 65.1%

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

   1. *-commutative 65.1%

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

6. Simplified 65.1%

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

   1. div-inv 65.0%

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

8. Applied egg-rr 65.0%

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

9. Taylor expanded in x around 0 65.1%

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

10. Final simplification 65.1%

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

Alternative 13: 65.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in tau around 0 65.1%

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

   1. *-commutative 65.1%

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

6. Simplified 65.1%

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

   1. div-inv 65.0%

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

8. Applied egg-rr 65.0%

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

9. Taylor expanded in x around 0 65.1%

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

    1. +-commutative 65.1%

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

    2. unpow2 65.1%

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

    3. *-commutative 65.1%

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

    4. unpow2 65.1%

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

    5. swap-sqr 65.1%

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

    6. unpow2 65.1%

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

    7. fma-def 65.1%

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

    8. *-commutative 65.1%

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

11. Simplified 65.1%

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

12. Final simplification 65.1%

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

Alternative 14: 65.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in tau around 0 65.1%

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

   1. *-commutative 65.1%

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

6. Simplified 65.1%

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

   1. div-inv 65.0%

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

8. Applied egg-rr 65.0%

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

9. Taylor expanded in x around 0 65.1%

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

    1. unpow2 65.1%

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

    2. *-commutative 65.1%

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

    3. unpow2 65.1%

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

    4. swap-sqr 65.1%

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

    5. unpow2 65.1%

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

    6. *-commutative 65.1%

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

11. Simplified 65.1%

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

12. Final simplification 65.1%

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

Alternative 15: 64.2% 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. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.4%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around 0 64.4%

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

5. Final simplification 64.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023182 
(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))))