Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 14.3s

Alternatives: 18

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: 98.0% 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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := tau \cdot \left(x \cdot \pi\right)\\ \frac{\sin t_1}{t_1} \cdot e^{{\left(\sqrt[3]{\log \left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* tau (* x PI))))
   (*
    (/ (sin t_1) t_1)
    (exp (pow (cbrt (log (/ (sin (* x PI)) (* x PI)))) 3.0)))))

C

float code(float x, float tau) {
	float t_1 = tau * (x * ((float) M_PI));
	return (sinf(t_1) / t_1) * expf(powf(cbrtf(logf((sinf((x * ((float) M_PI))) / (x * ((float) M_PI))))), 3.0f));
}

Julia

function code(x, tau)
	t_1 = Float32(tau * Float32(x * Float32(pi)))
	return Float32(Float32(sin(t_1) / t_1) * exp((cbrt(log(Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))) ^ Float32(3.0))))
end

Derivation

1. Initial program 98.1%

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

   1. add-exp-log 98.1%

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

   2. *-commutative 98.1%

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

   3. *-commutative 98.1%

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

3. Applied egg-rr 98.1%

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

   1. add-cube-cbrt 98.1%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x}\right)} \cdot \sqrt[3]{\log \left(\frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x}\right)}}} \]

   2. pow3 98.1%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot e^{\color{blue}{{\left(\sqrt[3]{\log \left(\frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x}\right)}\right)}^{3}}} \]

5. Applied egg-rr 98.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot e^{\color{blue}{{\left(\sqrt[3]{\log \left(\frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x}\right)}\right)}^{3}}} \]

6. Final simplification 98.1%

   \[\leadsto \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau \cdot \left(x \cdot \pi\right)} \cdot e^{{\left(\sqrt[3]{\log \left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}\right)}^{3}} \]

Alternative 2: 98.0% 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 98.1%

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

   1. associate-*l* 97.6%

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

   2. associate-*l* 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around inf 96.7%

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

   1. associate-*r* 96.9%

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

   2. *-commutative 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. unpow2 96.9%

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

   6. unpow2 96.9%

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

   7. swap-sqr 97.1%

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

   8. unpow2 97.1%

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

6. Simplified 97.1%

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

7. Final simplification 97.1%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

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(x * Float32(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 98.1%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.8%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.8%

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

   5. associate-*l* 97.3%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.1%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.0%

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

3. Simplified 97.0%

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

   1. add-cube-cbrt 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \left(x \cdot \left(\pi \cdot \pi\right)\right)\right)}} \cdot \sqrt[3]{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \left(x \cdot \left(\pi \cdot \pi\right)\right)\right)}}\right) \cdot \sqrt[3]{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \left(x \cdot \left(\pi \cdot \pi\right)\right)\right)}}\right)} \]

   2. pow3 96.3%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{\sin \left(x \cdot \pi\right)}{tau \cdot \left(x \cdot \left(x \cdot \left(\pi \cdot \pi\right)\right)\right)}}\right)}^{3}} \]

   3. associate-/r* 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot {\left(\sqrt[3]{\color{blue}{\frac{\frac{\sin \left(x \cdot \pi\right)}{tau}}{x \cdot \left(x \cdot \left(\pi \cdot \pi\right)\right)}}}\right)}^{3} \]

   4. *-commutative 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot {\left(\sqrt[3]{\frac{\frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{tau}}{x \cdot \left(x \cdot \left(\pi \cdot \pi\right)\right)}}\right)}^{3} \]

   5. associate-*r* 96.1%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot {\left(\sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\pi \cdot \pi\right)}}}\right)}^{3} \]

   6. swap-sqr 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot {\left(\sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{\color{blue}{\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)}}}\right)}^{3} \]

   7. pow2 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot {\left(\sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{\color{blue}{{\left(x \cdot \pi\right)}^{2}}}}\right)}^{3} \]

   8. *-commutative 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot {\left(\sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{{\color{blue}{\left(\pi \cdot x\right)}}^{2}}}\right)}^{3} \]

5. Applied egg-rr 96.2%

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

   1. unpow3 96.2%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{{\left(\pi \cdot x\right)}^{2}}} \cdot \sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{{\left(\pi \cdot x\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\sin \left(\pi \cdot x\right)}{tau}}{{\left(\pi \cdot x\right)}^{2}}}\right)} \]

   2. add-cube-cbrt 97.1%

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

   3. associate-/l/ 97.2%

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

7. Applied egg-rr 97.2%

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

8. Final simplification 97.2%

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

Alternative 5: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. clear-num 96.8%

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

   2. inv-pow 96.8%

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

   3. *-commutative 96.8%

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

   4. associate-/l* 96.9%

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

   5. associate-*r* 96.8%

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

   6. swap-sqr 97.2%

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

   7. pow2 97.2%

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

   8. *-commutative 97.2%

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

   9. associate-*r* 97.5%

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

   10. *-commutative 97.5%

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

   11. associate-*l* 97.2%

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

5. Applied egg-rr 97.2%

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

   1. unpow-1 97.2%

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

   2. associate-/r/ 97.0%

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

   3. *-commutative 97.0%

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

7. Simplified 97.0%

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

   1. expm1-log1p-u 96.9%

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

   2. expm1-udef 96.7%

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

9. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin \left(\pi \cdot x\right)}{tau} \cdot \left(\sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot {\left(\pi \cdot x\right)}^{-2}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 97.0%

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

    2. expm1-log1p 97.1%

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

    3. associate-*r* 97.3%

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

    4. *-commutative 97.3%

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

    5. *-commutative 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 6: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.8%

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

   5. *-commutative 96.8%

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

   6. associate-*l* 97.1%

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

   7. associate-*r* 97.1%

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

   8. swap-sqr 97.1%

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

   9. pow2 97.1%

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

   10. *-commutative 97.1%

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

5. Applied egg-rr 97.1%

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

   1. pow-prod-down 97.1%

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

   2. *-commutative 97.1%

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

   3. associate-*r* 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-/r* 96.8%

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

   6. *-commutative 96.8%

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

   7. *-commutative 96.8%

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

   8. times-frac 96.8%

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

7. Applied egg-rr 97.3%

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

8. Final simplification 97.3%

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

Alternative 7: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 96.9%

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

   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* 96.8%

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

   7. associate-/l* 97.0%

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

   8. *-commutative 97.0%

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

   9. associate-*r* 97.1%

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

   10. swap-sqr 97.4%

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

   11. pow2 97.4%

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

   12. *-commutative 97.4%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 8: 97.4% accurate, 1.0× speedup

Math

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

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(Float32(sin(Float32(x * Float32(pi))) * 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 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. associate-*r/ 96.9%

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

   2. associate-/r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.8%

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

   5. *-commutative 96.8%

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

   6. associate-*l* 97.1%

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

   7. associate-*r* 97.1%

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

   8. swap-sqr 97.1%

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

   9. pow2 97.1%

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

   10. *-commutative 97.1%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in x around -inf 97.4%

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

7. Final simplification 97.4%

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

Alternative 9: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in x around 0 85.6%

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

   1. *-commutative 85.6%

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

   2. unpow2 85.6%

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

4. Simplified 85.6%

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

   1. pow2 85.6%

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

   2. unpow-prod-down 85.6%

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

6. Applied egg-rr 85.6%

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

7. Final simplification 85.6%

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

Alternative 10: 84.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.8%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.8%

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

   5. associate-*l* 97.3%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.1%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 85.0%

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

   1. distribute-lft-in 84.9%

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

   2. associate-*r* 84.9%

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

   3. *-commutative 84.9%

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

   4. associate-*r* 84.9%

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

   5. associate-/l* 84.9%

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

   6. associate-*r* 85.4%

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

   7. *-commutative 85.4%

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

   8. associate-*r* 84.9%

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

   9. *-commutative 84.9%

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

   10. *-commutative 84.9%

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

   11. associate-*r* 85.4%

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

6. Applied egg-rr 85.4%

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

   1. distribute-lft-out 85.3%

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

   2. fma-udef 85.3%

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

   3. associate-/r/ 85.3%

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

   4. associate-/r* 85.1%

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

8. Simplified 85.1%

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

9. Final simplification 85.1%

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

Alternative 11: 84.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.8%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.8%

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

   5. associate-*l* 97.3%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.1%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 85.0%

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

5. Taylor expanded in tau around inf 85.1%

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

6. Final simplification 85.1%

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

Alternative 12: 78.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

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%

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

Alternative 13: 70.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.8%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.8%

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

   5. associate-*l* 97.3%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.1%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 70.0%

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

5. Final simplification 70.0%

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

Alternative 14: 64.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 63.7%

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

   1. *-commutative 63.7%

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

6. Simplified 63.7%

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

7. Taylor expanded in x around 0 64.0%

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

   1. unpow2 64.0%

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

   2. *-commutative 64.0%

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

   3. unpow2 64.0%

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

   4. swap-sqr 64.0%

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

   5. unpow2 64.0%

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

   6. *-commutative 64.0%

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

9. Simplified 64.0%

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

    1. unpow-prod-down 64.0%

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

    2. *-commutative 64.0%

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

    3. unpow2 64.0%

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

11. Applied egg-rr 64.0%

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

    1. add-exp-log 64.0%

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

    2. log-pow 64.0%

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

13. Applied egg-rr 64.0%

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

14. Final simplification 64.0%

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

Alternative 15: 70.9% 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 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. times-frac 97.5%

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

   3. *-commutative 97.5%

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

3. Applied egg-rr 97.5%

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

4. Taylor expanded in x around 0 70.2%

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

5. Final simplification 70.2%

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

Alternative 16: 64.5% 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 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 63.7%

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

   1. *-commutative 63.7%

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

6. Simplified 63.7%

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

7. Taylor expanded in x around 0 64.0%

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

   1. +-commutative 64.0%

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

   2. unpow2 64.0%

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

   3. *-commutative 64.0%

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

   4. unpow2 64.0%

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

   5. swap-sqr 64.0%

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

   6. unpow2 64.0%

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

   7. fma-def 64.0%

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

   8. *-commutative 64.0%

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

9. Simplified 64.0%

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

10. Final simplification 64.0%

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

Alternative 17: 64.5% 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 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 63.7%

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

   1. *-commutative 63.7%

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

6. Simplified 63.7%

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

7. Taylor expanded in x around 0 64.0%

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

   1. unpow2 64.0%

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

   2. *-commutative 64.0%

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

   3. unpow2 64.0%

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

   4. swap-sqr 64.0%

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

   5. unpow2 64.0%

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

   6. *-commutative 64.0%

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

9. Simplified 64.0%

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

10. Final simplification 64.0%

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

Alternative 18: 63.6% 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 98.1%

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

   1. *-commutative 98.1%

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

   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/ 97.8%

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

   4. associate-*r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.3%

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

   7. associate-*l* 97.0%

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

   8. swap-sqr 96.7%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around 0 62.8%

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

5. Final simplification 62.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023228 
(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))))