Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 24.9s

Alternatives: 19

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative 97.6%

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

   2. clear-num 97.7%

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

   3. un-div-inv 97.8%

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

   4. associate-/l* 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 97.1%

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

   4. unpow2 97.1%

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

   5. unpow2 97.1%

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

   6. swap-sqr 97.4%

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

   7. unpow2 97.4%

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

   8. times-frac 97.3%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. add-log-exp 97.0%

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

   2. frac-times 97.1%

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

   3. associate-/l* 97.0%

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

   4. associate-*r* 96.7%

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

   5. associate-*r* 96.7%

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

   6. pow2 96.7%

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

5. Applied egg-rr 96.7%

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

   1. add-log-exp 97.1%

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

   2. associate-/r/ 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l/ 97.4%

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

   5. times-frac 97.3%

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

7. Applied egg-rr 97.3%

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

8. Final simplification 97.3%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.2%

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

5. Applied egg-rr 96.9%

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

   1. rem-cube-cbrt 97.1%

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

   2. associate-/r/ 97.2%

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

   3. *-commutative 97.2%

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

7. Applied egg-rr 97.2%

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

8. Taylor expanded in x around inf 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.2%

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

   3. *-commutative 97.2%

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

   4. unpow2 97.2%

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

   5. unpow2 97.2%

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

   6. swap-sqr 97.2%

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

   7. unpow2 97.2%

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

   8. associate-/r* 97.2%

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

   9. associate-*r* 97.3%

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

   10. *-commutative 97.3%

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

10. Simplified 97.3%

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

11. Final simplification 97.3%

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

Alternative 6: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. frac-times 97.9%

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

   2. associate-*r* 97.4%

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

   3. associate-*r* 97.4%

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

   4. pow2 97.4%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 7: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 80.5%

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

   1. unpow2 80.5%

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

   2. unpow2 80.5%

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

6. Simplified 80.5%

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

7. Taylor expanded in tau around 0 80.5%

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

   1. unpow2 80.2%

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

   2. unpow2 80.2%

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

   3. swap-sqr 80.2%

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

   4. unpow2 80.2%

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

   5. associate-*r* 80.2%

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

   6. unpow2 80.2%

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

9. Simplified 80.5%

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

10. Final simplification 80.5%

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

Alternative 8: 85.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt 97.8%

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

   2. pow2 97.8%

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

5. Applied egg-rr 97.8%

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

6. Taylor expanded in x around 0 85.9%

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

   1. unpow2 85.9%

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

   2. unpow2 85.9%

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

   3. swap-sqr 85.9%

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

   4. unpow2 85.9%

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

8. Simplified 85.9%

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

9. Final simplification 85.9%

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

Alternative 9: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 80.5%

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

   1. unpow2 80.5%

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

   2. unpow2 80.5%

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

6. Simplified 80.5%

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

7. Taylor expanded in tau around 0 80.5%

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

   1. unpow2 80.2%

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

   2. unpow2 80.2%

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

   3. unpow2 80.2%

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

   4. swap-sqr 80.2%

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

   5. swap-sqr 80.2%

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

   6. *-commutative 80.2%

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

   7. associate-*r* 80.2%

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

   8. *-commutative 80.2%

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

   9. associate-*r* 80.2%

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

   10. unpow2 80.2%

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

9. Simplified 80.5%

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

10. Final simplification 80.5%

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

Alternative 10: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. add-cube-cbrt 97.1%

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

   2. pow3 97.2%

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

5. Applied egg-rr 96.9%

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

   1. rem-cube-cbrt 97.1%

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

   2. associate-/r/ 97.2%

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

   3. *-commutative 97.2%

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

7. Applied egg-rr 97.2%

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

8. Taylor expanded in x around 0 80.4%

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

   1. fma-def 80.4%

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

   2. associate-*r* 80.4%

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

   3. unpow2 80.4%

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

10. Simplified 80.4%

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

11. Final simplification 80.4%

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

Alternative 11: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 80.5%

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

   1. unpow2 80.5%

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

   2. unpow2 80.5%

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

6. Simplified 80.5%

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

7. Taylor expanded in x around 0 80.2%

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

   1. associate-*r* 65.5%

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

   2. unpow2 65.5%

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

9. Simplified 80.2%

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

10. Taylor expanded in tau around 0 80.2%

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

    1. unpow2 80.2%

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

    2. unpow2 80.2%

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

    3. swap-sqr 80.2%

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

    4. unpow2 80.2%

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

    5. associate-*r* 80.2%

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

    6. unpow2 80.2%

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

12. Simplified 80.2%

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

13. Final simplification 80.2%

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

Alternative 12: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 80.5%

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

   1. unpow2 80.5%

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

   2. unpow2 80.5%

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

6. Simplified 80.5%

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

7. Taylor expanded in x around 0 80.2%

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

   1. associate-*r* 65.5%

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

   2. unpow2 65.5%

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

9. Simplified 80.2%

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

10. Taylor expanded in tau around 0 80.2%

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

    1. unpow2 80.2%

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

    2. unpow2 80.2%

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

    3. unpow2 80.2%

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

    4. swap-sqr 80.2%

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

    5. swap-sqr 80.2%

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

    6. *-commutative 80.2%

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

    7. associate-*r* 80.2%

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

    8. *-commutative 80.2%

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

    9. associate-*r* 80.2%

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

    10. unpow2 80.2%

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

12. Simplified 80.2%

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

13. Final simplification 80.2%

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

Alternative 13: 78.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-/r* 97.4%

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

   3. frac-times 97.6%

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

5. Applied egg-rr 97.6%

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

   1. add-log-exp 97.6%

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

7. Applied egg-rr 97.6%

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

8. Taylor expanded in x around 0 79.8%

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

   1. *-commutative 79.8%

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

   2. distribute-lft-out 79.8%

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

   3. associate-*r* 79.8%

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

   4. distribute-lft1-in 79.8%

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

   5. associate-*l* 79.8%

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

   6. unpow2 79.8%

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

   7. fma-def 79.8%

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

   8. *-commutative 79.8%

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

   9. unpow2 79.8%

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

   10. unpow2 79.8%

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

   11. swap-sqr 79.8%

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

   12. unpow2 79.8%

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

   13. *-commutative 79.8%

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

10. Simplified 79.8%

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

11. Final simplification 79.8%

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

Alternative 14: 78.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative 97.6%

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

   2. clear-num 97.7%

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

   3. un-div-inv 97.8%

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

   4. associate-/l* 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 79.8%

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

   1. unpow2 79.8%

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

   2. distribute-lft-out 79.8%

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

   3. *-commutative 79.8%

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

   4. unpow2 79.8%

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

8. Simplified 79.8%

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

9. Taylor expanded in x around 0 79.8%

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

    1. associate-*r* 79.8%

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

    2. unpow2 79.8%

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

    3. distribute-lft1-in 79.8%

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

    4. unpow2 79.8%

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

11. Simplified 79.8%

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

12. Final simplification 79.8%

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

Alternative 15: 69.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative 97.6%

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

   2. clear-num 97.7%

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

   3. un-div-inv 97.8%

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

   4. associate-/l* 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 79.8%

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

   1. unpow2 79.8%

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

   2. distribute-lft-out 79.8%

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

   3. *-commutative 79.8%

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

   4. unpow2 79.8%

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

8. Simplified 79.8%

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

9. Taylor expanded in tau around inf 70.6%

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

    1. *-commutative 70.6%

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

    2. unpow2 70.6%

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

    3. unpow2 70.6%

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

    4. swap-sqr 70.6%

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

    5. unpow2 70.6%

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

11. Simplified 70.6%

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

12. Final simplification 70.6%

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

Alternative 16: 69.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative 97.6%

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

   2. clear-num 97.7%

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

   3. un-div-inv 97.8%

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

   4. associate-/l* 97.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 79.8%

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

   1. unpow2 79.8%

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

   2. distribute-lft-out 79.8%

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

   3. *-commutative 79.8%

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

   4. unpow2 79.8%

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

8. Simplified 79.8%

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

9. Taylor expanded in tau around inf 70.6%

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

    1. associate-*r* 70.6%

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

    2. *-commutative 70.6%

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

    3. unpow2 70.6%

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

    4. unpow2 70.6%

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

    5. swap-sqr 70.6%

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

    6. unpow2 70.6%

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

    7. *-commutative 70.6%

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

    8. unpow2 70.6%

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

11. Simplified 70.6%

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

12. Final simplification 70.6%

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

Alternative 17: 64.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.0%

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

5. Taylor expanded in x around 0 65.5%

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

   1. associate-*r* 65.5%

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

   2. unpow2 65.5%

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

7. Simplified 65.5%

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

8. Taylor expanded in x around 0 65.5%

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

   1. *-commutative 65.5%

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

   2. unpow2 65.5%

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

10. Simplified 65.5%

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

11. Final simplification 65.5%

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

Alternative 18: 64.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.0%

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

5. Taylor expanded in x around 0 65.5%

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

   1. associate-*r* 65.5%

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

   2. unpow2 65.5%

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

7. Simplified 65.5%

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

8. Taylor expanded in x around 0 65.5%

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

   1. unpow2 65.5%

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

   2. unpow2 65.5%

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

   3. swap-sqr 65.5%

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

   4. unpow2 65.5%

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

10. Simplified 65.5%

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

11. Final simplification 65.5%

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

Alternative 19: 63.3% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. times-frac 97.4%

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

   3. associate-*l* 97.0%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 64.2%

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

5. Final simplification 64.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023275 
(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))))