Lanczos kernel

Percentage Accurate: 97.9% → 99.2%

Time: 20.7s

Alternatives: 21

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \langle \left( \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{t_1} \end{array} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (let* ((t_1 (* x (* PI tau))))
     (* (/ (sin t_1) (* x PI)) (/ (sin (* x PI)) t_1))))))

C

float code(float x, float tau) {
	double t_1_1 = ((double) x) * (((double) M_PI) * ((double) tau));
	double tmp = (sin(t_1_1) / (((double) x) * ((double) M_PI))) * (sin((((double) x) * ((double) M_PI))) / t_1_1);
	return (float) tmp;
}

Julia

function code(x, tau)
	t_1_1 = Float64(Float64(x) * Float64(pi * Float64(tau)))
	tmp = Float64(Float64(sin(t_1_1) / Float64(Float64(x) * pi)) * Float64(sin(Float64(Float64(x) * pi)) / t_1_1))
	return Float32(tmp)
end

MATLAB

function tmp_3 = code(x, tau)
	t_1_2 = double(x) * (pi * double(tau));
	tmp = (sin(t_1_2) / (double(x) * pi)) * (sin((double(x) * pi)) / t_1_2);
	tmp_3 = single(tmp);
end

Derivation

1. Initial program 99.1%

   \[\langle \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \rangle_{\text{binary64}} \]

2. Final simplification 99.1%

   \[\leadsto \langle \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \left(\pi \cdot tau\right)} \rangle_{\text{binary64}} \]

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = sin((x * single(pi))) * ((sin(t_1) / t_1) / (x * single(pi)));
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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin((x * single(pi))) / (x * single(pi))) * (sin(t_1) / 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. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * single(pi))) * (sin((tau * (x * single(pi)))) / (tau * ((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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around inf 96.5%

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

   1. associate-/l* 96.4%

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

   2. associate-*r* 96.6%

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

   3. *-commutative 96.6%

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

   4. *-commutative 96.6%

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

   5. unpow2 96.6%

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

   6. unpow2 96.6%

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

   7. swap-sqr 96.9%

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

   8. unpow2 96.9%

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

   9. associate-/r/ 96.9%

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

   10. remove-double-neg 96.9%

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

   11. distribute-frac-neg 96.9%

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

6. Simplified 97.2%

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

7. Final simplification 97.2%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

   1. associate-/l/ 97.5%

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

   2. associate-*r/ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 97.5%

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

   5. associate-/r* 97.2%

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

   6. associate-*r/ 97.2%

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

5. Applied egg-rr 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-*l/ 96.8%

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

   3. times-frac 96.9%

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

   4. associate-/r* 96.9%

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

   5. associate-/l/ 96.8%

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

   6. unpow2 96.8%

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

   7. *-commutative 96.8%

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

   8. *-commutative 96.8%

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

   9. associate-*r* 97.1%

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

7. Simplified 97.1%

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

   1. add-log-exp_binary32 96.3%

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

9. Applied rewrite-once 96.3%

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

10. Taylor expanded in tau around inf 96.5%

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

    1. associate-/r* 96.4%

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

    2. associate-*r* 96.5%

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

    3. *-commutative 96.5%

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

    4. *-commutative 96.5%

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

    5. *-commutative 96.5%

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

    6. associate-*l/ 96.7%

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

    7. *-commutative 96.7%

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

    8. *-commutative 96.7%

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

    9. *-commutative 96.7%

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

    10. *-commutative 96.7%

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

    11. associate-*r* 96.4%

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

    12. *-commutative 96.4%

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

12. Simplified 97.2%

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

13. Final simplification 97.2%

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

Alternative 6: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (x * single(pi)))) / ((x * single(pi)) ^ single(2.0))) / (tau / sin((x * single(pi))));
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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

   1. associate-/l/ 97.5%

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

   2. associate-*r/ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 97.5%

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

   5. associate-*r* 97.0%

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

   6. associate-/r* 97.0%

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

   7. associate-*r/ 97.2%

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

5. Applied egg-rr 97.0%

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

   1. associate-*r/ 96.9%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. unpow2 97.0%

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

   6. *-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. associate-*r* 97.3%

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 7: 84.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. associate-*l* 97.2%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 85.8%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

6. Simplified 85.8%

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

7. Final simplification 85.8%

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

Alternative 8: 84.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (single(1.0) + (((x * single(pi)) ^ single(2.0)) * single(-0.16666666666666666))) * (sin(t_1) / 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. *-commutative 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-*l* 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 85.8%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

6. Simplified 85.8%

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

7. Final simplification 85.8%

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

Alternative 9: 78.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * single(pi))) * ((single(1.0) + (single(-0.16666666666666666) * ((tau * (x * single(pi))) ^ single(2.0)))) / (x * single(pi)));
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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 80.2%

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

   1. associate-*r* 80.4%

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

   2. unpow2 80.4%

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

   3. unpow2 80.4%

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

   4. swap-sqr 80.4%

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

   5. *-commutative 80.4%

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

   6. *-commutative 80.4%

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

   7. unpow2 80.4%

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

   8. swap-sqr 80.4%

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

   9. *-commutative 80.4%

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

   10. *-commutative 80.4%

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

   11. unpow1 80.4%

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

   12. unpow1 80.4%

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

   13. pow-sqr 80.4%

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

   14. *-commutative 80.4%

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

   15. *-commutative 80.4%

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

   16. associate-*r* 80.4%

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

   17. metadata-eval 80.4%

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

6. Simplified 80.2%

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

7. Final simplification 80.2%

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

Alternative 10: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. associate-*l* 97.2%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 80.4%

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

   1. associate-*r* 80.4%

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

   2. unpow2 80.4%

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

   3. unpow2 80.4%

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

   4. swap-sqr 80.4%

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

   5. *-commutative 80.4%

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

   6. *-commutative 80.4%

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

   7. unpow2 80.4%

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

   8. swap-sqr 80.4%

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

   9. *-commutative 80.4%

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

   10. *-commutative 80.4%

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

   11. unpow1 80.4%

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

   12. unpow1 80.4%

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

   13. pow-sqr 80.4%

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

   14. *-commutative 80.4%

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

   15. *-commutative 80.4%

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

   16. associate-*r* 80.4%

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

   17. metadata-eval 80.4%

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

6. Simplified 80.4%

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

7. Final simplification 80.4%

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

Alternative 11: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. associate-*l* 97.2%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 80.4%

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

   1. associate-*r* 80.4%

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

   2. unpow2 80.4%

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

   3. unpow2 80.4%

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

   4. swap-sqr 80.4%

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

   5. *-commutative 80.4%

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

   6. *-commutative 80.4%

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

   7. unpow2 80.4%

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

   8. swap-sqr 80.4%

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

   9. *-commutative 80.4%

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

   10. *-commutative 80.4%

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

   11. unpow1 80.4%

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

   12. unpow1 80.4%

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

   13. pow-sqr 80.4%

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

   14. *-commutative 80.4%

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

   15. *-commutative 80.4%

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

   16. associate-*r* 80.4%

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

   17. metadata-eval 80.4%

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

6. Simplified 80.4%

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

7. Taylor expanded in x around 0 80.0%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

9. Simplified 80.0%

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

10. Final simplification 80.0%

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

Alternative 12: 78.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((x * x) * (single(-0.16666666666666666) * ((single(pi) ^ single(2.0)) + ((tau * tau) * (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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 79.6%

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

   1. unpow2 79.6%

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

   2. distribute-lft-out 79.6%

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

   3. unpow2 79.6%

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

6. Simplified 79.6%

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

7. Final simplification 79.6%

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

Alternative 13: 78.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

   1. associate-/l/ 97.5%

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

   2. associate-*r/ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 97.5%

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

   5. associate-/r* 97.2%

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

   6. associate-*r/ 97.2%

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

5. Applied egg-rr 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-*l/ 96.8%

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

   3. times-frac 96.9%

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

   4. associate-/r* 96.9%

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

   5. associate-/l/ 96.8%

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

   6. unpow2 96.8%

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

   7. *-commutative 96.8%

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

   8. *-commutative 96.8%

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

   9. associate-*r* 97.1%

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

7. Simplified 97.1%

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

   1. add-log-exp_binary32 96.3%

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

9. Applied rewrite-once 96.3%

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

    1. rem-log-exp 97.1%

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

    2. associate-*l/ 97.0%

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

    3. *-commutative 97.0%

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

    4. unpow2 97.0%

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

    5. *-commutative 97.0%

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

    6. associate-*l* 96.8%

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

    7. times-frac 96.8%

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

    8. *-commutative 96.8%

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

    9. *-commutative 96.8%

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

11. Applied egg-rr 96.4%

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

12. Taylor expanded in x around 0 79.6%

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

    1. +-commutative 79.6%

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

    2. *-commutative 79.6%

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

    3. unpow2 79.6%

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

    4. fma-def 79.6%

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

    5. distribute-lft-out 79.6%

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

    6. distribute-lft1-in 79.6%

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

    7. unpow2 79.6%

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

14. Simplified 79.6%

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

15. Final simplification 79.6%

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

Alternative 14: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(1.0) / Float32(Float32(1.0) / fma(Float32(-0.16666666666666666), (Float32(x * Float32(pi)) ^ Float32(2.0)), 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.2%

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

5. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

7. Simplified 65.6%

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

   1. flip3-+ 65.6%

      \[\leadsto \color{blue}{\frac{{1}^{3} + {\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)}^{3}}{1 \cdot 1 + \left(\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) - 1 \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)\right)}} \]

   2. clear-num 65.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 \cdot 1 + \left(\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) - 1 \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)\right)}{{1}^{3} + {\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)}^{3}}}} \]

   3. clear-num 65.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{1}^{3} + {\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)}^{3}}{1 \cdot 1 + \left(\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) - 1 \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)\right)}}}} \]

   4. flip3-+ 65.7%

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

   5. +-commutative 65.7%

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

   6. fma-def 65.7%

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

   7. *-commutative 65.7%

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

9. Applied egg-rr 65.7%

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

10. Final simplification 65.7%

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

Alternative 15: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

   2. associate-*l* 97.2%

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

   3. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. clear-num 97.7%

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

   2. inv-pow 97.7%

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

   3. div-inv 97.6%

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

   4. metadata-eval 97.6%

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

   5. unpow-prod-down 97.5%

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

   6. metadata-eval 97.5%

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

   7. inv-pow 97.5%

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

   8. metadata-eval 97.5%

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

5. Applied egg-rr 97.5%

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

   1. unpow-1 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in x around 0 71.5%

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

9. Final simplification 71.5%

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

Alternative 16: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * Float32(x * x)), (Float32(pi) ^ Float32(2.0)), 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.2%

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

5. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

7. Simplified 65.6%

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

8. Taylor expanded in x around 0 65.6%

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

   1. +-commutative 65.6%

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

   2. unpow2 65.6%

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

   3. associate-*r* 65.6%

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

   4. fma-udef 65.6%

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

10. Simplified 65.6%

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

11. Final simplification 65.6%

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

Alternative 17: 64.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma((Float32(x * Float32(pi)) ^ Float32(2.0)), Float32(-0.16666666666666666), 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.2%

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

5. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

7. Simplified 65.6%

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

   1. +-commutative 65.6%

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

   2. *-commutative 65.6%

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

   3. fma-def 65.6%

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

   4. *-commutative 65.6%

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

9. Applied egg-rr 65.6%

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

10. Final simplification 65.6%

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

Alternative 18: 64.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.2%

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

5. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

7. Simplified 65.6%

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

8. Final simplification 65.6%

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

Alternative 19: 64.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((single(pi) ^ single(2.0)) * (single(-0.16666666666666666) * (x * 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.2%

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

5. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

7. Simplified 65.6%

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

8. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

   2. associate-*r* 65.6%

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

10. Simplified 65.6%

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

11. Final simplification 65.6%

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

Alternative 20: 64.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in tau around 0 65.2%

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

5. Taylor expanded in x around 0 65.6%

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

   1. unpow2 65.6%

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

   2. unpow2 65.6%

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

   3. swap-sqr 65.6%

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

   4. unpow2 65.6%

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

7. Simplified 65.6%

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

8. Final simplification 65.6%

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

Alternative 21: 63.1% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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-*r/ 97.8%

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

   2. associate-*l* 97.1%

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

   3. *-commutative 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-/l* 97.8%

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

   9. associate-/r/ 97.6%

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

   10. *-commutative 97.6%

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

3. Simplified 97.6%

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

   1. associate-/l/ 97.5%

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

   2. associate-*r/ 97.6%

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

   3. *-commutative 97.6%

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

   4. frac-times 97.5%

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

   5. associate-/r* 97.2%

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

   6. associate-*r/ 97.2%

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

5. Applied egg-rr 96.9%

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

   1. associate-*r/ 96.9%

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

   2. associate-*l/ 96.8%

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

   3. times-frac 96.9%

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

   4. associate-/r* 96.9%

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

   5. associate-/l/ 96.8%

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

   6. unpow2 96.8%

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

   7. *-commutative 96.8%

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

   8. *-commutative 96.8%

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

   9. associate-*r* 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in x around 0 64.4%

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

9. Final simplification 64.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(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))))