Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.8s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*l* 97.0%

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

   2. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around inf 97.6%

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

   1. associate-*r* 97.2%

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

   2. *-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. *-commutative 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*r* 97.7%

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

   7. *-commutative 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*l* 97.0%

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

   2. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around -inf 97.2%

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

5. Final simplification 97.2%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*l* 97.0%

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

   2. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-/l/ 97.3%

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

   4. associate-*r/ 97.2%

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

   5. associate-*l* 96.7%

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

   6. associate-*r* 96.8%

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

   7. associate-/r* 96.8%

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

   8. associate-/l/ 96.8%

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

   9. swap-sqr 96.6%

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

   10. associate-*r* 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in x around -inf 96.4%

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

5. Taylor expanded in tau around inf 96.5%

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

   1. times-frac 96.3%

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

   2. *-commutative 96.3%

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

   3. *-commutative 96.3%

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

   4. associate-*r* 96.6%

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

   5. *-commutative 96.6%

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

   6. *-commutative 96.6%

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

   7. unpow2 96.6%

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

   8. unpow2 96.6%

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

   9. swap-sqr 96.8%

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

   10. unpow2 96.8%

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

   11. times-frac 96.9%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.8%

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

   4. associate-*r* 96.6%

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

   5. *-commutative 96.6%

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

   6. associate-*l* 96.7%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.6%

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

   10. swap-sqr 96.8%

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

   11. pow2 96.8%

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

   12. *-commutative 96.8%

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

5. Applied egg-rr 96.8%

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

   1. expm1-log1p-u 96.8%

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

7. Applied egg-rr 96.8%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 96.1%

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

9. Applied egg-rr 96.3%

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

    1. expm1-def 96.7%

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

    2. expm1-log1p 97.0%

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

    3. *-commutative 97.0%

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

    4. associate-*r* 97.1%

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

    5. *-commutative 97.1%

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

11. Simplified 97.1%

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

12. Final simplification 97.1%

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

Alternative 6: 85.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*l* 97.0%

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

   2. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 82.7%

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

   1. *-commutative 82.7%

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

   2. unpow2 82.7%

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

6. Simplified 82.7%

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

7. Final simplification 82.7%

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

Alternative 7: 85.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

2. Taylor expanded in x around 0 82.7%

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

   1. *-commutative 82.7%

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

   2. unpow2 82.7%

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

4. Simplified 82.7%

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

5. Final simplification 82.7%

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

Alternative 8: 84.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-/l/ 97.3%

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

   4. associate-*r/ 97.2%

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

   5. associate-*l* 96.7%

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

   6. associate-*r* 96.8%

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

   7. associate-/r* 96.8%

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

   8. associate-/l/ 96.8%

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

   9. swap-sqr 96.6%

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

   10. associate-*r* 96.5%

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

3. Simplified 96.5%

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

   1. add-sqr-sqrt 96.3%

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

   2. sqrt-unprod 96.5%

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

   3. swap-sqr 96.5%

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

   4. associate-*r* 96.7%

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

   5. expm1-log1p-u 96.6%

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

   6. associate-*r* 96.4%

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

   7. swap-sqr 96.4%

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

   8. sqrt-unprod 96.2%

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

   9. add-sqr-sqrt 96.4%

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

   10. *-commutative 96.4%

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

5. Applied egg-rr 96.4%

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

6. Taylor expanded in x around inf 96.6%

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

   1. *-commutative 96.6%

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

   2. *-commutative 96.6%

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

   3. unpow2 96.6%

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

   4. unpow2 96.6%

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

   5. swap-sqr 96.8%

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

   6. unpow2 96.8%

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

8. Simplified 96.8%

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

9. Taylor expanded in x around 0 81.8%

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

    1. fma-def 81.8%

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

    2. associate-/l* 81.8%

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

    3. *-commutative 81.8%

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

    4. associate-*r* 82.3%

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

11. Simplified 82.3%

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

12. Final simplification 82.3%

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

Alternative 9: 84.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-/l/ 97.3%

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

   4. associate-*r/ 97.2%

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

   5. associate-*l* 96.7%

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

   6. associate-*r* 96.8%

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

   7. associate-/r* 96.8%

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

   8. associate-/l/ 96.8%

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

   9. swap-sqr 96.6%

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

   10. associate-*r* 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in x around -inf 96.4%

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

5. Taylor expanded in x around 0 82.3%

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

6. Final simplification 82.3%

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

Alternative 10: 78.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in x around 0 75.4%

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

   1. +-commutative 75.4%

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

   2. fma-def 75.4%

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

   3. associate-*r* 75.4%

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

   4. distribute-rgt-out 75.4%

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

   5. unpow2 75.4%

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

   6. unpow2 75.4%

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

6. Simplified 75.4%

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

7. Final simplification 75.4%

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

Alternative 11: 71.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-/l/ 97.3%

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

   4. associate-*r/ 97.2%

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

   5. associate-*l* 96.7%

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

   6. associate-*r* 96.8%

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

   7. associate-/r* 96.8%

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

   8. associate-/l/ 96.8%

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

   9. swap-sqr 96.6%

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

   10. associate-*r* 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in x around 0 67.6%

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

   1. associate-/r* 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in x around -inf 67.8%

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

8. Final simplification 67.8%

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

Alternative 12: 71.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. associate-*l* 97.0%

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

   2. associate-*l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around inf 97.6%

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

   1. associate-*r* 97.2%

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

   2. *-commutative 97.2%

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

   3. *-commutative 97.2%

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

   4. *-commutative 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*r* 97.7%

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

   7. *-commutative 97.7%

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

6. Simplified 97.7%

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

   1. expm1-log1p-u 97.3%

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

   2. expm1-udef 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

8. Applied egg-rr 97.3%

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

9. Taylor expanded in x around 0 67.8%

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

10. Final simplification 67.8%

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

Alternative 13: 69.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.6%

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

   1. associate-*r/ 97.5%

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

   2. associate-*l/ 97.5%

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

   3. associate-/l/ 97.3%

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

   4. associate-*r/ 97.2%

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

   5. associate-*l* 96.7%

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

   6. associate-*r* 96.8%

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

   7. associate-/r* 96.8%

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

   8. associate-/l/ 96.8%

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

   9. swap-sqr 96.6%

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

   10. associate-*r* 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in x around 0 67.6%

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

   1. associate-/r* 67.5%

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

6. Simplified 67.5%

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

7. Taylor expanded in x around 0 66.3%

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

   1. +-commutative 66.3%

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

   2. *-commutative 66.3%

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

   3. fma-def 66.3%

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

   4. *-commutative 66.3%

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

   5. associate-*l* 66.3%

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

   6. unpow2 66.3%

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

   7. unpow2 66.3%

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

   8. unpow2 66.3%

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

   9. unswap-sqr 66.3%

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

   10. swap-sqr 66.3%

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

   11. unpow2 66.3%

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

9. Simplified 66.3%

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

10. Final simplification 66.3%

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

Alternative 14: 64.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 60.2%

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

   1. *-commutative 60.2%

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

6. Simplified 60.2%

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

7. Taylor expanded in x around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. unpow2 60.5%

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

   3. unpow2 60.5%

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

   4. swap-sqr 60.5%

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

   5. unpow2 60.5%

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

9. Simplified 60.5%

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

    1. expm1-log1p-u 60.5%

       \[\leadsto 1 + -0.16666666666666666 \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}}^{2} \]

    2. expm1-udef 60.5%

       \[\leadsto 1 + -0.16666666666666666 \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot x\right)} - 1\right)}}^{2} \]

    3. log1p-udef 60.5%

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

    4. add-exp-log 60.5%

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

11. Applied egg-rr 60.5%

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

12. Final simplification 60.5%

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

Alternative 15: 64.6% accurate, 2.9× speedup

Math

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

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(x * Float32(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 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 60.2%

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

   1. *-commutative 60.2%

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

6. Simplified 60.2%

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

7. Taylor expanded in x around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. unpow2 60.5%

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

   3. unpow2 60.5%

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

   4. swap-sqr 60.5%

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

   5. unpow2 60.5%

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

9. Simplified 60.5%

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

    1. *-commutative 60.5%

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

    2. unpow-prod-down 60.5%

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

    3. pow2 60.5%

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

    4. pow2 60.5%

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

    5. associate-*r* 60.5%

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

    6. *-commutative 60.5%

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

    7. pow2 60.5%

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

11. Applied egg-rr 60.5%

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

12. Final simplification 60.5%

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

Alternative 16: 64.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 60.2%

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

   1. *-commutative 60.2%

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

6. Simplified 60.2%

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

7. Taylor expanded in x around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. unpow2 60.5%

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

   3. unpow2 60.5%

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

   4. swap-sqr 60.5%

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

   5. unpow2 60.5%

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

9. Simplified 60.5%

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

10. Taylor expanded in x around 0 60.5%

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

    1. *-commutative 60.5%

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

    2. unpow2 60.5%

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

12. Simplified 60.5%

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

13. Final simplification 60.5%

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

Alternative 17: 64.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 60.2%

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

   1. *-commutative 60.2%

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

6. Simplified 60.2%

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

7. Taylor expanded in x around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. unpow2 60.5%

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

   3. unpow2 60.5%

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

   4. swap-sqr 60.5%

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

   5. unpow2 60.5%

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

9. Simplified 60.5%

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

10. Final simplification 60.5%

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

Alternative 18: 63.6% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.6%

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

   1. *-commutative 97.6%

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

   2. times-frac 97.3%

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

   3. associate-*r/ 97.3%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.8%

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

   8. swap-sqr 96.6%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in x around 0 59.4%

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

5. Final simplification 59.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023227 
(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))))