Lanczos kernel

Percentage Accurate: 97.9% → 97.8%

Time: 18.4s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (/ (sin (expm1 (log1p (* 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(expm1f(log1pf((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(expm1(log1p(Float32(x * Float32(pi))))) / Float32(x * Float32(pi))))
end

Derivation

1. Initial program 97.5%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-undefine 82.7%

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

4. Applied egg-rr 82.7%

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

   1. expm1-define 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}{x \cdot \pi} \]
8. Add Preprocessing

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \sin t\_1 \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{t\_1}}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

   1. associate-*r* 96.2%

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

   2. associate-*r* 97.0%

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

   3. associate-*r* 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*r* 96.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-*r/ 97.4%

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

6. Applied egg-rr 97.3%

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

   1. associate-/l* 97.1%

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

   2. associate-*r* 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-*r* 97.2%

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

   6. *-commutative 97.2%

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

   7. *-commutative 97.2%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \sin t\_1 \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

5. Taylor expanded in x around 0 96.5%

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

   1. associate-*r* 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-*l* 96.7%

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

7. Simplified 96.7%

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

   1. associate-*r* 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-*r* 96.3%

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

   4. associate-/r* 96.5%

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

   5. associate-*r* 96.6%

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

   6. unpow2 96.6%

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

   7. associate-*r* 96.8%

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

   8. associate-*r* 96.3%

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

   9. *-commutative 96.3%

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

   10. associate-/l/ 96.4%

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

   11. associate-/r* 96.5%

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

9. Applied egg-rr 97.3%

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

    1. associate-/l* 97.3%

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

    2. associate-*r* 96.7%

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

    3. *-commutative 96.7%

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

    4. associate-*r* 97.3%

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

    5. *-commutative 97.3%

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \sin t\_1 \cdot \frac{\sin \left(x \cdot \pi\right)}{\left(x \cdot \pi\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. associate-*l* 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 5: 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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing

3. Final simplification 97.5%

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

Alternative 6: 84.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-/r* 97.3%

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

   2. div-inv 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.6%

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

4. Applied egg-rr 96.6%

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

   1. associate-*r/ 96.8%

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

   2. *-rgt-identity 96.8%

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

   3. associate-*r* 97.3%

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

   4. *-commutative 97.3%

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

   5. *-commutative 97.3%

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

6. Simplified 97.3%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-undefine 82.7%

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

8. Applied egg-rr 82.5%

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

   1. expm1-define 97.6%

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

10. Simplified 97.3%

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

11. Taylor expanded in x around 0 81.8%

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

    1. unpow2 81.8%

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

    2. unpow2 81.8%

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

    3. swap-sqr 81.8%

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

    4. unpow2 81.8%

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

13. Simplified 81.8%

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

    1. unpow2 81.8%

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

15. Applied egg-rr 81.8%

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

16. Final simplification 81.8%

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

Alternative 7: 84.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. associate-*l* 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in x around 0 81.4%

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

6. Final simplification 81.4%

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

Alternative 8: 70.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

   1. associate-*r* 96.2%

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

   2. associate-*r* 97.0%

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

   3. associate-*r* 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*r* 96.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-*r/ 97.4%

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

6. Applied egg-rr 97.3%

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

   1. associate-/l* 97.1%

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

   2. associate-*r* 96.9%

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

   3. *-commutative 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-*r* 97.2%

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

   6. *-commutative 97.2%

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

   7. *-commutative 97.2%

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

8. Simplified 97.2%

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

9. Taylor expanded in x around 0 68.1%

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

10. Final simplification 68.1%

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

Alternative 9: 70.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-undefine 82.7%

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

4. Applied egg-rr 82.7%

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

   1. expm1-define 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in x around 0 68.3%

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

8. Final simplification 68.3%

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

Alternative 10: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-/r* 97.3%

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

   2. div-inv 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 96.6%

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

4. Applied egg-rr 96.6%

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

   1. associate-*r/ 96.8%

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

   2. *-rgt-identity 96.8%

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

   3. associate-*r* 97.3%

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

   4. *-commutative 97.3%

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

   5. *-commutative 97.3%

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

6. Simplified 97.3%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-undefine 82.7%

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

8. Applied egg-rr 82.5%

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

   1. expm1-define 97.6%

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

10. Simplified 97.3%

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

11. Taylor expanded in x around 0 68.3%

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

12. Final simplification 68.3%

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

Alternative 11: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

5. Taylor expanded in tau around 0 61.4%

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

6. Taylor expanded in x around 0 61.6%

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

   1. +-commutative 61.6%

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

   2. fma-define 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 61.6%

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

   6. unpow2 61.6%

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

8. Simplified 61.6%

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

   1. unpow2 61.6%

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

   2. associate-*r* 61.6%

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

10. Applied egg-rr 61.6%

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

11. Final simplification 61.6%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \pi \cdot \left(x \cdot \left(x \cdot \pi\right)\right), 1\right) \]
12. Add Preprocessing

Alternative 12: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

5. Taylor expanded in tau around 0 61.4%

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

6. Taylor expanded in x around 0 61.6%

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

   1. +-commutative 61.6%

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

   2. fma-define 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 61.6%

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

   6. unpow2 61.6%

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

8. Simplified 61.6%

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

   1. fma-undefine 61.6%

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

10. Applied egg-rr 61.6%

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

11. Final simplification 61.6%

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

Alternative 13: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

5. Taylor expanded in tau around 0 61.4%

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

6. Final simplification 61.4%

   \[\leadsto \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
7. Add Preprocessing

Alternative 14: 63.4% accurate, 219.0× 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 97.5%

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

   1. associate-*l/ 97.3%

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

   2. associate-/l* 97.3%

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

   3. associate-*l* 96.8%

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

   4. associate-/l/ 96.8%

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

   5. remove-double-neg 96.8%

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

   6. distribute-rgt-neg-out 96.8%

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

   7. distribute-lft-neg-out 96.8%

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

   8. distribute-rgt-neg-out 96.8%

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

   9. distribute-lft-neg-out 96.8%

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

   10. distribute-lft-neg-out 96.8%

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

   11. distribute-rgt-neg-out 96.8%

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

3. Simplified 96.9%

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

5. Taylor expanded in x around 0 60.6%

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

6. Final simplification 60.6%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))