Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.0s

Alternatives: 10

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: 97.9% 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 98.2%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

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 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

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(x * Float32(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 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. remove-double-neg 97.3%

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

   6. distribute-rgt-neg-out 97.3%

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

   7. distribute-lft-neg-out 97.3%

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

   8. distribute-rgt-neg-out 97.3%

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

   9. distribute-lft-neg-out 97.3%

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

   10. distribute-lft-neg-out 97.3%

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

   11. distribute-rgt-neg-out 97.3%

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

3. Simplified 97.6%

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

Alternative 4: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in x around 0 78.4%

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

   1. unpow2 78.4%

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

   2. *-commutative 78.4%

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

   3. unpow2 78.4%

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

   4. unpow2 78.4%

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

   5. swap-sqr 78.4%

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

   6. swap-sqr 78.4%

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

   7. unpow2 78.4%

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

   8. *-commutative 78.4%

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

5. Simplified 78.4%

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

6. Final simplification 78.4%

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

Alternative 5: 70.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return ((sinf(((x * ((float) M_PI)) * tau)) / (x * ((float) M_PI))) / tau) * ((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(x * Float32(pi)) / Float32(x * Float32(pi))))
end

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-/r* 98.1%

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

   2. div-inv 98.0%

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

   3. *-commutative 98.0%

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

   4. *-commutative 98.0%

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

   5. *-commutative 98.0%

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

4. Applied egg-rr 98.0%

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

   1. un-div-inv 98.1%

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

   2. associate-*r* 97.7%

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

   3. *-commutative 97.7%

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

   4. *-commutative 97.7%

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

   5. rem-cube-cbrt 96.3%

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

   6. rem-cube-cbrt 97.7%

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

   7. *-commutative 97.7%

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

6. Applied egg-rr 97.7%

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

7. Taylor expanded in x around 0 69.8%

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

8. Taylor expanded in x around inf 69.9%

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

9. Final simplification 69.9%

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

Alternative 6: 70.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. remove-double-neg 97.3%

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

   6. distribute-rgt-neg-out 97.3%

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

   7. distribute-lft-neg-out 97.3%

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

   8. distribute-rgt-neg-out 97.3%

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

   9. distribute-lft-neg-out 97.3%

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

   10. distribute-lft-neg-out 97.3%

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

   11. distribute-rgt-neg-out 97.3%

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

3. Simplified 97.6%

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

   1. *-commutative 97.6%

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

   2. associate-*r* 97.9%

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

   3. associate-/r* 97.8%

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

   4. associate-*r* 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*r* 97.4%

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

   7. associate-/r* 97.3%

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

   8. associate-*l/ 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in x around 0 69.8%

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

8. Final simplification 69.8%

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

Alternative 7: 70.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return sinf((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(1.0) / Float32(Float32(x * Float32(pi)) * tau)))
end

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. remove-double-neg 97.3%

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

   6. distribute-rgt-neg-out 97.3%

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

   7. distribute-lft-neg-out 97.3%

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

   8. distribute-rgt-neg-out 97.3%

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

   9. distribute-lft-neg-out 97.3%

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

   10. distribute-lft-neg-out 97.3%

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

   11. distribute-rgt-neg-out 97.3%

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

3. Simplified 97.6%

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

5. Taylor expanded in x around 0 69.8%

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

6. Final simplification 69.8%

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

Alternative 8: 64.4% 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 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. remove-double-neg 97.3%

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

   6. distribute-rgt-neg-out 97.3%

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

   7. distribute-lft-neg-out 97.3%

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

   8. distribute-rgt-neg-out 97.3%

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

   9. distribute-lft-neg-out 97.3%

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

   10. distribute-lft-neg-out 97.3%

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

   11. distribute-rgt-neg-out 97.3%

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

3. Simplified 97.6%

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

5. Taylor expanded in tau around 0 64.0%

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

Alternative 9: 64.6% accurate, 19.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. remove-double-neg 97.3%

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

   6. distribute-rgt-neg-out 97.3%

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

   7. distribute-lft-neg-out 97.3%

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

   8. distribute-rgt-neg-out 97.3%

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

   9. distribute-lft-neg-out 97.3%

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

   10. distribute-lft-neg-out 97.3%

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

   11. distribute-rgt-neg-out 97.3%

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

3. Simplified 97.6%

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

5. Taylor expanded in tau around 0 64.0%

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

6. Taylor expanded in x around 0 63.9%

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

   1. *-commutative 63.9%

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

   2. unpow2 63.9%

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

   3. unpow2 63.9%

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

   4. swap-sqr 63.9%

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

   5. unpow2 63.9%

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

   6. *-commutative 63.9%

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

8. Simplified 63.9%

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

   1. unpow2 63.9%

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

10. Applied egg-rr 63.9%

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

Alternative 10: 63.6% 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 98.2%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.9%

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

   3. associate-*l* 97.4%

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

   4. associate-/l/ 97.3%

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

   5. remove-double-neg 97.3%

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

   6. distribute-rgt-neg-out 97.3%

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

   7. distribute-lft-neg-out 97.3%

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

   8. distribute-rgt-neg-out 97.3%

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

   9. distribute-lft-neg-out 97.3%

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

   10. distribute-lft-neg-out 97.3%

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

   11. distribute-rgt-neg-out 97.3%

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

3. Simplified 97.6%

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

5. Taylor expanded in x around 0 63.1%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))