Lanczos kernel

Percentage Accurate: 98.0% → 97.9%

Time: 15.5s

Alternatives: 16

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: 98.0% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t\_1}{t\_1} \cdot {\left({\left({\left({\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (*
    (/ (sin t_1) t_1)
    (pow
     (pow
      (pow
       (pow (pow (pow (/ (sin (* x PI)) (* x PI)) 3.0) 0.3333333333333333) 3.0)
       0.3333333333333333)
      3.0)
     0.3333333333333333))))

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf(t_1) / t_1) * powf(powf(powf(powf(powf(powf((sinf((x * ((float) M_PI))) / (x * ((float) M_PI))), 3.0f), 0.3333333333333333f), 3.0f), 0.3333333333333333f), 3.0f), 0.3333333333333333f);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * ((((((Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))) ^ Float32(3.0)) ^ Float32(0.3333333333333333)) ^ Float32(3.0)) ^ Float32(0.3333333333333333)) ^ Float32(3.0)) ^ Float32(0.3333333333333333)))
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))) ^ single(3.0)) ^ single(0.3333333333333333)) ^ single(3.0)) ^ single(0.3333333333333333)) ^ single(3.0)) ^ single(0.3333333333333333));
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

6. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

8. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\left({\color{blue}{\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]
9. Step-by-step derivation

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

10. Applied egg-rr 98.0%

    \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\left({\left({\left({\color{blue}{\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]

11. Final simplification 98.0%

    \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\left({\left({\left({\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]
12. Add Preprocessing

Alternative 2: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t\_1}{t\_1} \cdot {\left({\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (*
    (/ (sin t_1) t_1)
    (pow
     (pow (pow (pow (/ (sin (* x PI)) (* x PI)) 3.0) 0.3333333333333333) 3.0)
     0.3333333333333333))))

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf(t_1) / t_1) * powf(powf(powf(powf((sinf((x * ((float) M_PI))) / (x * ((float) M_PI))), 3.0f), 0.3333333333333333f), 3.0f), 0.3333333333333333f);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * ((((Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))) ^ Float32(3.0)) ^ Float32(0.3333333333333333)) ^ Float32(3.0)) ^ Float32(0.3333333333333333)))
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))) ^ single(3.0)) ^ single(0.3333333333333333)) ^ single(3.0)) ^ single(0.3333333333333333));
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

6. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

8. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\left({\color{blue}{\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333} \]

9. Final simplification 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\left({\left({\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333} \]
10. Add Preprocessing

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around inf 97.4%

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

6. Final simplification 97.4%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

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(x * Float32(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 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 97.6%

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

   3. *-commutative 97.6%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 7: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := tau \cdot \left(x \cdot \pi\right)\\ {1}^{0.3333333333333333} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* tau (* x PI))))
   (* (pow 1.0 0.3333333333333333) (/ (sin t_1) t_1))))

C

float code(float x, float tau) {
	float t_1 = tau * (x * ((float) M_PI));
	return powf(1.0f, 0.3333333333333333f) * (sinf(t_1) / t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(tau * Float32(x * Float32(pi)))
	return Float32((Float32(1.0) ^ Float32(0.3333333333333333)) * Float32(sin(t_1) / t_1))
end

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (x * single(pi));
	tmp = (single(1.0) ^ single(0.3333333333333333)) * (sin(t_1) / t_1);
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

6. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}} \]

7. Taylor expanded in x around 0 71.5%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{1}}^{0.3333333333333333} \]

8. Taylor expanded in x around inf 71.5%

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau \cdot \left(x \cdot \pi\right)}} \cdot {1}^{0.3333333333333333} \]

9. Final simplification 71.5%

   \[\leadsto {1}^{0.3333333333333333} \cdot \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau \cdot \left(x \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 8: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t\_1}{t\_1} \cdot {1}^{0.3333333333333333} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (* (/ (sin t_1) t_1) (pow 1.0 0.3333333333333333))))

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf(t_1) / t_1) * powf(1.0f, 0.3333333333333333f);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(t_1) / t_1) * (Float32(1.0) ^ Float32(0.3333333333333333)))
end

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin(t_1) / t_1) * (single(1.0) ^ single(0.3333333333333333));
end

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. add-cbrt-cube 98.0%

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

   2. pow1/3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left(\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{0.3333333333333333}} \]

   3. pow3 98.0%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \]

6. Applied egg-rr 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{{\left({\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}^{3}\right)}^{0.3333333333333333}} \]

7. Taylor expanded in x around 0 71.5%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {\color{blue}{1}}^{0.3333333333333333} \]

8. Final simplification 71.5%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot {1}^{0.3333333333333333} \]
9. Add Preprocessing

Alternative 9: 70.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. frac-2neg 98.0%

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

   2. div-inv 97.9%

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

   3. associate-*r* 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-*r* 97.4%

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

   6. associate-*r* 97.5%

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

   7. *-commutative 97.5%

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

   8. associate-*r* 97.9%

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

   9. distribute-rgt-neg-in 97.9%

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in x around 0 71.5%

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

8. Final simplification 71.5%

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

Alternative 10: 70.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. associate-*r* 97.4%

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

   2. associate-*r* 98.0%

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

   3. associate-/r* 97.7%

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

   4. div-inv 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*r* 97.2%

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

6. Applied egg-rr 97.2%

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

7. Taylor expanded in x around 0 71.4%

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

8. Final simplification 71.4%

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

Alternative 11: 70.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. associate-*r* 97.4%

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

   2. associate-*r* 98.0%

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

   3. associate-/r* 97.7%

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

   4. div-inv 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*r* 97.2%

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

6. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.2%

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

   2. un-div-inv 97.4%

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

   3. associate-*r* 97.8%

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

   4. *-commutative 97.8%

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

   5. associate-*l* 97.5%

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

8. Applied egg-rr 97.5%

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

9. Taylor expanded in x around 0 71.2%

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

10. Final simplification 71.2%

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

Alternative 12: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. times-frac 65.1%

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

7. Applied egg-rr 65.1%

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

   1. clear-num 65.0%

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

   2. associate-/r/ 65.2%

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

9. Applied egg-rr 65.2%

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

10. Final simplification 65.2%

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

Alternative 13: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. times-frac 65.1%

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

7. Applied egg-rr 65.1%

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

8. Final simplification 65.1%

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

Alternative 14: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. times-frac 65.1%

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

7. Applied egg-rr 65.1%

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

   1. frac-times 65.1%

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

   2. associate-/l* 65.1%

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

9. Applied egg-rr 65.1%

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

10. Final simplification 65.1%

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

Alternative 15: 64.3% 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 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 65.1%

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

6. Final simplification 65.1%

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

Alternative 16: 63.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 65.1%

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

   1. *-un-lft-identity 65.1%

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

   2. times-frac 65.1%

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

7. Applied egg-rr 65.1%

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

8. Taylor expanded in x around 0 64.3%

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

9. Final simplification 64.3%

   \[\leadsto x \cdot \frac{1}{x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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))))