Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 10.9s

Alternatives: 12

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 98.3%

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

2. Final simplification 98.3%

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

Alternative 2: 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 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*l* 97.8%

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

   3. *-commutative 97.8%

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

   4. associate-*l* 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. frac-times 98.0%

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

   2. *-commutative 98.0%

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

   3. *-commutative 98.0%

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

   4. associate-*l* 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*l* 97.4%

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

   7. pow2 97.4%

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

   8. *-commutative 97.4%

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

3. Applied egg-rr 97.4%

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

   1. div-inv 97.4%

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

   2. *-commutative 97.4%

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

   3. associate-*r* 97.7%

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

   4. *-commutative 97.7%

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

   5. associate-*l* 97.6%

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

   6. *-commutative 97.6%

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

   7. *-commutative 97.6%

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

   8. associate-/r* 97.6%

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

5. Applied egg-rr 97.7%

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

6. Final simplification 97.7%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in x around inf 97.3%

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

   1. times-frac 97.2%

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

   2. *-commutative 97.2%

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

   3. unpow2 97.2%

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

   4. unpow2 97.2%

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

   5. swap-sqr 97.9%

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

   6. unpow2 97.9%

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

   7. *-commutative 97.9%

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

4. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 5: 84.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in x around inf 97.3%

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

   1. times-frac 97.2%

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

   2. *-commutative 97.2%

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

   3. unpow2 97.2%

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

   4. unpow2 97.2%

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

   5. swap-sqr 97.9%

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

   6. unpow2 97.9%

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

   7. *-commutative 97.9%

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

4. Simplified 97.9%

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

5. Taylor expanded in x around 0 86.5%

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

6. Final simplification 86.5%

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

Alternative 6: 70.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in x around inf 97.3%

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

   1. times-frac 97.2%

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

   2. *-commutative 97.2%

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

   3. unpow2 97.2%

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

   4. unpow2 97.2%

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

   5. swap-sqr 97.9%

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

   6. unpow2 97.9%

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

   7. *-commutative 97.9%

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

4. Simplified 97.9%

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

5. Taylor expanded in x around 0 71.5%

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

6. Final simplification 71.5%

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

Alternative 7: 70.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*l* 97.8%

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

   3. *-commutative 97.8%

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

   4. associate-*l* 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in x around 0 71.5%

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

5. Final simplification 71.5%

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

Alternative 8: 69.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*l* 97.8%

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

   3. *-commutative 97.8%

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

   4. associate-*l* 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in x around 0 81.1%

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

   1. +-commutative 81.1%

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

   2. fma-def 81.1%

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

   3. *-commutative 81.1%

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

   4. associate-*l* 81.1%

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

   5. unpow2 81.1%

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

   6. unpow2 81.1%

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

   7. unpow2 81.1%

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

   8. unswap-sqr 81.1%

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

   9. swap-sqr 81.1%

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

   10. unpow2 81.1%

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

   11. associate-*r* 81.1%

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

   12. *-commutative 81.1%

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

   13. *-commutative 81.1%

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

   14. *-commutative 81.1%

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

6. Simplified 81.1%

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

7. Taylor expanded in x around 0 70.8%

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

8. Final simplification 70.8%

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

Alternative 9: 69.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*l* 97.8%

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

   3. *-commutative 97.8%

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

   4. associate-*l* 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in x around 0 81.1%

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

   1. +-commutative 81.1%

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

   2. fma-def 81.1%

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

   3. *-commutative 81.1%

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

   4. associate-*l* 81.1%

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

   5. unpow2 81.1%

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

   6. unpow2 81.1%

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

   7. unpow2 81.1%

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

   8. unswap-sqr 81.1%

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

   9. swap-sqr 81.1%

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

   10. unpow2 81.1%

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

   11. associate-*r* 81.1%

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

   12. *-commutative 81.1%

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

   13. *-commutative 81.1%

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

   14. *-commutative 81.1%

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

6. Simplified 81.1%

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

7. Taylor expanded in x around 0 70.8%

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

8. Taylor expanded in tau around 0 70.8%

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

   1. *-commutative 70.8%

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

   2. unpow2 70.8%

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

   3. unpow2 70.8%

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

   4. swap-sqr 70.8%

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

   5. unpow2 70.8%

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

   6. swap-sqr 70.8%

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

   7. *-commutative 70.8%

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

   8. associate-*r* 70.8%

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

   9. *-commutative 70.8%

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

   10. associate-*r* 70.8%

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

   11. unpow2 70.8%

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

10. Simplified 70.8%

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

11. Final simplification 70.8%

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

Alternative 10: 69.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*l* 97.8%

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

   3. *-commutative 97.8%

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

   4. associate-*l* 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in x around 0 81.1%

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

   1. +-commutative 81.1%

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

   2. fma-def 81.1%

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

   3. *-commutative 81.1%

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

   4. associate-*l* 81.1%

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

   5. unpow2 81.1%

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

   6. unpow2 81.1%

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

   7. unpow2 81.1%

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

   8. unswap-sqr 81.1%

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

   9. swap-sqr 81.1%

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

   10. unpow2 81.1%

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

   11. associate-*r* 81.1%

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

   12. *-commutative 81.1%

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

   13. *-commutative 81.1%

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

   14. *-commutative 81.1%

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

6. Simplified 81.1%

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

7. Taylor expanded in x around 0 70.8%

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

   1. fma-udef 70.8%

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

9. Applied egg-rr 70.8%

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

10. Final simplification 70.8%

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

Alternative 11: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in tau around 0 64.7%

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

3. Taylor expanded in x around 0 65.1%

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

   1. +-commutative 65.1%

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

   2. fma-def 65.1%

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

   3. *-commutative 65.1%

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

   4. unpow2 65.1%

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

   5. unpow2 65.1%

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

   6. swap-sqr 65.1%

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

   7. unpow2 65.1%

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

   8. *-commutative 65.1%

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

5. Simplified 65.1%

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

   1. fma-udef 65.1%

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

   2. *-commutative 65.1%

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

   3. *-commutative 65.1%

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

7. Applied egg-rr 65.1%

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

8. Final simplification 65.1%

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

Alternative 12: 63.5% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ 98.2%

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

   2. associate-/l* 98.3%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 98.1%

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

   5. *-commutative 98.1%

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

   6. *-commutative 98.1%

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

3. Applied egg-rr 98.1%

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

4. Taylor expanded in x around 0 63.9%

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

5. Final simplification 63.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(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))))