Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 13.6s

Alternatives: 13

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 97.9%

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

2. Final simplification 97.9%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r/ 97.1%

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

   3. associate-*r* 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*r* 97.1%

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

   6. associate-*r* 97.3%

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

   7. pow2 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around inf 96.7%

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

   1. *-commutative 96.7%

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

   2. *-commutative 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*r* 96.9%

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

   5. associate-*r/ 97.0%

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

   6. unpow2 97.0%

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

   7. unpow2 97.0%

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

   8. swap-sqr 97.2%

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

   9. unpow2 97.2%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 3: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. times-frac 97.3%

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

3. Applied egg-rr 97.3%

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

4. Taylor expanded in x around 0 82.3%

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

   1. unpow2 82.3%

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

   2. unpow2 82.3%

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

   3. swap-sqr 82.3%

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

   4. unpow2 82.3%

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

6. Simplified 82.3%

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

7. Final simplification 82.3%

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

Alternative 4: 84.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 81.6%

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

5. Taylor expanded in x around 0 81.8%

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

   1. *-commutative 81.8%

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

   2. associate-*r* 81.8%

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

   3. *-commutative 81.8%

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

7. Simplified 81.8%

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

8. Final simplification 81.8%

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

Alternative 5: 84.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 81.6%

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

5. Taylor expanded in tau around inf 81.8%

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

6. Final simplification 81.8%

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

Alternative 6: 78.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 75.5%

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

   1. +-commutative 75.5%

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

   2. fma-def 75.5%

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

   3. distribute-lft-out 75.5%

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

   4. distribute-lft1-in 75.5%

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

6. Simplified 75.5%

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

   1. fma-udef 75.5%

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

   2. associate-*r* 75.5%

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

   3. add-sqr-sqrt 75.5%

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

   4. pow2 75.5%

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

   5. *-commutative 75.5%

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

   6. sqrt-prod 75.5%

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

   7. unpow2 75.5%

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

   8. sqrt-prod 75.5%

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

   9. add-sqr-sqrt 75.5%

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

   10. +-commutative 75.5%

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

   11. unpow2 75.5%

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

   12. hypot-1-def 75.5%

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

8. Applied egg-rr 75.5%

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

9. Final simplification 75.5%

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

Alternative 7: 78.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 75.5%

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

   1. +-commutative 75.5%

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

   2. fma-def 75.5%

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

   3. distribute-lft-out 75.5%

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

   4. distribute-lft1-in 75.5%

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

6. Simplified 75.5%

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

7. Taylor expanded in x around 0 75.5%

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

   1. +-commutative 75.5%

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

   2. fma-def 75.5%

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

   3. associate-*r* 75.5%

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

   4. unpow2 75.5%

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

   5. unpow2 75.5%

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

   6. swap-sqr 75.5%

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

   7. unpow2 75.5%

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

   8. +-commutative 75.5%

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

   9. unpow2 75.5%

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

   10. fma-udef 75.5%

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

9. Simplified 75.5%

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

10. Final simplification 75.5%

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

Alternative 8: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. times-frac 97.3%

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

3. Applied egg-rr 97.3%

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

4. Taylor expanded in x around 0 67.6%

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

5. Final simplification 67.6%

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

Alternative 9: 70.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

   1. *-commutative 97.0%

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

   2. clear-num 97.0%

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

   3. un-div-inv 97.1%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. associate-*r* 97.2%

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

   7. associate-*r* 97.3%

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

   8. pow2 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 67.5%

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

7. Final simplification 67.5%

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

Alternative 10: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in tau around 0 60.8%

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

5. Taylor expanded in x around 0 61.2%

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

   1. +-commutative 61.2%

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

   2. fma-def 61.2%

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

   3. unpow2 61.2%

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

   4. unpow2 61.2%

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

   5. swap-sqr 61.2%

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

   6. unpow2 61.2%

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

7. Simplified 61.2%

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

   1. unpow2 61.2%

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

   2. associate-*r* 61.2%

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

9. Applied egg-rr 61.2%

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

10. Final simplification 61.2%

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

Alternative 11: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in tau around 0 60.8%

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

5. Taylor expanded in x around 0 61.2%

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

   1. +-commutative 61.2%

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

   2. fma-def 61.2%

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

   3. unpow2 61.2%

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

   4. unpow2 61.2%

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

   5. swap-sqr 61.2%

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

   6. unpow2 61.2%

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

7. Simplified 61.2%

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

8. Final simplification 61.2%

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

Alternative 12: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in tau around 0 60.8%

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

5. Taylor expanded in x around 0 61.2%

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

   1. +-commutative 61.2%

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

   2. fma-def 61.2%

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

   3. unpow2 61.2%

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

   4. unpow2 61.2%

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

   5. swap-sqr 61.2%

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

   6. unpow2 61.2%

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

7. Simplified 61.2%

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

   1. fma-udef 61.2%

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

9. Applied egg-rr 61.2%

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

10. Final simplification 61.2%

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

Alternative 13: 63.4% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.8%

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

   2. *-commutative 97.8%

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

   3. times-frac 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 60.0%

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

5. Final simplification 60.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023298 
(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))))