Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 19.8s

Alternatives: 15

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 98.1%

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

2. Final simplification 98.1%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * 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 = x * (single(pi) * tau);
	tmp = (sin((x * single(pi))) / (x * single(pi))) * (sin(t_1) / t_1);
end

Derivation

1. Initial program 98.1%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 4: 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 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-*l* 97.4%

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

   3. *-commutative 97.4%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 5: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. frac-times 97.8%

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

   2. associate-/l* 97.9%

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

   3. associate-*r* 97.4%

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

   4. associate-*r* 97.4%

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

   5. pow2 97.4%

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

5. Applied egg-rr 97.4%

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

   1. expm1-log1p-u 97.2%

      \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}^{2} \cdot tau}{\sin \left(x \cdot \pi\right)}} \]

7. Applied egg-rr 97.2%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}^{2} \cdot tau}{\sin \left(x \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. div-inv 97.1%

      \[\leadsto \color{blue}{\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{1}{\frac{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}^{2} \cdot tau}{\sin \left(x \cdot \pi\right)}}} \]

   2. clear-num 97.1%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}^{2} \cdot tau}} \]

   3. *-un-lft-identity 97.1%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\color{blue}{1 \cdot \sin \left(x \cdot \pi\right)}}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}^{2} \cdot tau} \]

   4. times-frac 96.8%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \color{blue}{\left(\frac{1}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}^{2}} \cdot \frac{\sin \left(x \cdot \pi\right)}{tau}\right)} \]

   5. metadata-eval 96.8%

      \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \left(\frac{\color{blue}{1 \cdot 1}}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}^{2}} \cdot \frac{\sin \left(x \cdot \pi\right)}{tau}\right) \]

   6. expm1-log1p-u 97.0%

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

   7. unpow2 97.0%

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

   8. frac-times 96.8%

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

   9. inv-pow 96.8%

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

   10. inv-pow 96.8%

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

   11. pow-prod-up 97.3%

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

   12. metadata-eval 97.3%

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

9. Applied egg-rr 97.3%

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

10. Final simplification 97.3%

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

Alternative 6: 97.4% 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 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. frac-times 97.8%

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

   2. associate-/l* 97.9%

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

   3. associate-*r* 97.4%

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

   4. associate-*r* 97.4%

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

   5. pow2 97.4%

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

5. Applied egg-rr 97.4%

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

6. Taylor expanded in x around inf 96.9%

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

   1. times-frac 97.0%

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

   2. *-commutative 97.0%

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

   3. *-commutative 97.0%

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

   4. unpow2 97.0%

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

   5. unpow2 97.0%

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

   6. swap-sqr 97.5%

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

   7. unpow2 97.5%

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

   8. associate-*r/ 97.7%

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

   9. associate-*l/ 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 7: 84.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 84.8%

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

   1. associate-*r* 84.8%

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

   2. unpow2 84.8%

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

6. Simplified 84.8%

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

7. Taylor expanded in x around 0 84.8%

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

   1. unpow2 84.8%

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

   2. unpow2 84.8%

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

   3. swap-sqr 84.8%

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

   4. unpow2 84.8%

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

9. Simplified 84.8%

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

10. Final simplification 84.8%

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

Alternative 8: 78.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. frac-times 97.8%

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

   2. associate-/l* 97.9%

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

   3. associate-*r* 97.4%

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

   4. associate-*r* 97.4%

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

   5. pow2 97.4%

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

5. Applied egg-rr 97.4%

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

6. Taylor expanded in x around inf 96.9%

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

   1. times-frac 97.0%

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

   2. *-commutative 97.0%

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

   3. *-commutative 97.0%

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

   4. unpow2 97.0%

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

   5. unpow2 97.0%

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

   6. swap-sqr 97.5%

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

   7. unpow2 97.5%

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

   8. associate-*r/ 97.7%

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

   9. associate-*l/ 97.6%

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

8. Simplified 97.6%

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

9. Taylor expanded in tau around 0 78.3%

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

10. Final simplification 78.3%

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

Alternative 9: 78.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. frac-times 97.8%

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

   2. associate-/l* 97.9%

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

   3. associate-*r* 97.4%

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

   4. associate-*r* 97.4%

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

   5. pow2 97.4%

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

5. Applied egg-rr 97.4%

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

6. Taylor expanded in x around inf 96.9%

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

   1. times-frac 97.0%

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

   2. *-commutative 97.0%

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

   3. *-commutative 97.0%

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

   4. unpow2 97.0%

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

   5. unpow2 97.0%

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

   6. swap-sqr 97.5%

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

   7. unpow2 97.5%

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

   8. associate-*r/ 97.7%

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

   9. associate-*l/ 97.6%

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

8. Simplified 97.6%

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

9. Taylor expanded in tau around 0 78.3%

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

    1. +-commutative 78.3%

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

    2. *-commutative 78.3%

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

    3. fma-def 78.3%

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

    4. unpow2 78.3%

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

11. Simplified 78.3%

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

12. Final simplification 78.3%

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

Alternative 10: 78.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in x around 0 77.7%

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

   1. +-commutative 77.7%

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

   2. fma-def 77.7%

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

   3. distribute-lft-out 77.7%

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

   4. distribute-rgt1-in 77.7%

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

   5. unpow2 77.7%

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

   6. unpow2 77.7%

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

6. Simplified 77.7%

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

7. Final simplification 77.7%

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

Alternative 11: 70.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. *-commutative 97.7%

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

   2. clear-num 97.5%

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

   3. associate-/r* 97.2%

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

   4. frac-times 97.4%

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

   5. *-un-lft-identity 97.4%

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

   6. associate-/l* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 70.3%

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

7. Taylor expanded in x around -inf 70.5%

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

8. Final simplification 70.5%

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

Alternative 12: 69.4% accurate, 2.0× speedup

Math

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

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(x * Float32(Float32(pi) * tau)) ^ Float32(2.0)), Float32(1.0))
end

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. *-commutative 97.7%

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

   2. clear-num 97.5%

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

   3. associate-/r* 97.2%

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

   4. frac-times 97.4%

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

   5. *-un-lft-identity 97.4%

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

   6. associate-/l* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in x around 0 70.3%

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

7. Taylor expanded in x around 0 69.0%

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

   1. +-commutative 69.0%

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

   2. unpow2 69.0%

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

   3. *-commutative 69.0%

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

   4. unpow2 69.0%

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

   5. fma-def 69.0%

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

9. Simplified 69.0%

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

10. Final simplification 69.0%

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

Alternative 13: 64.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in tau around 0 64.1%

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

   1. *-commutative 64.1%

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

6. Simplified 64.1%

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

7. Final simplification 64.1%

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

Alternative 14: 64.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in x around 0 63.9%

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

5. Taylor expanded in x around 0 64.0%

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

   1. associate-/l* 64.0%

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

   2. unpow2 64.0%

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

7. Simplified 64.0%

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

8. Final simplification 64.0%

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

Alternative 15: 63.2% 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 98.1%

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

   1. associate-*l/ 98.0%

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

   2. times-frac 97.5%

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

   3. associate-*l* 97.3%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. associate-*l* 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in x around 0 63.3%

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

5. Final simplification 63.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023274 
(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))))