Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 13.2s

Alternatives: 14

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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* 97.3%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (x * single(pi)))) * (((x * single(pi)) ^ single(-2.0)) * (sin((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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around inf 97.0%

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

   1. associate-/l* 96.8%

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

   2. *-commutative 96.8%

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

   3. *-commutative 96.8%

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

   4. associate-/l* 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. times-frac 96.8%

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

6. Simplified 97.0%

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

   1. *-commutative 97.0%

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

   2. div-inv 96.9%

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

   3. associate-*l* 96.8%

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

   4. associate-*r* 96.6%

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

   5. *-commutative 96.6%

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

   6. times-frac 96.9%

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

   7. *-un-lft-identity 96.9%

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

   8. pow2 96.9%

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

   9. unpow-prod-down 97.0%

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

   10. associate-/l/ 97.1%

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

   11. add-log-exp 52.7%

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

8. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sin \left(\pi \cdot \left(tau \cdot x\right)\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{tau}\right) \cdot {\left(\pi \cdot x\right)}^{-2}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 97.3%

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

   2. expm1-log1p 97.4%

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

   3. associate-*l* 97.2%

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

   4. *-commutative 97.2%

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

   5. associate-*r* 97.3%

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

   6. *-commutative 97.3%

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

10. Simplified 97.3%

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

11. Final simplification 97.3%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (x * single(pi)))) * ((x * single(pi)) ^ single(-2.0))) * (sin((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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around inf 97.0%

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

   1. associate-/l* 96.8%

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

   2. *-commutative 96.8%

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

   3. *-commutative 96.8%

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

   4. associate-/l* 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. times-frac 96.8%

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

6. Simplified 97.0%

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

   1. add-log-exp 96.4%

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

   2. exp-prod 78.2%

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

   3. div-inv 78.2%

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

   4. pow2 78.2%

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

   5. unpow-prod-down 78.3%

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

   6. pow-flip 78.3%

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

   7. metadata-eval 78.3%

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

8. Applied egg-rr 78.3%

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

   1. log-pow 78.5%

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

   2. *-commutative 78.5%

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

   3. associate-*r* 78.6%

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

   4. *-commutative 78.6%

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

   5. rem-log-exp 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 4: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

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(((float) M_PI), 2.0f) * (x * x))));
}

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((Float32(pi) ^ Float32(2.0)) * Float32(x * x)))))
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) * ((single(pi) ^ single(2.0)) * (x * 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. associate-*l* 97.3%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around 0 85.3%

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

   1. *-commutative 85.3%

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

   2. unpow2 85.3%

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

6. Simplified 85.3%

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

7. Final simplification 85.3%

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

Alternative 5: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = tau * (x * ((float) M_PI));
	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(tau * Float32(x * Float32(pi)))
	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 = tau * (x * single(pi));
	tmp = (sin(t_1) / t_1) * (single(1.0) + (single(-0.16666666666666666) * ((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. Taylor expanded in x around 0 85.3%

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

   1. *-commutative 85.3%

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

   2. unpow2 85.3%

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

4. Simplified 85.3%

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

5. Taylor expanded in x around 0 85.3%

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

   1. unpow2 85.3%

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

   2. *-commutative 85.3%

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

   3. unpow2 85.3%

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

   4. swap-sqr 85.3%

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

   5. unpow2 85.3%

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

7. Simplified 85.3%

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

8. Final simplification 85.3%

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

Alternative 6: 84.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(x * Float32(pi)))) * 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((tau * (x * single(pi)))) * (((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-*r/ 98.0%

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

   2. associate-*l/ 97.8%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.8%

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

   5. associate-*l* 97.2%

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

   6. associate-*r* 97.0%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.0%

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

   9. swap-sqr 96.9%

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

   10. associate-*r* 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in x around 0 84.8%

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

5. Taylor expanded in tau around inf 85.0%

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

6. Final simplification 85.0%

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

Alternative 7: 79.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = exp(((x * x) * ((single(pi) ^ single(2.0)) * (single(-0.16666666666666666) + (single(-0.16666666666666666) * (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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

   1. add-exp-log 93.9%

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

   2. associate-*r/ 93.8%

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

   3. associate-*l/ 93.8%

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

   4. *-commutative 93.8%

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

   5. associate-*r* 93.6%

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

   6. *-commutative 93.6%

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

   7. associate-*l* 93.9%

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

   8. associate-/r* 93.9%

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

   9. *-commutative 93.9%

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

5. Applied egg-rr 93.9%

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

6. Taylor expanded in x around 0 81.0%

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

   1. associate-*r* 81.0%

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

   2. distribute-rgt-out 81.0%

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

   3. unpow2 81.0%

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

   4. unpow2 81.0%

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

8. Simplified 81.0%

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

9. Final simplification 81.0%

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

Alternative 8: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (x * single(pi));
	tmp = sin(t_1) / t_1;
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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around inf 97.0%

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

   1. associate-/l* 96.8%

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

   2. *-commutative 96.8%

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

   3. *-commutative 96.8%

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

   4. associate-/l* 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. times-frac 96.8%

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

6. Simplified 97.0%

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

7. Taylor expanded in x around 0 71.7%

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

8. Taylor expanded in x around -inf 72.0%

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

9. Final simplification 72.0%

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

Alternative 9: 69.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around inf 97.0%

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

   1. associate-/l* 96.8%

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

   2. *-commutative 96.8%

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

   3. *-commutative 96.8%

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

   4. associate-/l* 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. times-frac 96.8%

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

6. Simplified 97.0%

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

7. Taylor expanded in x around 0 71.7%

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

8. Taylor expanded in x around 0 70.6%

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

   1. +-commutative 70.6%

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

   2. fma-def 70.6%

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

   3. unpow2 70.6%

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

   4. associate-*r* 70.6%

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

   5. unpow2 70.6%

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

   6. unpow2 70.6%

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

   7. unswap-sqr 70.6%

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

   8. swap-sqr 70.6%

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

   9. associate-*r* 70.6%

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

   10. *-commutative 70.6%

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

   11. associate-*r* 70.6%

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

   12. associate-*r* 70.6%

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

   13. *-commutative 70.6%

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

   14. associate-*r* 70.6%

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

   15. unpow2 70.6%

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

10. Simplified 70.6%

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

11. Final simplification 70.6%

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

Alternative 10: 69.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * ((x * x) * ((single(pi) ^ single(2.0)) * (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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around inf 97.0%

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

   1. associate-/l* 96.8%

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

   2. *-commutative 96.8%

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

   3. *-commutative 96.8%

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

   4. associate-/l* 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. *-commutative 97.0%

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

   8. times-frac 96.8%

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

6. Simplified 97.0%

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

7. Taylor expanded in x around 0 71.7%

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

8. Taylor expanded in x around 0 70.6%

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

   1. unpow2 70.6%

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

   2. associate-*r* 70.6%

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

   3. *-commutative 70.6%

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

   4. unpow2 70.6%

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

10. Simplified 70.6%

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

11. Final simplification 70.6%

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

Alternative 11: 64.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 66.4%

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

   1. *-commutative 66.4%

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

6. Simplified 66.4%

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

7. Taylor expanded in x around 0 66.4%

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

   1. unpow2 66.4%

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

   2. *-commutative 66.4%

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

   3. unpow2 66.4%

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

   4. swap-sqr 66.4%

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

   5. unpow2 66.4%

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

9. Simplified 66.4%

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

    1. unpow2 66.4%

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

    2. associate-*r* 66.4%

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

11. Applied egg-rr 66.4%

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

12. Final simplification 66.4%

    \[\leadsto 1 + -0.16666666666666666 \cdot \left(x \cdot \left(\pi \cdot \left(x \cdot \pi\right)\right)\right) \]

Alternative 12: 64.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * (x * (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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 66.4%

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

   1. *-commutative 66.4%

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

6. Simplified 66.4%

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

7. Taylor expanded in x around 0 66.4%

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

   1. unpow2 66.4%

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

   2. *-commutative 66.4%

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

   3. unpow2 66.4%

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

   4. swap-sqr 66.4%

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

   5. unpow2 66.4%

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

9. Simplified 66.4%

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

    1. unpow-prod-down 66.4%

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

    2. pow2 66.4%

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

    3. associate-*r* 66.4%

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

11. Applied egg-rr 66.4%

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

12. Final simplification 66.4%

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

Alternative 13: 64.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 66.4%

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

   1. *-commutative 66.4%

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

6. Simplified 66.4%

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

7. Taylor expanded in x around 0 66.4%

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

   1. unpow2 66.4%

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

   2. *-commutative 66.4%

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

   3. unpow2 66.4%

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

   4. swap-sqr 66.4%

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

   5. unpow2 66.4%

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

9. Simplified 66.4%

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

10. Final simplification 66.4%

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

Alternative 14: 63.1% 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. *-commutative 97.9%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.7%

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

   4. associate-*r* 97.5%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 97.5%

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

   7. associate-*l* 97.3%

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

   8. swap-sqr 97.1%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around 0 65.6%

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

5. Final simplification 65.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023172 
(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))))