Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 13.9s

Alternatives: 13

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% 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 97.7%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.7%

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

3. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. associate-*r/ 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 96.6%

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

   4. *-commutative 96.6%

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

   5. associate-*l* 96.8%

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

   6. associate-*r* 96.8%

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

   7. swap-sqr 96.9%

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

   8. pow2 96.9%

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

   9. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

   1. *-commutative 96.9%

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

   2. expm1-log1p-u 96.8%

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

7. Applied egg-rr 96.8%

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

8. Taylor expanded in x around inf 96.6%

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

   1. *-commutative 96.6%

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

   2. *-commutative 96.6%

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

   3. associate-*r* 96.8%

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

   4. *-commutative 96.8%

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

   5. *-commutative 96.8%

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

   6. *-commutative 96.8%

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

   7. unpow2 96.8%

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

   8. unpow2 96.8%

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

   9. swap-sqr 96.9%

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

   10. unpow2 96.9%

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

   11. associate-*r/ 96.8%

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

10. Simplified 97.1%

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

11. Final simplification 97.1%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. associate-*r/ 96.9%

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

   2. *-commutative 96.9%

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

   3. associate-*r* 96.6%

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

   4. *-commutative 96.6%

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

   5. associate-*l* 96.8%

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

   6. associate-*r* 96.8%

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

   7. swap-sqr 96.9%

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

   8. pow2 96.9%

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

   9. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

7. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{{\left(x \cdot \pi\right)}^{2}}\right)\right)} \]
8. Step-by-step derivation

   1. log1p-def 97.0%

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

   2. log1p-expm1-u 97.0%

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

   3. associate-*r* 97.3%

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

   4. *-commutative 97.3%

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

   5. *-commutative 97.3%

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

   6. div-inv 97.3%

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

   7. *-commutative 97.3%

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

   8. *-commutative 97.3%

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

   9. pow-flip 97.4%

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

   10. metadata-eval 97.4%

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

9. Applied egg-rr 97.4%

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

10. Final simplification 97.4%

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

Alternative 4: 85.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. associate-*l* 97.2%

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

   2. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 97.4%

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

   2. expm1-udef 97.4%

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

   3. associate-*r* 97.0%

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

   4. associate-*r* 97.3%

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

   5. *-commutative 97.3%

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

   6. associate-*l* 96.9%

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

   7. *-commutative 96.9%

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

   8. associate-*l* 97.3%

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

5. Applied egg-rr 97.3%

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

   1. sub-neg 97.3%

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

   2. log1p-udef 97.4%

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

   3. add-exp-log 97.4%

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

   4. metadata-eval 97.4%

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

7. Applied egg-rr 97.4%

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

   1. associate-+l+ 97.6%

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

   2. associate-*r* 97.1%

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

   3. *-commutative 97.1%

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

   4. associate-*r* 97.6%

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

   5. *-commutative 97.6%

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

9. Simplified 97.6%

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

10. Taylor expanded in x around 0 87.1%

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

    1. associate-*r* 87.1%

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

    2. unpow2 87.1%

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

12. Simplified 87.1%

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

13. Final simplification 87.1%

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

Alternative 5: 85.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-unprod 97.7%

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

   3. swap-sqr 97.5%

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

   4. associate-*r* 97.5%

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

   5. expm1-log1p-u 97.4%

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

   6. associate-*r* 97.4%

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

   7. swap-sqr 97.6%

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

   8. sqrt-unprod 96.8%

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

   9. add-sqr-sqrt 97.6%

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

   10. *-commutative 97.6%

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

3. Applied egg-rr 97.6%

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

4. Taylor expanded in x around 0 87.1%

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

   1. unpow2 87.1%

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

   2. unpow2 87.1%

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

   3. swap-sqr 87.1%

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

   4. unpow2 87.1%

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

6. Simplified 87.1%

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

7. Final simplification 87.1%

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

Alternative 6: 84.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 97.0%

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

   4. associate-*r* 96.7%

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

   5. *-commutative 96.7%

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

   6. associate-*l* 97.0%

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

   7. associate-/l* 96.7%

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

   8. *-commutative 96.7%

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

   9. associate-*r* 96.6%

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

   10. swap-sqr 96.7%

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

   11. pow2 96.7%

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

   12. *-commutative 96.7%

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

5. Applied egg-rr 96.7%

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

6. Taylor expanded in x around 0 86.3%

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

   1. *-commutative 86.3%

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

8. Simplified 86.3%

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

9. Final simplification 86.3%

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

Alternative 7: 78.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around 0 80.3%

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

   1. +-commutative 80.3%

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

   2. fma-def 80.3%

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

   3. distribute-lft-out 80.3%

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

   4. distribute-rgt1-in 80.3%

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

   5. unpow2 80.3%

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

   6. unpow2 80.3%

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

6. Simplified 80.3%

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

7. Final simplification 80.3%

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

Alternative 8: 70.8% 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 97.7%

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

   1. add-sqr-sqrt 96.9%

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

   2. sqrt-unprod 97.7%

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

   3. swap-sqr 97.5%

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

   4. associate-*r* 97.5%

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

   5. expm1-log1p-u 97.4%

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

   6. associate-*r* 97.4%

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

   7. swap-sqr 97.6%

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

   8. sqrt-unprod 96.8%

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

   9. add-sqr-sqrt 97.6%

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

   10. *-commutative 97.6%

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

3. Applied egg-rr 97.6%

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

4. Taylor expanded in x around 0 73.1%

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

5. Final simplification 73.1%

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

Alternative 9: 64.2% 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 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 66.3%

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

   1. *-commutative 66.3%

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

6. Simplified 66.3%

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

7. Final simplification 66.3%

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

Alternative 10: 64.4% 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 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 66.3%

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

   1. *-commutative 66.3%

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

6. Simplified 66.3%

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

   1. expm1-log1p-u 66.3%

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

   2. expm1-udef 56.0%

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

   3. log1p-udef 56.0%

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

   4. add-exp-log 56.0%

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

   5. +-commutative 56.0%

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

   6. *-commutative 56.0%

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

8. Applied egg-rr 56.0%

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

9. Taylor expanded in x around 0 66.2%

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

    1. *-commutative 66.2%

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

    2. unpow2 66.2%

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

    3. unpow2 66.2%

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

    4. swap-sqr 66.2%

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

    5. unpow2 66.2%

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

11. Simplified 66.2%

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

    1. unpow2 66.2%

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

    2. associate-*r* 66.2%

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

    3. *-commutative 66.2%

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

    4. associate-*r* 66.2%

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

    5. pow2 66.2%

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

13. Applied egg-rr 66.2%

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

14. Final simplification 66.2%

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

Alternative 11: 64.4% accurate, 2.9× speedup

Math

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

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(Float32(x * 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 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 66.3%

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

   1. *-commutative 66.3%

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

6. Simplified 66.3%

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

   1. expm1-log1p-u 66.3%

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

   2. expm1-udef 56.0%

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

   3. log1p-udef 56.0%

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

   4. add-exp-log 56.0%

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

   5. +-commutative 56.0%

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

   6. *-commutative 56.0%

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

8. Applied egg-rr 56.0%

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

9. Taylor expanded in x around 0 66.2%

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

    1. *-commutative 66.2%

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

    2. unpow2 66.2%

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

    3. unpow2 66.2%

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

    4. swap-sqr 66.2%

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

    5. unpow2 66.2%

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

11. Simplified 66.2%

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

    1. pow-prod-down 66.2%

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

    2. unpow2 66.2%

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

13. Applied egg-rr 66.2%

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

14. Final simplification 66.2%

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

Alternative 12: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 66.3%

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

   1. *-commutative 66.3%

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

6. Simplified 66.3%

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

   1. expm1-log1p-u 66.3%

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

   2. expm1-udef 56.0%

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

   3. log1p-udef 56.0%

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

   4. add-exp-log 56.0%

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

   5. +-commutative 56.0%

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

   6. *-commutative 56.0%

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

8. Applied egg-rr 56.0%

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

9. Taylor expanded in x around 0 66.2%

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

    1. *-commutative 66.2%

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

    2. unpow2 66.2%

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

    3. unpow2 66.2%

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

    4. swap-sqr 66.2%

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

    5. unpow2 66.2%

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

11. Simplified 66.2%

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

12. Final simplification 66.2%

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

Alternative 13: 63.4% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. *-commutative 97.7%

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

   2. times-frac 97.7%

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

   3. associate-*r/ 97.4%

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

   4. associate-*r* 97.1%

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

   5. associate-/r* 97.0%

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

   6. associate-/l/ 97.1%

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

   7. associate-*l* 96.9%

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

   8. swap-sqr 96.5%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around 0 65.6%

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

5. Final simplification 65.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(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))))