Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 15.0s

Alternatives: 20

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

2. Final simplification 97.4%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. associate-*l* 96.7%

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

   2. associate-*l* 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

   1. associate-/r/ 96.5%

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

   2. associate-/r* 96.7%

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

   3. un-div-inv 96.4%

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

   4. clear-num 96.4%

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

   5. associate-*r/ 96.6%

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

   6. expm1-log1p-u 96.3%

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

   7. expm1-udef 95.8%

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

7. Applied egg-rr 96.0%

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

   1. expm1-def 96.5%

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

   2. expm1-log1p 96.7%

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

   3. associate-*r/ 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-/r/ 96.6%

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

   6. associate-*r/ 96.6%

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

   7. *-commutative 96.6%

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

   8. associate-*r* 97.0%

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

   9. *-commutative 97.0%

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

   10. *-commutative 97.0%

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

9. Simplified 96.7%

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

10. Final simplification 96.7%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

   1. associate-/r/ 96.5%

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

   2. associate-/r* 96.7%

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

   3. un-div-inv 96.4%

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

   4. clear-num 96.4%

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

   5. div-inv 96.2%

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

7. Applied egg-rr 96.7%

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

   1. add-log-exp 60.0%

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

   2. *-commutative 60.0%

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

   3. *-commutative 60.0%

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

   4. associate-*r* 60.1%

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

   5. *-commutative 60.1%

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

9. Applied egg-rr 60.1%

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

    1. expm1-log1p-u 60.1%

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

    2. expm1-udef 59.5%

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

    3. add-log-exp 96.2%

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

    4. associate-/r/ 96.2%

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

    5. *-commutative 96.2%

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

11. Applied egg-rr 96.2%

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

    1. expm1-def 96.8%

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

    2. expm1-log1p 97.0%

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

    3. associate-*l/ 97.0%

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

    4. associate-*r/ 96.9%

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

    5. associate-*l* 96.9%

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

    6. *-commutative 96.9%

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

    7. associate-*l* 96.6%

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

    8. associate-*l/ 96.6%

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

    9. associate-/l* 96.8%

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

13. Simplified 96.8%

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

14. Final simplification 96.8%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

   1. associate-/r/ 96.5%

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

   2. associate-/r* 96.7%

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

   3. un-div-inv 96.4%

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

   4. clear-num 96.4%

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

   5. div-inv 96.2%

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

7. Applied egg-rr 96.7%

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

   1. add-log-exp 60.0%

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

   2. *-commutative 60.0%

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

   3. *-commutative 60.0%

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

   4. associate-*r* 60.1%

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

   5. *-commutative 60.1%

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

9. Applied egg-rr 60.1%

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

    1. add-log-exp 97.0%

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

    2. div-inv 96.8%

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

    3. clear-num 96.9%

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

11. Applied egg-rr 96.9%

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

12. Final simplification 96.9%

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

Alternative 6: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

   1. associate-/r/ 96.5%

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

   2. associate-/r* 96.7%

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

   3. un-div-inv 96.4%

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

   4. clear-num 96.4%

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

   5. div-inv 96.2%

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

7. Applied egg-rr 96.7%

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

   1. associate-/r/ 96.5%

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

   2. *-commutative 96.5%

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

   3. *-commutative 96.5%

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

   4. associate-*r* 97.0%

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

   5. *-commutative 97.0%

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

9. Applied egg-rr 97.0%

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

10. Final simplification 97.0%

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

Alternative 7: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

   1. associate-/r/ 96.5%

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

   2. associate-/r* 96.7%

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

   3. un-div-inv 96.4%

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

   4. clear-num 96.4%

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

   5. div-inv 96.2%

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

7. Applied egg-rr 96.7%

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

8. Taylor expanded in x around -inf 97.0%

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

9. Final simplification 97.0%

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

Alternative 8: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

6. Taylor expanded in x around -inf 97.0%

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

7. Final simplification 97.0%

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

Alternative 9: 85.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. add-sqr-sqrt 96.7%

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

   2. sqrt-unprod 97.4%

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

   3. swap-sqr 97.1%

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

   4. associate-*r* 97.2%

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

   5. expm1-log1p-u 97.1%

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

   6. associate-*r* 97.1%

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

   7. swap-sqr 97.4%

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

   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 97.4%

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

   10. *-commutative 97.4%

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

3. Applied egg-rr 97.4%

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

4. Taylor expanded in x around 0 81.3%

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

   1. unpow2 81.3%

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

   2. unpow2 81.3%

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

   3. swap-sqr 81.3%

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

   4. unpow2 81.3%

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

6. Simplified 81.3%

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

7. Final simplification 81.3%

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

Alternative 10: 84.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

6. Taylor expanded in x around -inf 97.0%

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

7. Taylor expanded in x around 0 80.8%

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

   1. +-commutative 80.8%

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

   2. *-commutative 80.8%

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

   3. fma-def 80.8%

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

   4. *-commutative 80.8%

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

   5. *-commutative 80.8%

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

9. Simplified 80.8%

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

10. Final simplification 80.8%

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

Alternative 11: 84.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

6. Taylor expanded in x around -inf 97.0%

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

7. Taylor expanded in x around 0 80.8%

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

8. Final simplification 80.8%

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

Alternative 12: 78.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in x around 0 75.1%

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

   1. distribute-lft-out 75.1%

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

   2. *-commutative 75.1%

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

   3. unpow2 75.1%

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

   4. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Final simplification 75.1%

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

Alternative 13: 78.5% 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 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in x around 0 75.1%

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

   1. +-commutative 75.1%

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

   2. fma-def 75.1%

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

   3. distribute-lft-out 75.1%

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

   4. distribute-rgt1-in 75.1%

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

   5. unpow2 75.1%

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

   6. unpow2 75.1%

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

6. Simplified 75.1%

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

7. Final simplification 75.1%

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

Alternative 14: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. associate-/l* 96.6%

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

   4. associate-*r* 96.3%

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

   5. *-commutative 96.3%

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

   6. associate-*l* 96.3%

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

   7. associate-/l* 96.5%

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

   8. *-commutative 96.5%

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

   9. associate-*r* 96.5%

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

   10. swap-sqr 96.6%

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

   11. pow2 96.6%

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

   12. *-commutative 96.6%

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

5. Applied egg-rr 96.6%

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

   1. add-sqr-sqrt 96.1%

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

   2. pow2 96.1%

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

   3. associate-*r* 96.3%

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

   4. *-commutative 96.3%

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

   5. associate-*l* 96.1%

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

7. Applied egg-rr 96.1%

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

8. Taylor expanded in x around 0 68.4%

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

   1. associate-/r* 68.5%

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

10. Simplified 68.5%

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

    1. expm1-log1p-u 68.6%

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

    2. expm1-udef 68.5%

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

    3. associate-/r/ 68.6%

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

    4. unpow2 68.6%

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

    5. add-sqr-sqrt 68.7%

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

    6. *-commutative 68.7%

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

    7. *-commutative 68.7%

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

    8. associate-*r* 68.7%

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

    9. associate-/l/ 68.8%

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

12. Applied egg-rr 68.8%

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

    1. expm1-def 68.8%

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

    2. expm1-log1p 68.8%

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

    3. associate-*l/ 68.7%

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

    4. associate-/l* 68.6%

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

    5. associate-/l/ 68.5%

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

    6. associate-/l* 68.6%

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

    7. associate-*l/ 68.5%

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

    8. *-commutative 68.5%

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

    9. associate-*l* 68.7%

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

    10. *-commutative 68.7%

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

    11. associate-/r/ 68.8%

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

    12. /-rgt-identity 68.8%

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

    13. *-commutative 68.8%

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

14. Simplified 68.8%

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

15. Final simplification 68.8%

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

Alternative 15: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. add-sqr-sqrt 96.7%

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

   2. sqrt-unprod 97.4%

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

   3. swap-sqr 97.1%

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

   4. associate-*r* 97.2%

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

   5. expm1-log1p-u 97.1%

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

   6. associate-*r* 97.1%

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

   7. swap-sqr 97.4%

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

   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 97.4%

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

   10. *-commutative 97.4%

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

3. Applied egg-rr 97.4%

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

4. Taylor expanded in x around 0 68.8%

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

5. Final simplification 68.8%

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

Alternative 16: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 62.1%

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

   1. expm1-log1p-u 62.1%

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

   2. expm1-udef 53.0%

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

   3. log1p-udef 53.0%

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

   4. add-exp-log 53.0%

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

   5. +-commutative 53.0%

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

   6. *-commutative 53.0%

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

6. Applied egg-rr 53.0%

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

7. Taylor expanded in x around 0 62.4%

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

   1. associate-*r* 62.4%

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

   2. unpow2 62.4%

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

9. Simplified 62.4%

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

    1. +-commutative 62.4%

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

    2. *-commutative 62.4%

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

    3. fma-def 62.4%

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

11. Applied egg-rr 62.4%

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

12. Final simplification 62.4%

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

Alternative 17: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 62.1%

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

   1. expm1-log1p-u 62.1%

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

   2. expm1-udef 53.0%

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

   3. log1p-udef 53.0%

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

   4. add-exp-log 53.0%

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

   5. +-commutative 53.0%

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

   6. *-commutative 53.0%

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

6. Applied egg-rr 53.0%

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

7. Taylor expanded in x around 0 62.4%

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

   1. +-commutative 62.4%

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

   2. *-commutative 62.4%

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

   3. fma-def 62.4%

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

   4. unpow2 62.4%

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

   5. unpow2 62.4%

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

   6. swap-sqr 62.4%

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

   7. unpow2 62.4%

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

   8. *-commutative 62.4%

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

9. Simplified 62.4%

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

10. Final simplification 62.4%

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

Alternative 18: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 62.1%

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

   1. expm1-log1p-u 62.1%

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

   2. expm1-udef 53.0%

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

   3. log1p-udef 53.0%

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

   4. add-exp-log 53.0%

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

   5. +-commutative 53.0%

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

   6. *-commutative 53.0%

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

6. Applied egg-rr 53.0%

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

7. Taylor expanded in x around 0 62.4%

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

   1. associate-*r* 62.4%

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

   2. unpow2 62.4%

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

9. Simplified 62.4%

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

10. Final simplification 62.4%

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

Alternative 19: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in tau around 0 62.1%

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

   1. expm1-log1p-u 62.1%

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

   2. expm1-udef 53.0%

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

   3. log1p-udef 53.0%

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

   4. add-exp-log 53.0%

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

   5. +-commutative 53.0%

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

   6. *-commutative 53.0%

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

6. Applied egg-rr 53.0%

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

7. Taylor expanded in x around 0 62.4%

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

   1. associate-*r* 62.4%

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

   2. unpow2 62.4%

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

9. Simplified 62.4%

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

10. Taylor expanded in x around 0 62.4%

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

    1. *-commutative 62.4%

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

    2. unpow2 62.4%

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

    3. unpow2 62.4%

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

    4. swap-sqr 62.4%

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

    5. unpow2 62.4%

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

12. Simplified 62.4%

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

13. Final simplification 62.4%

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

Alternative 20: 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 97.4%

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

   1. *-commutative 97.4%

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

   2. times-frac 97.4%

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

   3. associate-*r/ 97.2%

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

   4. associate-*r* 96.9%

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

   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/ 96.9%

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

   7. associate-*l* 96.6%

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

   8. swap-sqr 96.4%

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

   9. associate-*r* 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in x around 0 61.3%

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

5. Final simplification 61.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023238 
(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))))