Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.2s

Alternatives: 14

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, 0.8× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (exp (log (/ (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) * expf(logf((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) * exp(log(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) * exp(log((sin((x * single(pi))) / (x * single(pi)))));
end

Derivation

1. Initial program 97.8%

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

   1. add-exp-log 97.8%

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

   2. *-commutative 97.8%

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

   3. *-commutative 97.8%

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

3. Applied egg-rr 97.8%

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

4. Final simplification 97.8%

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

Alternative 2: 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 97.8%

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

2. Final simplification 97.8%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. associate-*r/ 96.5%

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

   2. associate-*r* 96.7%

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

   3. associate-/r* 96.6%

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

   4. *-commutative 96.6%

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

   5. *-commutative 96.6%

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

   6. associate-*l* 96.9%

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

   7. *-commutative 96.9%

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

   8. *-commutative 96.9%

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

   9. pow2 96.9%

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

5. Applied egg-rr 96.9%

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

   1. associate-/l/ 96.9%

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

   2. times-frac 96.8%

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

   3. *-commutative 96.8%

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

   4. *-commutative 96.8%

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

   5. *-commutative 96.8%

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

7. Applied egg-rr 96.8%

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

8. Taylor expanded in x around inf 96.6%

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

   1. times-frac 96.7%

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

   2. *-commutative 96.7%

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

   3. associate-*r* 96.7%

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

   4. *-commutative 96.7%

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

   5. *-commutative 96.7%

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

   6. *-commutative 96.7%

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

   7. unpow2 96.7%

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

   8. unpow2 96.7%

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

   9. swap-sqr 97.0%

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

   10. unpow2 97.0%

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

10. Simplified 97.1%

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

11. Final simplification 97.1%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(sin(Float32(x * Float32(pi))) * Float32(Float32(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 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around inf 96.6%

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

   1. associate-/r* 96.8%

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

   2. associate-*r* 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*l/ 96.7%

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

   5. unpow2 96.7%

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

   6. unpow2 96.7%

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

   7. swap-sqr 97.0%

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

   8. unpow2 97.0%

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

   9. associate-/l* 97.0%

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

   10. associate-/r/ 97.1%

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

6. Simplified 97.5%

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

   1. div-inv 97.4%

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

   2. *-commutative 97.4%

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

   3. associate-*r* 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-*r* 97.0%

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

   6. pow-flip 97.0%

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

   7. metadata-eval 97.0%

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

8. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*r* 97.5%

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

   3. *-commutative 97.5%

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

   4. *-commutative 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around inf 96.6%

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

   1. associate-/r* 96.8%

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

   2. associate-*r* 96.7%

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

   3. *-commutative 96.7%

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

   4. associate-*l/ 96.7%

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

   5. unpow2 96.7%

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

   6. unpow2 96.7%

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

   7. swap-sqr 97.0%

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

   8. unpow2 97.0%

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

   9. associate-/l* 97.0%

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

   10. associate-/r/ 97.1%

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

6. Simplified 97.5%

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

7. Taylor expanded in tau around inf 96.7%

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

   1. *-commutative 96.7%

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

   2. unpow2 96.7%

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

   3. *-commutative 96.7%

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

   4. unpow2 96.7%

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

   5. swap-sqr 97.6%

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

   6. unpow2 97.6%

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

   7. *-commutative 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

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

Alternative 6: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.5%

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

3. Simplified 97.5%

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

4. Taylor expanded in x around 0 83.7%

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

   1. *-commutative 83.7%

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

   2. unpow2 83.7%

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

6. Simplified 83.7%

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

7. Final simplification 83.7%

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

Alternative 7: 85.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in x around 0 83.7%

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

   1. *-commutative 83.7%

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

   2. unpow2 83.7%

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

4. Simplified 83.7%

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

5. Final simplification 83.7%

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

Alternative 8: 84.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-*l/ 96.5%

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

   3. associate-/l* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. associate-/l* 96.8%

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

   7. *-commutative 96.8%

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

   8. associate-*r* 96.7%

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

   9. swap-sqr 97.0%

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

   10. pow2 97.0%

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

   11. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

6. Taylor expanded in x around 0 83.0%

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

   1. +-commutative 83.0%

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

   2. *-commutative 83.0%

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

   3. fma-def 83.0%

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

   4. *-commutative 83.0%

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

   5. associate-/r* 83.1%

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

8. Simplified 83.1%

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

9. Final simplification 83.1%

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

Alternative 9: 84.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

   1. *-commutative 96.8%

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

   2. associate-*l/ 96.5%

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

   3. associate-/l* 96.7%

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

   4. *-commutative 96.7%

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

   5. associate-*l* 96.7%

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

   6. associate-/l* 96.8%

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

   7. *-commutative 96.8%

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

   8. associate-*r* 96.7%

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

   9. swap-sqr 97.0%

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

   10. pow2 97.0%

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

   11. *-commutative 97.0%

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

5. Applied egg-rr 97.0%

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

6. Taylor expanded in x around 0 83.0%

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

7. Final simplification 83.0%

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

Alternative 10: 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 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. fma-def 77.6%

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

   3. distribute-lft-out 77.6%

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

   4. distribute-rgt1-in 77.6%

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

   5. unpow2 77.6%

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

   6. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Final simplification 77.6%

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

Alternative 11: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + \sqrt[3]{{\left(x \cdot \pi\right)}^{6} \cdot -0.004629629629629629} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ 1.0 (cbrt (* (pow (* x PI) 6.0) -0.004629629629629629))))

C

float code(float x, float tau) {
	return 1.0f + cbrtf((powf((x * ((float) M_PI)), 6.0f) * -0.004629629629629629f));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + cbrt(Float32((Float32(x * Float32(pi)) ^ Float32(6.0)) * Float32(-0.004629629629629629))))
end

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 65.0%

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

   1. *-commutative 65.0%

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

6. Simplified 65.0%

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

7. Taylor expanded in x around 0 65.4%

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

   1. +-commutative 65.4%

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

   2. *-commutative 65.4%

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

   3. unpow2 65.4%

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

   4. unpow2 65.4%

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

   5. swap-sqr 65.4%

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

   6. unpow2 65.4%

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

   7. *-commutative 65.4%

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

9. Simplified 65.4%

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

    1. add-cbrt-cube 65.4%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)}} + 1 \]

    2. pow1/3 -0.0%

       \[\leadsto \color{blue}{{\left(\left(\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)\right) \cdot \left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)\right)}^{0.3333333333333333}} + 1 \]

    3. pow3 -0.0%

       \[\leadsto {\color{blue}{\left({\left(-0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right)}^{3}\right)}}^{0.3333333333333333} + 1 \]

    4. *-commutative -0.0%

       \[\leadsto {\left({\color{blue}{\left({\left(x \cdot \pi\right)}^{2} \cdot -0.16666666666666666\right)}}^{3}\right)}^{0.3333333333333333} + 1 \]

    5. *-commutative -0.0%

       \[\leadsto {\left({\left({\color{blue}{\left(\pi \cdot x\right)}}^{2} \cdot -0.16666666666666666\right)}^{3}\right)}^{0.3333333333333333} + 1 \]

    6. unpow-prod-down -0.0%

       \[\leadsto {\color{blue}{\left({\left({\left(\pi \cdot x\right)}^{2}\right)}^{3} \cdot {-0.16666666666666666}^{3}\right)}}^{0.3333333333333333} + 1 \]

    7. metadata-eval -0.0%

       \[\leadsto {\left({\left({\left(\pi \cdot x\right)}^{2}\right)}^{3} \cdot \color{blue}{-0.004629629629629629}\right)}^{0.3333333333333333} + 1 \]

11. Applied egg-rr -0.0%

    \[\leadsto \color{blue}{{\left({\left({\left(\pi \cdot x\right)}^{2}\right)}^{3} \cdot -0.004629629629629629\right)}^{0.3333333333333333}} + 1 \]
12. Step-by-step derivation

    1. unpow1/3 65.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\pi \cdot x\right)}^{2}\right)}^{3} \cdot -0.004629629629629629}} + 1 \]

    2. unpow2 65.4%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\pi \cdot x\right) \cdot \left(\pi \cdot x\right)\right)}}^{3} \cdot -0.004629629629629629} + 1 \]

    3. cube-prod 65.4%

       \[\leadsto \sqrt[3]{\color{blue}{\left({\left(\pi \cdot x\right)}^{3} \cdot {\left(\pi \cdot x\right)}^{3}\right)} \cdot -0.004629629629629629} + 1 \]

    4. pow-sqr 65.4%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\pi \cdot x\right)}^{\left(2 \cdot 3\right)}} \cdot -0.004629629629629629} + 1 \]

    5. *-commutative 65.4%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(x \cdot \pi\right)}}^{\left(2 \cdot 3\right)} \cdot -0.004629629629629629} + 1 \]

    6. metadata-eval 65.4%

       \[\leadsto \sqrt[3]{{\left(x \cdot \pi\right)}^{\color{blue}{6}} \cdot -0.004629629629629629} + 1 \]

13. Simplified 65.4%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot \pi\right)}^{6} \cdot -0.004629629629629629}} + 1 \]

14. Final simplification 65.4%

    \[\leadsto 1 + \sqrt[3]{{\left(x \cdot \pi\right)}^{6} \cdot -0.004629629629629629} \]

Alternative 12: 64.2% 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 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 65.0%

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

   1. *-commutative 65.0%

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

6. Simplified 65.0%

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

7. Taylor expanded in x around 0 65.4%

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

   1. +-commutative 65.4%

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

   2. *-commutative 65.4%

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

   3. unpow2 65.4%

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

   4. unpow2 65.4%

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

   5. swap-sqr 65.4%

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

   6. unpow2 65.4%

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

   7. fma-def 65.4%

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

   8. *-commutative 65.4%

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

9. Simplified 65.4%

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

10. Final simplification 65.4%

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

Alternative 13: 64.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in tau around 0 65.0%

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

   1. *-commutative 65.0%

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

6. Simplified 65.0%

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

7. Taylor expanded in x around 0 65.4%

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

   1. +-commutative 65.4%

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

   2. *-commutative 65.4%

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

   3. unpow2 65.4%

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

   4. unpow2 65.4%

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

   5. swap-sqr 65.4%

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

   6. unpow2 65.4%

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

   7. *-commutative 65.4%

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

9. Simplified 65.4%

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

10. Final simplification 65.4%

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

Alternative 14: 63.2% accurate, 615.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. times-frac 97.8%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.6%

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

   5. associate-/r* 97.5%

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

   6. associate-/l/ 97.6%

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

   7. associate-*l* 97.2%

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

   8. swap-sqr 96.8%

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

   9. associate-*r* 96.8%

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

3. Simplified 96.8%

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

4. Taylor expanded in x around 0 64.4%

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

5. Final simplification 64.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023257 
(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))))