Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 15.8s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.4%

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

   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/ 97.4%

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

   7. associate-*l* 97.1%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

   1. pow2 97.2%

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

   2. pow-to-exp 95.5%

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

5. Applied egg-rr 95.5%

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

6. Taylor expanded in x around inf 96.8%

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

   1. *-commutative 96.8%

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

   2. unpow2 96.8%

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

   3. unpow2 96.8%

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

   4. swap-sqr 97.4%

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

   5. unpow2 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.4%

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

   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/ 97.4%

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

   7. associate-*l* 97.1%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-/r* 97.2%

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

   3. associate-*l/ 97.2%

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

   4. associate-*r* 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*l* 97.2%

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

   7. *-commutative 97.2%

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

   8. associate-*r* 97.3%

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

   9. swap-sqr 97.1%

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

   10. pow2 97.1%

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

   11. *-commutative 97.1%

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

5. Applied egg-rr 97.1%

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

   1. expm1-log1p-u 97.0%

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

   2. expm1-udef 96.7%

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

   3. div-inv 96.5%

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

   4. *-commutative 96.5%

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

   5. pow-flip 96.5%

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

   6. metadata-eval 96.5%

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

7. Applied egg-rr 96.5%

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

   1. expm1-def 96.9%

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

   2. expm1-log1p 97.1%

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

   3. associate-*l* 97.2%

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

   4. associate-*r* 97.6%

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

   5. *-commutative 97.6%

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

   6. *-commutative 97.6%

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

   7. *-commutative 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

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

Alternative 4: 85.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in x around 0 87.3%

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

   1. *-commutative 87.3%

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

   2. unpow2 87.3%

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

6. Simplified 87.3%

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

7. Final simplification 87.3%

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

Alternative 5: 85.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0 87.3%

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

   1. *-commutative 87.3%

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

   2. unpow2 87.3%

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

4. Simplified 87.3%

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

5. Final simplification 87.3%

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

Alternative 6: 84.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

   1. expm1-log1p-u 86.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{\frac{tau}{x}}, \frac{1}{\pi \cdot \left(tau \cdot x\right)}\right)\right)\right)} \]

   2. expm1-udef 86.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{\frac{tau}{x}}, \frac{1}{\pi \cdot \left(tau \cdot x\right)}\right)\right)} - 1} \]

8. Applied egg-rr 86.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau} \cdot x, \frac{1}{\pi \cdot \left(x \cdot tau\right)}\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 86.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau} \cdot x, \frac{1}{\pi \cdot \left(x \cdot tau\right)}\right)\right)\right)} \]

   2. expm1-log1p 87.1%

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

   3. associate-*r* 86.5%

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

   4. *-commutative 86.5%

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

   5. *-commutative 86.5%

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

   6. associate-*r* 87.1%

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

   7. *-commutative 87.1%

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

10. Simplified 87.1%

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

11. Final simplification 87.1%

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

Alternative 7: 84.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

   1. expm1-log1p-u 86.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{\frac{tau}{x}}, \frac{1}{\pi \cdot \left(tau \cdot x\right)}\right)\right)\right)} \]

   2. expm1-udef 86.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{\frac{tau}{x}}, \frac{1}{\pi \cdot \left(tau \cdot x\right)}\right)\right)} - 1} \]

8. Applied egg-rr 86.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau} \cdot x, \frac{1}{\pi \cdot \left(x \cdot tau\right)}\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 86.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau} \cdot x, \frac{1}{\pi \cdot \left(x \cdot tau\right)}\right)\right)\right)} \]

   2. expm1-log1p 87.1%

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

   3. *-commutative 87.1%

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

   4. associate-*l/ 87.1%

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

   5. *-commutative 87.1%

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

   6. associate-/l* 87.1%

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

   7. associate-/r/ 87.1%

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

   8. *-commutative 87.1%

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

   9. associate-/r* 86.7%

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

   10. *-commutative 86.7%

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

10. Simplified 86.7%

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

11. Taylor expanded in tau around inf 86.7%

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

    1. associate-/l* 86.8%

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

    2. associate-*r* 86.5%

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

    3. *-commutative 86.5%

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

    4. *-commutative 86.5%

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

    5. associate-/l/ 86.6%

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

    6. associate-*r* 86.6%

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

13. Simplified 86.6%

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

14. Final simplification 86.6%

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

Alternative 8: 84.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

7. Taylor expanded in tau around inf 86.7%

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

   1. *-commutative 86.7%

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

9. Simplified 86.7%

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

10. Final simplification 86.7%

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

Alternative 9: 84.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

7. Taylor expanded in tau around -inf 86.7%

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

   1. mul-1-neg 86.7%

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

   2. associate-/l* 86.8%

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

   3. *-commutative 86.8%

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

   4. *-commutative 86.8%

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

   5. *-commutative 86.8%

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

9. Simplified 86.8%

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

    1. associate-/r/ 86.9%

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

    2. *-commutative 86.9%

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

    3. associate-*l* 86.9%

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

11. Applied egg-rr 86.9%

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

12. Final simplification 86.9%

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

Alternative 10: 70.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 73.9%

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

5. Final simplification 73.9%

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

Alternative 11: 70.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 73.9%

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

   1. *-commutative 73.9%

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

   2. associate-*r* 74.0%

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

   3. *-commutative 74.0%

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

6. Simplified 74.0%

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

7. Final simplification 74.0%

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

Alternative 12: 78.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

7. Taylor expanded in x around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. unpow2 82.4%

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

   3. distribute-lft-out 82.4%

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

   4. unpow2 82.4%

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

9. Simplified 82.4%

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

10. Taylor expanded in x around 0 82.4%

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

    1. *-commutative 82.4%

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

    2. distribute-lft1-in 82.4%

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

    3. unpow2 82.4%

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

    4. fma-udef 82.4%

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

    5. *-commutative 82.4%

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

    6. associate-*r* 82.4%

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

    7. *-commutative 82.4%

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

    8. unpow2 82.4%

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

    9. unpow2 82.4%

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

    10. swap-sqr 82.4%

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

    11. unpow2 82.4%

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

    12. *-commutative 82.4%

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

12. Simplified 82.4%

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

13. Final simplification 82.4%

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

Alternative 13: 70.8% accurate, 2.0× speedup

Math

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

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(x * Float32(Float32(pi) * tau))
	return Float32(sin(t_1) / t_1)
end

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*l* 97.4%

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

   2. associate-*l* 98.1%

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

3. Simplified 98.1%

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

   1. expm1-log1p-u 97.8%

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

   2. expm1-udef 97.8%

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

   3. *-commutative 97.8%

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

   4. *-commutative 97.8%

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

5. Applied egg-rr 97.8%

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

6. Taylor expanded in x around 0 74.1%

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

7. Final simplification 74.1%

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

Alternative 14: 69.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

7. Taylor expanded in x around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. unpow2 82.4%

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

   3. distribute-lft-out 82.4%

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

   4. unpow2 82.4%

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

9. Simplified 82.4%

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

10. Taylor expanded in tau around inf 73.3%

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

    1. *-commutative 73.3%

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

    2. unpow2 73.3%

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

12. Simplified 73.3%

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

13. Final simplification 73.3%

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

Alternative 15: 69.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

7. Taylor expanded in x around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. unpow2 82.4%

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

   3. distribute-lft-out 82.4%

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

   4. unpow2 82.4%

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

9. Simplified 82.4%

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

10. Taylor expanded in tau around inf 73.3%

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

    1. associate-*r* 73.3%

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

    2. *-commutative 73.3%

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

    3. *-commutative 73.3%

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

    4. unpow2 73.3%

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

    5. unpow2 73.3%

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

    6. swap-sqr 73.3%

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

    7. unpow2 73.3%

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

    8. *-commutative 73.3%

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

    9. unpow2 73.3%

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

12. Simplified 73.3%

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

13. Final simplification 73.3%

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

Alternative 16: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r/ 97.9%

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

   2. associate-*l/ 97.9%

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

   3. associate-/l/ 97.9%

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

   4. associate-*r/ 97.9%

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

   5. associate-*l* 97.5%

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

   6. associate-*r* 97.2%

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

   7. associate-/r* 97.2%

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

   8. associate-/l/ 97.2%

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

   9. swap-sqr 97.0%

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

   10. associate-*r* 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 86.6%

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

   1. fma-def 86.6%

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

   2. associate-/l* 86.6%

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

   3. associate-*r* 86.7%

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

   4. *-commutative 86.7%

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

   5. *-commutative 86.7%

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

   6. *-commutative 86.7%

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

6. Simplified 86.7%

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

7. Taylor expanded in x around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. unpow2 82.4%

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

   3. distribute-lft-out 82.4%

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

   4. unpow2 82.4%

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

9. Simplified 82.4%

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

10. Taylor expanded in tau around 0 67.4%

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

11. Final simplification 67.4%

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

Alternative 17: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.4%

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

   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/ 97.4%

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

   7. associate-*l* 97.1%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in tau around 0 67.7%

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

   1. *-commutative 67.7%

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

6. Simplified 67.7%

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

7. Taylor expanded in x around 0 67.4%

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

   1. *-commutative 67.4%

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

   2. unpow2 67.4%

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

   3. unpow2 67.4%

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

   4. swap-sqr 67.4%

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

   5. unpow2 67.4%

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

9. Simplified 67.4%

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

10. Final simplification 67.4%

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

Alternative 18: 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 98.0%

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

   1. *-commutative 98.0%

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

   2. times-frac 97.9%

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

   3. associate-*r/ 97.8%

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

   4. associate-*r* 97.4%

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

   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/ 97.4%

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

   7. associate-*l* 97.1%

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

   8. swap-sqr 97.0%

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

   9. associate-*r* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in x around 0 66.9%

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

5. Final simplification 66.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023258 
(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))))