Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 22.1s

Alternatives: 18

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	t_1 = Float32(tau * Float32(x * Float32(pi)))
	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 = tau * (x * single(pi));
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 98.2%

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

2. Final simplification 98.2%

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

Alternative 2: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in x around inf 97.1%

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

   1. *-commutative 97.1%

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

   2. *-commutative 97.1%

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

   3. *-commutative 97.1%

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

   4. associate-*r* 97.4%

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

   5. unpow2 97.4%

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

   6. unpow2 97.4%

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

   7. swap-sqr 97.6%

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

   8. unpow2 97.6%

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

   9. *-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r/ 97.3%

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

   2. *-commutative 97.3%

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

   3. associate-*r* 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-*l* 97.1%

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

   6. *-commutative 97.1%

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

   7. associate-*r* 97.3%

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

   8. swap-sqr 97.3%

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

   9. pow2 97.3%

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

   10. *-commutative 97.3%

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

5. Applied egg-rr 97.3%

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

   1. expm1-log1p-u 97.2%

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

   2. expm1-udef 96.8%

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

7. Applied egg-rr 96.7%

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

   1. expm1-def 97.1%

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

   2. expm1-log1p 97.5%

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

   3. *-commutative 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*r* 97.8%

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

   6. *-commutative 97.8%

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

9. Simplified 97.8%

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

10. Final simplification 97.8%

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

Alternative 4: 84.8% 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 98.2%

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

2. Taylor expanded in x around 0 87.1%

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

   1. *-commutative 87.1%

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

   2. unpow2 87.1%

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

4. Simplified 87.1%

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

5. Final simplification 87.1%

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

Alternative 5: 84.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 86.4%

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

   1. fma-def 86.4%

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

   2. associate-/l* 86.4%

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

   3. *-commutative 86.4%

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

   4. *-commutative 86.4%

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

   5. *-commutative 86.4%

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

   6. associate-*r* 87.0%

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

6. Simplified 87.0%

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

7. Final simplification 87.0%

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

Alternative 6: 84.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return sinf((x * (((float) M_PI) * tau))) * ((-0.16666666666666666f * ((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(Float32(-0.16666666666666666) * Float32(Float32(x * 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(-0.16666666666666666) * ((x * single(pi)) / tau)) + (single(1.0) / (tau * (x * single(pi)))));
end

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 86.4%

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

5. Final simplification 86.4%

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

Alternative 7: 78.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in x around 0 81.1%

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

   1. +-commutative 81.1%

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

   2. *-commutative 81.1%

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

   3. fma-def 81.1%

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

   4. unpow2 81.1%

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

   5. distribute-lft-out 81.1%

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

   6. distribute-rgt1-in 81.1%

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

   7. unpow2 81.1%

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

6. Simplified 81.1%

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

7. Final simplification 81.1%

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

Alternative 8: 78.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in x around 0 81.4%

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

   1. fma-def 81.4%

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

   2. *-commutative 81.4%

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

   3. associate-*l* 81.4%

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

   4. unpow2 81.4%

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

   5. *-commutative 81.4%

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

6. Simplified 81.4%

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

7. Taylor expanded in x around 0 81.1%

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

   1. +-commutative 81.1%

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

   2. *-commutative 81.1%

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

   3. fma-def 81.1%

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

   4. unpow2 81.1%

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

   5. associate-*r* 81.1%

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

   6. distribute-rgt-out 81.1%

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

   7. unpow2 81.1%

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

9. Simplified 81.1%

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

10. Final simplification 81.1%

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

Alternative 9: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = sin(t_1) * (single(1.0) / t_1);
end

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. *-commutative 71.8%

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

   3. associate-*r* 71.9%

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

6. Simplified 71.9%

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

7. Final simplification 71.9%

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

Alternative 10: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = single(1.0) / (t_1 / sin(t_1));
end

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. associate-*r* 71.8%

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

6. Simplified 71.8%

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

   1. associate-*r/ 71.8%

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

   2. clear-num 71.8%

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

   3. *-rgt-identity 71.8%

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

   4. *-commutative 71.8%

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

   5. associate-*r* 71.9%

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

   6. *-commutative 71.9%

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

8. Applied egg-rr 71.9%

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

9. Final simplification 71.9%

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

Alternative 11: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. associate-*r* 71.8%

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

6. Simplified 71.8%

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

7. Taylor expanded in x around -inf 71.9%

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

8. Final simplification 71.9%

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

Alternative 12: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. associate-*r* 71.8%

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

6. Simplified 71.8%

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

7. Taylor expanded in x around inf 71.9%

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

   1. *-commutative 71.9%

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

   2. *-commutative 71.9%

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

   3. associate-*r* 71.7%

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

   4. *-commutative 71.7%

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

   5. associate-*r* 71.9%

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

9. Simplified 71.9%

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

10. Final simplification 71.9%

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

Alternative 13: 69.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. associate-*r* 71.8%

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

6. Simplified 71.8%

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

7. Taylor expanded in x around 0 71.0%

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

   1. +-commutative 71.0%

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

   2. *-commutative 71.0%

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

   3. fma-def 71.0%

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

   4. associate-*r* 71.0%

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

   5. unpow2 71.0%

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

   6. unpow2 71.0%

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

   7. unswap-sqr 71.0%

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

   8. *-commutative 71.0%

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

   9. *-commutative 71.0%

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

   10. *-commutative 71.0%

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

   11. unpow2 71.0%

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

   12. swap-sqr 71.0%

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

   13. unpow2 71.0%

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

9. Simplified 71.0%

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

10. Final simplification 71.0%

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

Alternative 14: 69.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. associate-*r/ 98.1%

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

   2. associate-*l/ 98.0%

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

   3. associate-/l/ 98.0%

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

   4. associate-*r/ 98.1%

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

   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* 97.5%

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

   7. associate-/r* 97.4%

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

   8. associate-/l/ 97.5%

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

   9. swap-sqr 97.3%

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

   10. associate-*r* 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around 0 71.8%

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

   1. *-commutative 71.8%

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

   2. associate-*r* 71.8%

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

6. Simplified 71.8%

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

   1. expm1-log1p-u 71.9%

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

   2. expm1-udef 71.8%

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

   3. un-div-inv 71.8%

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

   4. *-commutative 71.8%

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

   5. associate-*r* 71.9%

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

   6. *-commutative 71.9%

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

8. Applied egg-rr 71.9%

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

9. Taylor expanded in x around 0 71.0%

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

    1. unpow2 71.0%

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

    2. unpow2 71.0%

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

11. Simplified 71.0%

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

12. Final simplification 71.0%

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

Alternative 15: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in tau around 0 65.6%

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

5. Taylor expanded in x around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. unpow2 65.8%

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

   3. unpow2 65.8%

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

   4. swap-sqr 65.8%

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

   5. unpow2 65.8%

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

7. Simplified 65.8%

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

   1. expm1-log1p-u 65.8%

      \[\leadsto 1 + -0.16666666666666666 \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}}^{2} \]

   2. expm1-udef 65.8%

      \[\leadsto 1 + -0.16666666666666666 \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot x\right)} - 1\right)}}^{2} \]

   3. log1p-udef 65.8%

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

   4. add-exp-log 65.8%

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

9. Applied egg-rr 65.8%

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

10. Final simplification 65.8%

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

Alternative 16: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in tau around 0 65.6%

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

5. Taylor expanded in x around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. unpow2 65.8%

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

   3. unpow2 65.8%

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

   4. swap-sqr 65.8%

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

   5. unpow2 65.8%

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

7. Simplified 65.8%

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

   1. unpow-prod-down 65.8%

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

   2. pow2 65.8%

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

9. Applied egg-rr 65.8%

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

10. Final simplification 65.8%

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

Alternative 17: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in tau around 0 65.6%

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

5. Taylor expanded in x around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. unpow2 65.8%

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

   3. unpow2 65.8%

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

   4. swap-sqr 65.8%

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

   5. unpow2 65.8%

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

7. Simplified 65.8%

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

8. Final simplification 65.8%

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

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 98.2%

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

   1. *-commutative 98.2%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*r* 97.7%

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

   5. associate-/r* 97.6%

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

   6. associate-/l/ 97.7%

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

   7. associate-*l* 97.6%

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

   8. swap-sqr 97.4%

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

   9. associate-*r* 97.4%

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

3. Simplified 97.4%

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

4. Taylor expanded in x around 0 64.7%

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

5. Final simplification 64.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(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))))