Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 14.8s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin \left(x \cdot \pi\right) \cdot \sin t\_1}{\left(x \cdot \pi\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

6. Applied egg-rr 97.2%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. PI-lowering-PI.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 97.9

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

8. Applied egg-rr 97.9%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. div-inv N/A

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

   7. frac-2neg N/A

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

   8. associate-*r/ N/A

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

   9. /-lowering-/.f32 N/A

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

10. Applied egg-rr 98.1%

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

11. Final simplification 98.1%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \sin \left(x \cdot \pi\right) \cdot \frac{\sin t\_1}{x \cdot \left(\pi \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

6. Applied egg-rr 97.2%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. PI-lowering-PI.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 97.9

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

8. Applied egg-rr 97.9%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. sin-lowering-sin.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. /-lowering-/.f32 N/A

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

10. Applied egg-rr 97.8%

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

11. Final simplification 97.8%

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

Alternative 4: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{\pi \cdot \left(\pi \cdot \pi\right)}{tau}, \left(x \cdot x\right) \cdot 0.008333333333333333, \frac{\pi \cdot -0.16666666666666666}{tau}\right), \frac{1}{\pi \cdot tau}\right)}{x} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (sin (* x (* PI tau)))
  (/
   (fma
    (* x x)
    (fma
     (/ (* PI (* PI PI)) tau)
     (* (* x x) 0.008333333333333333)
     (/ (* PI -0.16666666666666666) tau))
    (/ 1.0 (* PI tau)))
   x)))

C

float code(float x, float tau) {
	return sinf((x * (((float) M_PI) * tau))) * (fmaf((x * x), fmaf(((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / tau), ((x * x) * 0.008333333333333333f), ((((float) M_PI) * -0.16666666666666666f) / tau)), (1.0f / (((float) M_PI) * tau))) / x);
}

Julia

function code(x, tau)
	return Float32(sin(Float32(x * Float32(Float32(pi) * tau))) * Float32(fma(Float32(x * x), fma(Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / tau), Float32(Float32(x * x) * Float32(0.008333333333333333)), Float32(Float32(Float32(pi) * Float32(-0.16666666666666666)) / tau)), Float32(Float32(1.0) / Float32(Float32(pi) * tau))) / x))
end

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. sin-lowering-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. associate-/r* N/A

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

   14. associate-/l/ N/A

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

   15. /-lowering-/.f32 N/A

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

5. Simplified 97.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]

8. Simplified 93.6%

   \[\leadsto \sin \left(x \cdot \left(tau \cdot \pi\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{\pi \cdot \left(\pi \cdot \pi\right)}{tau}, \left(x \cdot x\right) \cdot 0.008333333333333333, \frac{-0.16666666666666666 \cdot \pi}{tau}\right), \frac{1}{tau \cdot \pi}\right)}{x}} \]

9. Final simplification 93.6%

   \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{\pi \cdot \left(\pi \cdot \pi\right)}{tau}, \left(x \cdot x\right) \cdot 0.008333333333333333, \frac{\pi \cdot -0.16666666666666666}{tau}\right), \frac{1}{\pi \cdot tau}\right)}{x} \]
10. Add Preprocessing

Alternative 5: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau}, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{tau} \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)\right), \frac{1}{\pi \cdot tau}\right)}{x} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (sin (* x (* PI tau)))
  (/
   (fma
    (* x x)
    (fma
     -0.16666666666666666
     (/ PI tau)
     (* (/ (* PI (* PI PI)) tau) (* (* x x) 0.008333333333333333)))
    (/ 1.0 (* PI tau)))
   x)))

C

float code(float x, float tau) {
	return sinf((x * (((float) M_PI) * tau))) * (fmaf((x * x), fmaf(-0.16666666666666666f, (((float) M_PI) / tau), (((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / tau) * ((x * x) * 0.008333333333333333f))), (1.0f / (((float) M_PI) * tau))) / x);
}

Julia

function code(x, tau)
	return Float32(sin(Float32(x * Float32(Float32(pi) * tau))) * Float32(fma(Float32(x * x), fma(Float32(-0.16666666666666666), Float32(Float32(pi) / tau), Float32(Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / tau) * Float32(Float32(x * x) * Float32(0.008333333333333333)))), Float32(Float32(1.0) / Float32(Float32(pi) * tau))) / x))
end

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. sin-lowering-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. associate-/r* N/A

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

   14. associate-/l/ N/A

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

   15. /-lowering-/.f32 N/A

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

5. Simplified 97.2%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. PI-lowering-PI.f32 97.6

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

7. Applied egg-rr 97.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\mathsf{PI}\left(\right)}{tau} + \frac{1}{120} \cdot \frac{{x}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}{tau}\right) + \frac{1}{tau \cdot \mathsf{PI}\left(\right)}}{x}} \]

10. Simplified 93.6%

    \[\leadsto \sin \left(x \cdot \left(tau \cdot \pi\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau}, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{tau} \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)\right), \frac{1}{tau \cdot \pi}\right)}{x}} \]

11. Final simplification 93.6%

    \[\leadsto \sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.16666666666666666, \frac{\pi}{tau}, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{tau} \cdot \left(\left(x \cdot x\right) \cdot 0.008333333333333333\right)\right), \frac{1}{\pi \cdot tau}\right)}{x} \]
12. Add Preprocessing

Alternative 6: 85.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left(\pi \cdot \pi, \left(x \cdot x\right) \cdot -0.16666666666666666, 1\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {x}^{2}\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \left(1 + \color{blue}{\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}}\right) \]

   3. +-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2} + 1\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{6}\right)} \cdot {x}^{2} + 1\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} + 1\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, \frac{-1}{6} \cdot {x}^{2}, 1\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{6} \cdot {x}^{2}, 1\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{6} \cdot {x}^{2}, 1\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{-1}{6} \cdot {x}^{2}, 1\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \frac{-1}{6} \cdot {x}^{2}, 1\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, 1\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, 1\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, 1\right) \]

   14. *-lowering-*.f32 88.3

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

5. Simplified 88.3%

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

Alternative 7: 84.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(tau \cdot tau, 0.008333333333333333, 0.027777777777777776\right), 0.008333333333333333\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (fma
  (* x x)
  (fma
   (* x x)
   (*
    (* PI (* PI (* PI PI)))
    (fma
     (* tau tau)
     (fma (* tau tau) 0.008333333333333333 0.027777777777777776)
     0.008333333333333333))
   (* (* PI PI) (fma -0.16666666666666666 (* tau tau) -0.16666666666666666)))
  1.0))

C

float code(float x, float tau) {
	return fmaf((x * x), fmaf((x * x), ((((float) M_PI) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * fmaf((tau * tau), fmaf((tau * tau), 0.008333333333333333f, 0.027777777777777776f), 0.008333333333333333f)), ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(x * x), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * fma(Float32(tau * tau), fma(Float32(tau * tau), Float32(0.008333333333333333), Float32(0.027777777777777776)), Float32(0.008333333333333333))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

7. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\mathsf{PI}\left(\right)}^{\color{blue}{\left(3 + 1\right)}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   3. pow3 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   11. PI-lowering-PI.f32 88.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \color{blue}{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

9. Applied egg-rr 88.1%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

10. Taylor expanded in tau around 0

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\frac{1}{120} + {tau}^{2} \cdot \left(\frac{1}{36} + \frac{1}{120} \cdot {tau}^{2}\right)\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({tau}^{2} \cdot \left(\frac{1}{36} + \frac{1}{120} \cdot {tau}^{2}\right) + \frac{1}{120}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    2. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{fma}\left({tau}^{2}, \frac{1}{36} + \frac{1}{120} \cdot {tau}^{2}, \frac{1}{120}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\color{blue}{tau \cdot tau}, \frac{1}{36} + \frac{1}{120} \cdot {tau}^{2}, \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\color{blue}{tau \cdot tau}, \frac{1}{36} + \frac{1}{120} \cdot {tau}^{2}, \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, \color{blue}{\frac{1}{120} \cdot {tau}^{2} + \frac{1}{36}}, \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, \color{blue}{{tau}^{2} \cdot \frac{1}{120}} + \frac{1}{36}, \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    7. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, \color{blue}{\mathsf{fma}\left({tau}^{2}, \frac{1}{120}, \frac{1}{36}\right)}, \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(\color{blue}{tau \cdot tau}, \frac{1}{120}, \frac{1}{36}\right), \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    9. *-lowering-*.f32 88.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(\color{blue}{tau \cdot tau}, 0.008333333333333333, 0.027777777777777776\right), 0.008333333333333333\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

12. Simplified 88.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(tau \cdot tau, 0.008333333333333333, 0.027777777777777776\right), 0.008333333333333333\right)}, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

13. Final simplification 88.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, \mathsf{fma}\left(tau \cdot tau, 0.008333333333333333, 0.027777777777777776\right), 0.008333333333333333\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]
14. Add Preprocessing

Alternative 8: 80.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(tau \cdot \left(tau \cdot \left(0.008333333333333333 \cdot \left(tau \cdot tau\right)\right)\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (fma
  (* x x)
  (fma
   (* x x)
   (*
    (* PI (* PI (* PI PI)))
    (* tau (* tau (* 0.008333333333333333 (* tau tau)))))
   (* (* PI PI) (fma -0.16666666666666666 (* tau tau) -0.16666666666666666)))
  1.0))

C

float code(float x, float tau) {
	return fmaf((x * x), fmaf((x * x), ((((float) M_PI) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (tau * (tau * (0.008333333333333333f * (tau * tau))))), ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(x * x), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(tau * Float32(tau * Float32(Float32(0.008333333333333333) * Float32(tau * tau))))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

7. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\mathsf{PI}\left(\right)}^{\color{blue}{\left(3 + 1\right)}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   3. pow3 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   11. PI-lowering-PI.f32 88.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \color{blue}{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

9. Applied egg-rr 88.1%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

10. Taylor expanded in tau around inf

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {tau}^{4}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{120} \cdot {tau}^{\color{blue}{\left(2 \cdot 2\right)}}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({tau}^{2} \cdot {tau}^{2}\right)}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot {tau}^{2}\right) \cdot {tau}^{2}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({tau}^{2} \cdot \left(\frac{1}{120} \cdot {tau}^{2}\right)\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(tau \cdot tau\right)} \cdot \left(\frac{1}{120} \cdot {tau}^{2}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(tau \cdot \left(tau \cdot \left(\frac{1}{120} \cdot {tau}^{2}\right)\right)\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(tau \cdot \left(tau \cdot \left(\frac{1}{120} \cdot {tau}^{2}\right)\right)\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(tau \cdot \color{blue}{\left(tau \cdot \left(\frac{1}{120} \cdot {tau}^{2}\right)\right)}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(tau \cdot \left(tau \cdot \color{blue}{\left({tau}^{2} \cdot \frac{1}{120}\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(tau \cdot \left(tau \cdot \color{blue}{\left({tau}^{2} \cdot \frac{1}{120}\right)}\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(tau \cdot \left(tau \cdot \left(\color{blue}{\left(tau \cdot tau\right)} \cdot \frac{1}{120}\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    12. *-lowering-*.f32 83.6

        \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(tau \cdot \left(tau \cdot \left(\color{blue}{\left(tau \cdot tau\right)} \cdot 0.008333333333333333\right)\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

12. Simplified 83.6%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \color{blue}{\left(tau \cdot \left(tau \cdot \left(\left(tau \cdot tau\right) \cdot 0.008333333333333333\right)\right)\right)}, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

13. Final simplification 83.6%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(tau \cdot \left(tau \cdot \left(0.008333333333333333 \cdot \left(tau \cdot tau\right)\right)\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]
14. Add Preprocessing

Alternative 9: 79.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (fma
  (* x x)
  (fma
   (* x x)
   (*
    (* PI (* PI (* PI PI)))
    (fma (* tau tau) 0.027777777777777776 0.008333333333333333))
   (* (* PI PI) (fma -0.16666666666666666 (* tau tau) -0.16666666666666666)))
  1.0))

C

float code(float x, float tau) {
	return fmaf((x * x), fmaf((x * x), ((((float) M_PI) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * fmaf((tau * tau), 0.027777777777777776f, 0.008333333333333333f)), ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(x * x), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * fma(Float32(tau * tau), Float32(0.027777777777777776), Float32(0.008333333333333333))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

7. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\mathsf{PI}\left(\right)}^{\color{blue}{\left(3 + 1\right)}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   3. pow3 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   11. PI-lowering-PI.f32 88.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \color{blue}{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

9. Applied egg-rr 88.1%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

10. Taylor expanded in tau around 0

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\frac{1}{120} + \frac{1}{36} \cdot {tau}^{2}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\frac{1}{36} \cdot {tau}^{2} + \frac{1}{120}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{{tau}^{2} \cdot \frac{1}{36}} + \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    3. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{fma}\left({tau}^{2}, \frac{1}{36}, \frac{1}{120}\right)}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\color{blue}{tau \cdot tau}, \frac{1}{36}, \frac{1}{120}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

    5. *-lowering-*.f32 83.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\color{blue}{tau \cdot tau}, 0.027777777777777776, 0.008333333333333333\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

12. Simplified 83.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right)}, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

13. Final simplification 83.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]
14. Add Preprocessing

Alternative 10: 78.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (fma
  (* x x)
  (fma
   (* x x)
   (* 0.008333333333333333 (* PI (* PI (* PI PI))))
   (* (* PI PI) (fma -0.16666666666666666 (* tau tau) -0.16666666666666666)))
  1.0))

C

float code(float x, float tau) {
	return fmaf((x * x), fmaf((x * x), (0.008333333333333333f * (((float) M_PI) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))), ((((float) M_PI) * ((float) M_PI)) * fmaf(-0.16666666666666666f, (tau * tau), -0.16666666666666666f))), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(x * x), Float32(Float32(0.008333333333333333) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(Float32(-0.16666666666666666), Float32(tau * tau), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

7. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\mathsf{PI}\left(\right)}^{\color{blue}{\left(3 + 1\right)}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   2. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   3. pow3 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), \frac{1}{120}, \left(tau \cdot tau\right) \cdot \frac{1}{36}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]

   11. PI-lowering-PI.f32 88.1

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \color{blue}{\pi}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

9. Applied egg-rr 88.1%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, tau \cdot tau, 1\right), 0.008333333333333333, \left(tau \cdot tau\right) \cdot 0.027777777777777776\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

10. Taylor expanded in tau around 0

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{1}{120}}, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6}, tau \cdot tau, \frac{-1}{6}\right)\right), 1\right) \]
11. Step-by-step derivation

12. Simplified 82.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \color{blue}{0.008333333333333333}, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]

13. Final simplification 82.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.008333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right) \]
14. Add Preprocessing

Alternative 11: 78.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1 \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1 \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

7. Simplified 82.0%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right)\right)}, 1\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right)}, 1\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right)}, 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right), 1\right) \]

   5. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right), 1\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right)}, 1\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{6} \cdot \left(tau \cdot tau\right) + \frac{-1}{6}\right)\right), 1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot tau\right) \cdot tau} + \frac{-1}{6}\right)\right), 1\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{tau \cdot \left(\frac{-1}{6} \cdot tau\right)} + \frac{-1}{6}\right)\right), 1\right) \]

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(tau, \frac{-1}{6} \cdot tau, \frac{-1}{6}\right)}\right), 1\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(tau, \color{blue}{tau \cdot \frac{-1}{6}}, \frac{-1}{6}\right)\right), 1\right) \]

   12. *-lowering-*.f32 82.0

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

9. Applied egg-rr 82.0%

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

Alternative 12: 78.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1 \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1 \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

7. Simplified 82.0%

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

Alternative 13: 69.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1 \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1 \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

7. Simplified 82.0%

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

8. Taylor expanded in tau around inf

   \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, 1\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {tau}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left({tau}^{2} \cdot \frac{-1}{6}\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({tau}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left({tau}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, 1\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(tau \cdot tau\right)} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), 1\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(tau \cdot tau\right)} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), 1\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\left(tau \cdot tau\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right), 1\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\left(tau \cdot tau\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right), 1\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\left(tau \cdot tau\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right), 1\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\left(tau \cdot tau\right) \cdot \left(\frac{-1}{6} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 1\right) \]

   11. PI-lowering-PI.f32 71.9

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

10. Simplified 71.9%

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

11. Final simplification 71.9%

    \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\left(tau \cdot tau\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right)\right), 1\right) \]
12. Add Preprocessing

Alternative 14: 69.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 97.7

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around 0

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

7. Simplified 72.1%

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

   1. associate-*l/ N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. *-lowering-*.f32 N/A

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

   8. /-lowering-/.f32 N/A

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

   9. sin-lowering-sin.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. PI-lowering-PI.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

   14. *-lowering-*.f32 N/A

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

   15. PI-lowering-PI.f32 72.1

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

9. Applied egg-rr 72.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 1} \]

    2. accelerator-lowering-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {tau}^{2} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

12. Simplified 71.9%

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

Alternative 15: 64.3% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1 \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1 \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

7. Simplified 82.0%

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

8. Taylor expanded in tau around 0

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot \left(x \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot \left(x \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \color{blue}{\left(x \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

   5. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 1\right) \]

   6. PI-lowering-PI.f32 65.8

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

10. Simplified 65.8%

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

Alternative 16: 63.3% accurate, 258.0× 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 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 64.9%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))