Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 14.6s

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 97.8%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 97.7

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

4. Applied egg-rr 97.7%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. frac-times N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. frac-2neg N/A

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

6. Applied egg-rr 97.3%

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

7. Final simplification 97.3%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

4. Applied egg-rr 97.2%

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

5. Final simplification 97.2%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 97.7

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

4. Applied egg-rr 97.7%

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

   1. associate-*r* N/A

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

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

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

6. Applied egg-rr 97.1%

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

7. Final simplification 97.1%

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

Alternative 5: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. div-inv N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

4. Applied egg-rr 97.0%

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

5. Final simplification 97.0%

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

Alternative 6: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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 96.8%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f32 97.0

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

7. Applied egg-rr 97.0%

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

8. Final simplification 97.0%

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

Alternative 7: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 97.7

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}} \]
6. 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. *-lowering-*.f32 N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

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

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

7. Simplified 96.9%

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

8. Final simplification 96.9%

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

Alternative 8: 91.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (sin (* x (* PI tau)))
  (/
   (fma
    (* x x)
    (fma
     (* x x)
     (/ (* 0.008333333333333333 (* PI (* PI PI))) tau)
     (/ (* 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((x * x), ((0.008333333333333333f * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) / tau), ((((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(x * x), Float32(Float32(Float32(0.008333333333333333) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) / tau), Float32(Float32(Float32(pi) * Float32(-0.16666666666666666)) / tau)), Float32(Float32(1.0) / Float32(Float32(pi) * tau))) / x))
end

Derivation

1. Initial program 97.8%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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 96.8%

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

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 89.0%

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

9. Final simplification 89.0%

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

Alternative 9: 85.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (fma x (* x (* (* PI PI) -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(x, (x * ((((float) M_PI) * ((float) M_PI)) * -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(x, Float32(x * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), Float32(1.0)))
end

Derivation

1. Initial program 97.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 97.7

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 82.6

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

7. Simplified 82.6%

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

8. Final simplification 82.6%

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

Alternative 10: 79.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. 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. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 75.3

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

5. Simplified 75.3%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   9. swap-sqr N/A

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

   10. cancel-sign-sub-inv N/A

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

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

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

   12. metadata-eval N/A

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

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

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

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

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

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

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

   18. *-lowering-*.f32 75.3

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

7. Applied egg-rr 75.3%

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

8. Final simplification 75.3%

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

Alternative 11: 79.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. 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. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 75.3

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

5. Simplified 75.3%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. *-lowering-*.f32 75.3

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

7. Applied egg-rr 75.3%

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

8. Final simplification 75.3%

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

Alternative 12: 70.6% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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 96.8%

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

6. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. PI-lowering-PI.f32 69.4

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

8. Simplified 69.4%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{-1}{6} \cdot \left({tau}^{2} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
10. 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

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

    3. associate-*r* N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

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

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

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

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

    8. unpow2 N/A

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

    9. associate-*r* N/A

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

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

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

    11. unpow2 N/A

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

    12. associate-*l* N/A

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

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

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

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

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

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

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

    16. PI-lowering-PI.f32 67.4

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

11. Simplified 67.4%

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

12. Final simplification 67.4%

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

Alternative 13: 65.3% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. 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. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 75.3

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

5. Simplified 75.3%

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

6. Taylor expanded in tau around 0

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

8. Simplified 62.5%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. swap-sqr N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   16. PI-lowering-PI.f32 62.5

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

10. Applied egg-rr 62.5%

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

Alternative 14: 65.3% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. 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. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 75.3

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

5. Simplified 75.3%

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

6. Taylor expanded in tau around 0

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

8. Simplified 62.5%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. PI-lowering-PI.f32 62.5

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

10. Applied egg-rr 62.5%

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

Alternative 15: 65.3% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. 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. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

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

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

   13. +-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 75.3

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

5. Simplified 75.3%

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

6. Taylor expanded in tau around 0

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

8. Simplified 62.5%

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

Alternative 16: 64.4% 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 97.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 61.7%

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

Reproduce

herbie shell --seed 2024201 
(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))))