Lanczos kernel

Percentage Accurate: 97.9% → 97.7%

Time: 16.0s

Alternatives: 22

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

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

3. Final simplification 97.5%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. inv-pow N/A

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

   3. div-inv N/A

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

   4. unpow-prod-down N/A

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

   5. inv-pow N/A

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

   6. inv-pow N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f32 97.4

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

4. Applied egg-rr 97.4%

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

   1. *-commutative N/A

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

   2. remove-double-div N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. frac-times N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

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

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. inv-pow N/A

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

   3. div-inv N/A

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

   4. unpow-prod-down N/A

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

   5. inv-pow N/A

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

   6. inv-pow N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f32 97.4

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

4. Applied egg-rr 97.4%

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

   1. *-commutative N/A

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

   2. remove-double-div N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-*r* N/A

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

   6. frac-times N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

6. Applied egg-rr 97.3%

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

7. Final simplification 97.3%

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

Alternative 5: 85.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. unpow2 N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 82.4

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

5. Simplified 82.4%

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

6. Final simplification 82.4%

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

Alternative 6: 84.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

9. Applied egg-rr 80.2%

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

Alternative 7: 84.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

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

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

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

Alternative 8: 84.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

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

Alternative 9: 84.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

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

Alternative 10: 84.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. *-commutative N/A

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

9. Applied egg-rr 80.2%

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

Alternative 11: 84.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

   1. +-commutative N/A

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

9. Applied egg-rr 80.2%

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

Alternative 12: 79.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

9. Applied egg-rr 80.2%

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

10. Taylor expanded in tau around 0

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

12. Simplified 76.9%

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

13. Final simplification 76.9%

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

Alternative 13: 79.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

9. Applied egg-rr 80.2%

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

10. Taylor expanded in tau around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

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

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

    6. *-lowering-*.f32 76.9

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

12. Simplified 76.9%

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

13. Final simplification 76.9%

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

Alternative 14: 78.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 97.5

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

4. Applied egg-rr 97.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 80.2%

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

9. Applied egg-rr 80.2%

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

10. Taylor expanded in tau around 0

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

12. Simplified 75.6%

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

13. Final simplification 75.6%

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

Alternative 15: 78.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. distribute-rgt-in N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

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

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

7. Applied egg-rr 75.5%

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

Alternative 16: 78.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

Alternative 17: 69.8% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

6. Taylor expanded in tau around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 67.2

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

8. Simplified 67.2%

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

Alternative 18: 69.8% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

6. Taylor expanded in tau around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

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

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

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

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

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

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

   13. PI-lowering-PI.f32 67.2

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

8. Simplified 67.2%

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

Alternative 19: 64.6% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

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 62.6%

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

   1. associate-*r* N/A

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

   2. unswap-sqr N/A

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

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

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

10. Applied egg-rr 62.6%

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

Alternative 20: 64.6% 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 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

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 62.6%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \pi\right) \cdot \color{blue}{-0.16666666666666666}, 1\right) \]
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 62.6

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

10. Applied egg-rr 62.6%

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

Alternative 21: 64.6% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.5%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 75.5

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

5. Simplified 75.5%

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

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 62.6%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \left(\pi \cdot \pi\right) \cdot \color{blue}{-0.16666666666666666}, 1\right) \]
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. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 62.6

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

10. Applied egg-rr 62.6%

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

11. Final simplification 62.6%

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

Alternative 22: 63.6% 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 97.5%

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

3. Taylor expanded in x around 0

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

5. Simplified 61.4%

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

Reproduce

herbie shell --seed 2024199 
(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))))