Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 16.6s

Alternatives: 24

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 97.8

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

5. Applied rewrites 97.8%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

6. Applied rewrites 97.6%

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

7. Final simplification 97.6%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

6. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-*.f32 97.4

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 97.4

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

8. Applied rewrites 97.4%

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

9. Final simplification 97.4%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Final simplification 97.3%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

6. Applied rewrites 97.6%

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

8. Applied rewrites 97.3%

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

9. Final simplification 97.3%

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

Alternative 6: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Final simplification 97.3%

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

Alternative 7: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 61.1%

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

11. Taylor expanded in x around inf

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

    1. associate-/l* N/A

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

    2. lower-*.f32 N/A

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

    3. lower-sin.f32 N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. lower-*.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lower-PI.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. lower-sin.f32 N/A

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

    12. lower-*.f32 N/A

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

    13. lower-PI.f32 N/A

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

    14. lower-*.f32 N/A

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

    15. unpow2 N/A

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

    16. associate-*l* N/A

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

    17. lower-*.f32 N/A

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

    18. lower-*.f32 N/A

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

13. Applied rewrites 97.1%

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

Alternative 8: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. associate-/r* N/A

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

   14. associate-/l/ N/A

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

   15. lower-/.f32 N/A

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

5. Applied rewrites 96.9%

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

6. Final simplification 96.9%

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

Alternative 9: 91.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 83.4%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

10. Applied rewrites 90.3%

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

11. Final simplification 90.3%

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

Alternative 10: 91.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 90.3%

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

8. Final simplification 90.3%

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

Alternative 11: 85.8% 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 97.7%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-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. lower-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. lower-*.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. lower-*.f32 84.0

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

5. Applied rewrites 84.0%

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

6. Final simplification 84.0%

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

Alternative 12: 85.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* tau 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 = x * (tau * ((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(x * Float32(tau * 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 97.7%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 97.8

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 84.0

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

8. Applied rewrites 84.0%

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

Alternative 13: 79.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in tau around 0

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

   1. lower-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 76.6

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

7. Applied rewrites 76.6%

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

8. Final simplification 76.6%

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

Alternative 14: 79.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 83.4%

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

9. Applied rewrites 83.4%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. associate-*l* N/A

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

    5. lower-*.f32 N/A

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

    6. +-commutative N/A

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

    7. associate-*r* N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

12. Applied rewrites 75.9%

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

13. Final simplification 75.9%

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

Alternative 15: 79.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 83.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

10. Applied rewrites 75.9%

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

11. Final simplification 75.9%

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

Alternative 16: 79.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 83.4%

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

8. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

10. Applied rewrites 75.7%

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

11. Final simplification 75.7%

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

Alternative 17: 79.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

9. Applied rewrites 75.6%

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

10. Final simplification 75.6%

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

Alternative 18: 79.3% 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 97.7%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

5. Applied rewrites 75.6%

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

Alternative 19: 70.3% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in tau around inf

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

10. Applied rewrites 66.5%

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

Alternative 20: 65.0% accurate, 11.7× speedup

Math

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

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(Float32(pi) * Float32(pi)))), Float32(-0.16666666666666666), Float32(1.0))
end

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 61.1%

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

12. Applied rewrites 61.1%

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

Alternative 21: 65.0% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 61.1%

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

12. Applied rewrites 61.1%

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

Alternative 22: 65.0% accurate, 11.7× speedup

Math

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

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(x, Float32(Float32(x * Float32(pi)) * Float32(Float32(pi) * Float32(-0.16666666666666666))), Float32(1.0))
end

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 61.1%

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

12. Applied rewrites 61.1%

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

13. Final simplification 61.1%

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

Alternative 23: 65.0% accurate, 11.7× speedup

Math

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

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(x, Float32(x * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), Float32(1.0))
end

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-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. lower-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. lower-*.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 61.1%

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

11. Final simplification 61.1%

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

Alternative 24: 64.0% 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 97.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 60.1%

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

Reproduce

herbie shell --seed 2024222 
(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))))