Lanczos kernel

Percentage Accurate: 98.0% → 97.8%

Time: 14.8s

Alternatives: 16

Speedup: N/A×

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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-2neg N/A

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

   9. frac-times N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

4. Applied rewrites 98.0%

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

6. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 98.0

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 98.0

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

8. Applied rewrites 98.0%

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

9. Final simplification 98.0%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-2neg N/A

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

   9. frac-times N/A

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

   10. neg-mul-1 N/A

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

   11. remove-double-neg N/A

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

4. Applied rewrites 98.0%

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

6. Applied rewrites 97.7%

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

7. Final simplification 97.7%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = ((float) M_PI) * (x * tau);
	return sinf(t_1) * (sinf((((float) M_PI) * x)) / (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(Float32(pi) * x)) / 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((single(pi) * x)) / (x * (single(pi) * t_1)));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.6%

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

5. Final simplification 97.6%

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

Alternative 4: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-sin.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f32 N/A

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

7. Applied rewrites 97.2%

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

8. Final simplification 97.2%

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

Alternative 5: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.0%

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

6. Final simplification 97.0%

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

Alternative 6: 90.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 97.6%

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

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 90.8%

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

8. Final simplification 90.8%

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

Alternative 7: 85.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 85.5

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

7. Applied rewrites 85.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 86.2

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 86.2

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

9. Applied rewrites 86.2%

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

10. Final simplification 86.2%

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

Alternative 8: 85.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 86.2

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

5. Applied rewrites 86.2%

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

6. Final simplification 86.2%

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

Alternative 9: 85.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 85.5

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

7. Applied rewrites 85.5%

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

8. Taylor expanded in tau around inf

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 86.2

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

10. Applied rewrites 86.2%

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

11. Final simplification 86.2%

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

Alternative 10: 79.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 85.5

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

7. Applied rewrites 85.5%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 65.1%

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

11. Taylor expanded in tau around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. *-commutative N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f32 N/A

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

    14. lower-PI.f32 N/A

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

    15. lower-*.f32 N/A

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

    16. unpow2 N/A

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

    17. lower-*.f32 80.4

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

13. Applied rewrites 80.4%

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

14. Final simplification 80.4%

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

Alternative 11: 79.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 85.5

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

7. Applied rewrites 85.5%

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

8. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

10. Applied rewrites 80.4%

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

11. Final simplification 80.4%

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

Alternative 12: 78.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

4. Applied rewrites 97.4%

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

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. distribute-rgt-in N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-fma.f32 N/A

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

7. Applied rewrites 79.7%

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

8. Final simplification 79.7%

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

Alternative 13: 78.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 79.7

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

5. Applied rewrites 79.7%

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

Alternative 14: 64.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 85.5

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

7. Applied rewrites 85.5%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 65.1%

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

12. Applied rewrites 65.1%

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

Alternative 15: 64.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 85.5

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

7. Applied rewrites 85.5%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 65.1%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 N/A

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

    9. lower-*.f32 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f32 N/A

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

    12. lower-PI.f32 65.1

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

13. Applied rewrites 65.1%

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

14. Final simplification 65.1%

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

Alternative 16: 63.7% 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 97.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.2%

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

Reproduce

herbie shell --seed 2024221 
(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))))