Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 5.0s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 97.8%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.8%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-sin.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

7. Applied rewrites 97.4%

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

Alternative 6: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 97.0%

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

Alternative 7: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

4. Applied rewrites 97.0%

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

Alternative 8: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

6. Applied rewrites 96.8%

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

Alternative 9: 91.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(tau \cdot x\right) \cdot \pi\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* tau x) PI)))
   (*
    (/ (sin t_1) t_1)
    (fma
     (fma
      (* 0.008333333333333333 (* x x))
      (* (* PI PI) (* PI PI))
      (* (* PI PI) -0.16666666666666666))
     (* x x)
     1.0))))

C

float code(float x, float tau) {
	float t_1 = (tau * x) * ((float) M_PI);
	return (sinf(t_1) / t_1) * fmaf(fmaf((0.008333333333333333f * (x * x)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f)), (x * x), 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * x) * Float32(pi))
	return Float32(Float32(sin(t_1) / t_1) * fma(fma(Float32(Float32(0.008333333333333333) * Float32(x * x)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), Float32(x * x), Float32(1.0)))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.6%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-/.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 70.8

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

10. Applied rewrites 70.8%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot x\right) \cdot \pi} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

13. Applied rewrites 91.4%

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

Alternative 10: 85.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.6%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-/.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 70.8

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

10. Applied rewrites 70.8%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. pow2 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. associate-*r* N/A

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

    7. pow2 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(tau \cdot x\right) \cdot \pi} \cdot \left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \]

    8. lift-PI.f32 N/A

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

    9. lift-PI.f32 N/A

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

    10. lift-*.f32 N/A

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

13. Applied rewrites 85.1%

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

Alternative 11: 79.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot \left(tau \cdot tau\right), \left(\pi \cdot \pi\right) \cdot \left(x \cdot x\right), 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 79.6%

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

Alternative 12: 79.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 79.3

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

7. Applied rewrites 79.3%

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

Alternative 13: 78.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

5. Applied rewrites 97.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 78.5%

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

Alternative 14: 78.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.5%

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

Alternative 15: 69.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.6%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. unswap-sqr N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 69.6

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

9. Applied rewrites 69.6%

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

Alternative 16: 69.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.6%

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

9. Applied rewrites 69.6%

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

Alternative 17: 69.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.3

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

4. Applied rewrites 79.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.6%

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

8. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot \left(tau \cdot tau\right)\right) \cdot \left({x}^{2} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot 1 \]

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

10. Applied rewrites 4.3%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 69.6%

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

Alternative 18: 64.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.7%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. lower-/.f32 N/A

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

6. Applied rewrites 84.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 64.4%

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

Alternative 19: 63.4% accurate, 94.3× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(x, tau)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0
end function

Julia

function code(x, tau)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0);
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2025106 
(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))))