Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 4.3s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

Alternative 2: 97.7% 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 97.9%

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

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sin.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

5. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

7. Applied rewrites 97.7%

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

Alternative 3: 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 97.9%

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

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* 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(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 = (single(pi) * x) * tau;
	tmp = sin((single(pi) * x)) * (sin(t_1) / ((t_1 * single(pi)) * x));
end

Derivation

1. Initial program 97.9%

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

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sin.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

5. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

7. Applied rewrites 97.7%

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

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

9. Applied rewrites 97.3%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sin.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

5. Applied rewrites 97.4%

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

Alternative 6: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (fma (pow (* PI x) 2.0) -0.16666666666666666 1.0))))

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * fmaf(powf((((float) M_PI) * x), 2.0f), -0.16666666666666666f, 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * fma((Float32(Float32(pi) * x) ^ Float32(2.0)), Float32(-0.16666666666666666), Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 84.7

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \]

4. Applied rewrites 84.7%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right)} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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. associate-*l/ N/A

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

3. Applied rewrites 97.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 84.1

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{x} \cdot \frac{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right)}{\pi \cdot tau} \]

6. Applied rewrites 84.1%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{x} \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right)}}{\pi \cdot tau} \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 84.1

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

9. Applied rewrites 84.1%

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

Alternative 8: 84.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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. associate-*l/ N/A

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

3. Applied rewrites 97.2%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 84.1

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{x} \cdot \frac{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right)}{\pi \cdot tau} \]

6. Applied rewrites 84.1%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{x} \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right)}}{\pi \cdot tau} \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 84.1

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

8. Applied rewrites 84.1%

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

Alternative 9: 79.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \pi\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f32 N/A

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

   21. lift-PI.f32 79.0

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

6. Applied rewrites 79.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 79.0

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

8. Applied rewrites 79.0%

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

Alternative 10: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\pi \cdot x\right) \cdot tau\\ \mathsf{fma}\left(t\_1 \cdot t\_1, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* PI x) tau)))
   (*
    (fma (* t_1 t_1) -0.16666666666666666 1.0)
    (fma (pow (* PI x) 2.0) -0.16666666666666666 1.0))))

C

float code(float x, float tau) {
	float t_1 = (((float) M_PI) * x) * tau;
	return fmaf((t_1 * t_1), -0.16666666666666666f, 1.0f) * fmaf(powf((((float) M_PI) * x), 2.0f), -0.16666666666666666f, 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(Float32(pi) * x) * tau)
	return Float32(fma(Float32(t_1 * t_1), Float32(-0.16666666666666666), Float32(1.0)) * fma((Float32(Float32(pi) * x) ^ Float32(2.0)), Float32(-0.16666666666666666), Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \pi\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f32 N/A

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

   21. lift-PI.f32 79.0

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

6. Applied rewrites 79.0%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 78.9

      \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot x\right) \cdot tau\right) \cdot \left(\left(\pi \cdot x\right) \cdot tau\right), -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \]

9. Applied rewrites 78.9%

   \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot x\right) \cdot tau\right) \cdot \left(\left(\pi \cdot x\right) \cdot tau\right), -0.16666666666666666, 1\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right)} \]
10. Add Preprocessing

Alternative 11: 78.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left({\left(\pi \cdot tau\right)}^{2} + \pi \cdot \pi\right), x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (fma (* -0.16666666666666666 (+ (pow (* PI tau) 2.0) (* PI PI))) (* x x) 1.0))

C

float code(float x, float tau) {
	return fmaf((-0.16666666666666666f * (powf((((float) M_PI) * tau), 2.0f) + (((float) M_PI) * ((float) M_PI)))), (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * Float32((Float32(Float32(pi) * tau) ^ Float32(2.0)) + Float32(Float32(pi) * Float32(pi)))), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 97.9%

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

2. 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 78.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left({\left(\pi \cdot tau\right)}^{2} + \pi \cdot \pi\right), x \cdot x, 1\right)} \]
5. Add Preprocessing

Alternative 12: 69.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. 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. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \pi\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
5. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f32 N/A

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

   21. lift-PI.f32 79.0

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

6. Applied rewrites 79.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 69.5%

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

Alternative 13: 63.3% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

float code(float x, float tau) {
	return 1.0f;
}

Fortran

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 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 63.3%

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

Reproduce

herbie shell --seed 2025101 
(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))))