Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 5.7s

Alternatives: 17

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

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

FPCore

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

C

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

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(Float32(Float32(pi) * x) * t_1))
end

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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 97.6%

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. frac-times N/A

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

9. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot x\right) \cdot \sin \left(\left(\pi \cdot x\right) \cdot tau\right)}{\left(\pi \cdot x\right) \cdot \left(\left(\pi \cdot x\right) \cdot tau\right)}} \]
10. 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: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. 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 97.6%

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

5. Applied rewrites 97.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 96.7%

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

Alternative 5: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

4. Applied rewrites 96.9%

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

Alternative 6: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\pi \cdot \pi\right) \cdot \pi\\ t_2 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), t\_1 \cdot t\_1, \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.008333333333333333\right), x \cdot x, \left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* PI PI) PI)) (t_2 (* (* x PI) tau)))
   (*
    (/ (sin t_2) t_2)
    (fma
     (fma
      (fma
       (* -0.0001984126984126984 (* x x))
       (* t_1 t_1)
       (* (* (* PI PI) (* PI PI)) 0.008333333333333333))
      (* x x)
      (* (* PI PI) -0.16666666666666666))
     (* x x)
     1.0))))

C

float code(float x, float tau) {
	float t_1 = (((float) M_PI) * ((float) M_PI)) * ((float) M_PI);
	float t_2 = (x * ((float) M_PI)) * tau;
	return (sinf(t_2) / t_2) * fmaf(fmaf(fmaf((-0.0001984126984126984f * (x * x)), (t_1 * t_1), (((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))) * 0.008333333333333333f)), (x * x), ((((float) M_PI) * ((float) M_PI)) * -0.16666666666666666f)), (x * x), 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))
	t_2 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_2) / t_2) * fma(fma(fma(Float32(Float32(-0.0001984126984126984) * Float32(x * x)), Float32(t_1 * t_1), Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))) * Float32(0.008333333333333333))), Float32(x * x), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))), 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

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \left(\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) \cdot {x}^{2} + 1\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right), \color{blue}{{x}^{2}}, 1\right) \]

4. Applied rewrites 93.9%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984 \cdot \left(x \cdot x\right), \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.008333333333333333\right), x \cdot x, \left(\pi \cdot \pi\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 91.0%

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

Alternative 8: 84.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (fma (* (* PI PI) (* x x)) -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(((((float) M_PI) * ((float) M_PI)) * (x * x)), -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(Float32(pi) * Float32(pi)) * Float32(x * x)), Float32(-0.16666666666666666), Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow3 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-PI.f32 84.5

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

4. Applied rewrites 84.5%

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

5. Taylor expanded in x around 0

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

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 84.7

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

7. Applied rewrites 84.7%

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

Alternative 9: 84.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * fmaf((-0.16666666666666666f * (x * x)), (((float) M_PI) * ((float) M_PI)), 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(-0.16666666666666666) * Float32(x * x)), Float32(Float32(pi) * Float32(pi)), 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. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 84.7

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

4. Applied rewrites 84.7%

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

Alternative 10: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\pi \cdot tau, \pi \cdot tau, \pi \cdot \pi\right), \mathsf{fma}\left(0.008333333333333333 \cdot \left(\left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right)\right), t\_1, \mathsf{fma}\left(0.027777777777777776 \cdot \left(tau \cdot tau\right), t\_1, t\_1 \cdot 0.008333333333333333\right)\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* PI PI) (* PI PI))))
   (fma
    (fma
     -0.16666666666666666
     (fma (* PI tau) (* PI tau) (* PI PI))
     (*
      (fma
       (* 0.008333333333333333 (* (* tau tau) (* tau tau)))
       t_1
       (fma
        (* 0.027777777777777776 (* tau tau))
        t_1
        (* t_1 0.008333333333333333)))
      (* x x)))
    (* x x)
    1.0)))

C

float code(float x, float tau) {
	float t_1 = (((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI));
	return fmaf(fmaf(-0.16666666666666666f, fmaf((((float) M_PI) * tau), (((float) M_PI) * tau), (((float) M_PI) * ((float) M_PI))), (fmaf((0.008333333333333333f * ((tau * tau) * (tau * tau))), t_1, fmaf((0.027777777777777776f * (tau * tau)), t_1, (t_1 * 0.008333333333333333f))) * (x * x))), (x * x), 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi)))
	return fma(fma(Float32(-0.16666666666666666), fma(Float32(Float32(pi) * tau), Float32(Float32(pi) * tau), Float32(Float32(pi) * Float32(pi))), Float32(fma(Float32(Float32(0.008333333333333333) * Float32(Float32(tau * tau) * Float32(tau * tau))), t_1, fma(Float32(Float32(0.027777777777777776) * Float32(tau * tau)), t_1, Float32(t_1 * Float32(0.008333333333333333)))) * Float32(x * x))), 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

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

4. Applied rewrites 83.8%

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

Alternative 11: 82.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow3 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-PI.f32 84.5

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

4. Applied rewrites 84.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 82.9%

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

Alternative 12: 82.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow3 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-PI.f32 84.5

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

4. Applied rewrites 84.5%

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

5. Taylor expanded in tau around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 82.9%

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

Alternative 13: 78.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(fma(Float32(fma(Float32(Float32(0.008333333333333333) * Float32(x * x)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.16666666666666666))) / tau), Float32(x * x), Float32(Float32(1.0) / tau)) * Float32(fma(Float32(Float32(-0.16666666666666666) * Float32(tau * tau)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(x * x)), Float32(1.0)) * 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. 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 97.6%

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

4. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

6. Applied rewrites 90.2%

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

7. Taylor expanded in tau around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. pow2 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. pow2 N/A

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

   16. lift-*.f32 78.7

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

9. Applied rewrites 78.7%

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

Alternative 14: 78.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow3 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-PI.f32 84.5

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

4. Applied rewrites 84.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. pow2 N/A

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

   14. lift-*.f32 78.6

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

7. Applied rewrites 78.6%

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

Alternative 15: 78.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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%

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

Alternative 16: 63.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\pi\right) \cdot x\right) \cdot \frac{-1}{\pi \cdot x} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (-single(pi) * x) * (single(-1.0) / (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 78.9%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. cos-PI N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. sin-PI 70.8

      \[\leadsto \frac{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]

6. Applied rewrites 70.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-\pi, x, 0\right)} \cdot \sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \pi\right)}}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]

   3. lift-sin.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \color{blue}{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]

   4. lift-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)}} \]

   8. lower-*.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\pi, x, 0\right) \cdot \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(-\pi \cdot x\right) \cdot \left(\left(tau \cdot x\right) \cdot \pi\right)}} \]

8. Applied rewrites 70.7%

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

9. Taylor expanded in x around 0

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

    1. lower-/.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 63.3

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

11. Applied rewrites 63.3%

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

Alternative 17: 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))))