Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 5.3s

Alternatives: 15

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * x) * single(pi);
	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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (single(pi) * x);
	tmp = (sin((single(pi) * x)) / t_1) * (sin(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. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

3. Applied rewrites 97.6%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f32 N/A

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

7. Applied rewrites 97.7%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 97.3

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 97.3

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 97.9

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 97.9

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

7. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

9. Applied rewrites 97.3%

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

Alternative 6: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 97.3

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 97.3

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 97.9

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 97.9

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

7. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

9. Applied rewrites 97.3%

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

Alternative 7: 78.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

3. Applied rewrites 97.3%

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

4. Taylor expanded in tau around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 78.7

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

6. Applied rewrites 78.7%

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

Alternative 8: 78.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32((x ^ Float32(2.0)) * fma(Float32(-0.16666666666666666), Float32((tau ^ Float32(2.0)) * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-0.16666666666666666) * (Float32(pi) ^ Float32(2.0))))))
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. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-PI.f32 78.4

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

4. Applied rewrites 78.4%

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

Alternative 9: 78.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. associate-*l/ N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-PI.f32 78.0

       \[\leadsto \frac{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}{\pi} \]

6. Applied rewrites 78.0%

   \[\leadsto \frac{\color{blue}{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}}{\pi} \]
7. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. sum-to-mult N/A

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

   3. lower-unsound-*.f32 N/A

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

8. Applied rewrites 78.2%

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

Alternative 10: 78.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. associate-*l/ N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-PI.f32 78.0

       \[\leadsto \frac{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}{\pi} \]

6. Applied rewrites 78.0%

   \[\leadsto \frac{\color{blue}{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}}{\pi} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 78.0%

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

10. Applied rewrites 78.2%

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

Alternative 11: 78.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. associate-*l/ N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-PI.f32 78.0

       \[\leadsto \frac{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}{\pi} \]

6. Applied rewrites 78.0%

   \[\leadsto \frac{\color{blue}{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}}{\pi} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 78.0%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 78.0%

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

Alternative 12: 78.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. associate-*l/ N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-PI.f32 78.0

       \[\leadsto \frac{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}{\pi} \]

6. Applied rewrites 78.0%

   \[\leadsto \frac{\color{blue}{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}}{\pi} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 78.0%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 78.0%

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

Alternative 13: 78.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. associate-*l/ N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-PI.f32 78.0

       \[\leadsto \frac{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}{\pi} \]

6. Applied rewrites 78.0%

   \[\leadsto \frac{\color{blue}{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}}{\pi} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 78.0%

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

10. Applied rewrites 78.0%

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

Alternative 14: 78.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. associate-*l/ N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lower-PI.f32 78.0

       \[\leadsto \frac{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}{\pi} \]

6. Applied rewrites 78.0%

   \[\leadsto \frac{\color{blue}{\pi + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot {\pi}^{3}, -0.16666666666666666 \cdot {\pi}^{3}\right)}}{\pi} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 78.0%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 78.0%

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

Alternative 15: 63.5% accurate, 94.3× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 63.5%

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

Reproduce

herbie shell --seed 2025162 
(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))))