Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 13.5s

Alternatives: 16

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

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 98.0%

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

   1. lift-*.f32 N/A

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

   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 97.4%

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

3. Applied rewrites 97.4%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \pi\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 98.0%

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

5. Applied rewrites 98.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

5. Applied rewrites 97.8%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. *-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. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 97.2%

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

Alternative 4: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   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 97.4%

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

3. Applied rewrites 97.4%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \pi\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 98.0%

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

5. Applied rewrites 98.0%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \pi}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

7. Applied rewrites 97.0%

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

Alternative 5: 85.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  (let* ((t_1 (* tau (* x PI))))
  (*
   (/ (sin t_1) t_1)
   (/
    (*
     (+
      1.0
      (/
       (* (* x (* (* x x) -0.16666666666666666)) (* (* PI PI) PI))
       (* x PI)))
     (* 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) * (((1.0f + (((x * ((x * x) * -0.16666666666666666f)) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))) / (x * ((float) M_PI)))) * (x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. sum-to-mult N/A

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

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

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

9. Applied rewrites 64.6%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f32 N/A

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

    2. lower-sin.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 85.1%

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

12. Applied rewrites 85.1%

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

Alternative 6: 71.1% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  (let* ((t_1 (* tau (* x PI))))
  (* (/ (sin t_1) t_1) (/ (* 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) * ((x * ((float) M_PI)) / (x * ((float) M_PI)));
}

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]

8. Taylor expanded in x around 0

   \[\leadsto 1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]
9. Step-by-step derivation

   1. lower-PI.f32 63.6%

      \[\leadsto 1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]

10. Applied rewrites 63.6%

    \[\leadsto 1 \cdot \frac{x \cdot \pi}{x \cdot \pi} \]

11. Taylor expanded in x around inf

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

    1. lower-/.f32 N/A

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

    2. lower-sin.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 71.1%

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

13. Applied rewrites 71.1%

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

Alternative 7: 71.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (single(1.0) * sin(t_1)) / t_1;
end

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   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 97.4%

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

3. Applied rewrites 97.4%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \pi\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 98.0%

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

5. Applied rewrites 98.0%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \pi\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \pi}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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. lower-/.f32 N/A

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 71.1%

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

Alternative 8: 64.6% accurate, 2.8× speedup

Math

\[1 \cdot \frac{\frac{1}{\frac{x \cdot \pi}{x \cdot \left(\pi - \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}} \cdot \left(x \cdot \pi\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * (((single(1.0) / ((x * single(pi)) / (x * (single(pi) - ((single(0.16666666666666666) * (x * x)) * ((single(pi) * single(pi)) * single(pi))))))) * (x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. sum-to-mult N/A

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

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

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

9. Applied rewrites 64.6%

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

11. Applied rewrites 64.6%

    \[\leadsto 1 \cdot \frac{\frac{1}{\frac{x \cdot \pi}{x \cdot \left(\pi - \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)}} \cdot \left(\color{blue}{x} \cdot \pi\right)}{x \cdot \pi} \]
12. Add Preprocessing

Alternative 9: 64.6% accurate, 3.6× speedup

Math

\[1 \cdot \frac{\left(1 + x \cdot \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot \frac{\pi \cdot \pi}{x}\right)\right) \cdot \left(x \cdot \pi\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * (((single(1.0) + (x * (((single(-0.16666666666666666) * x) * x) * ((single(pi) * single(pi)) / x)))) * (x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. sum-to-mult N/A

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

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

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

9. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\left(1 + \frac{\left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{x \cdot \pi}\right) \cdot \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]
10. Step-by-step derivation

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-/l* N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. associate-*l* N/A

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

    11. *-commutative N/A

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

    12. lower-*.f32 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f32 N/A

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

    15. lift-*.f32 N/A

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

    16. *-commutative N/A

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

11. Applied rewrites 64.6%

    \[\leadsto 1 \cdot \frac{\left(1 + x \cdot \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot \frac{\pi \cdot \pi}{x}\right)\right) \cdot \left(x \cdot \pi\right)}{x \cdot \pi} \]
12. Add Preprocessing

Alternative 10: 64.6% accurate, 3.9× speedup

Math

\[\frac{1}{\frac{x \cdot \pi}{\left(\left(\pi - \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot x\right) \cdot 1}} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) / ((x * single(pi)) / (((single(pi) - ((single(0.16666666666666666) * (x * x)) * ((single(pi) * single(pi)) * single(pi)))) * x) * single(1.0)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. div-flip N/A

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

9. Applied rewrites 64.6%

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

Alternative 11: 64.6% accurate, 4.3× speedup

Math

\[1 \cdot \frac{\left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) + x \cdot \pi}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * ((((x * ((x * x) * single(-0.16666666666666666))) * ((single(pi) * single(pi)) * single(pi))) + (x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-+.f32 N/A

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

9. Applied rewrites 64.6%

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

Alternative 12: 64.6% accurate, 4.7× speedup

Math

\[1 \cdot \frac{\pi \cdot \left(x + \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * ((single(pi) * (x + (((x * x) * x) * (single(-0.16666666666666666) * (single(pi) * single(pi)))))) / (x * single(pi)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. sum-to-mult N/A

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

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

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

9. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\left(1 + \frac{\left(x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}{x \cdot \pi}\right) \cdot \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-+.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. sum-to-mult-rev N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. lift-*.f32 N/A

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

    8. associate-*r* N/A

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

    9. distribute-rgt-out N/A

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

    10. lower-*.f32 N/A

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

11. Applied rewrites 64.6%

    \[\leadsto 1 \cdot \frac{\pi \cdot \color{blue}{\left(x + \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]
12. Add Preprocessing

Alternative 13: 64.6% accurate, 4.7× speedup

Math

\[1 \cdot \frac{x \cdot \left(\pi + \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * ((x * (single(pi) + ((((x * x) * single(-0.16666666666666666)) * (single(pi) * single(pi))) * single(pi)))) / (x * single(pi)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f32 N/A

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

   5. unpow3 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-pow.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 64.6%

       \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right)}{x \cdot \pi} \]

9. Applied rewrites 64.6%

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

Alternative 14: 64.6% accurate, 4.7× speedup

Math

\[1 \cdot \frac{x \cdot \left(\pi + \left(x \cdot x\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -0.16666666666666666\right)\right)}{x \cdot \pi} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * ((x * (single(pi) + ((x * x) * (((single(pi) * single(pi)) * single(pi)) * single(-0.16666666666666666))))) / (x * single(pi)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-PI.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 64.6%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\pi + -0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{3}\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-pow.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 64.6%

      \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + \left(x \cdot x\right) \cdot \left({\pi}^{3} \cdot \color{blue}{-0.16666666666666666}\right)\right)}{x \cdot \pi} \]

   10. lift-pow.f32 N/A

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

   11. unpow3 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 64.6%

       \[\leadsto 1 \cdot \frac{x \cdot \left(\pi + \left(x \cdot x\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -0.16666666666666666\right)\right)}{x \cdot \pi} \]

9. Applied rewrites 64.6%

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

Alternative 15: 63.6% accurate, 258.0× speedup

Math

\[1 \]

FPCore

(FPCore (x tau)
  :precision binary32
  1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 63.6%

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

Reproduce

herbie shell --seed 2025258 
(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))))