Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 4.9s

Alternatives: 19

Speedup: N/A×

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: 97.9% 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 97.2%

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

9. Applied rewrites 97.2%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. lift-*.f32 N/A

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

    7. lower-*.f32 97.8%

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

11. Applied rewrites 97.8%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (tau * ((float) M_PI)) * x;
	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 * Float32(pi)) * x)
	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 * single(pi)) * x;
	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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

Alternative 3: 97.8% 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 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 97.2%

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

3. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \pi\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 97.2%

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

9. Applied rewrites 97.2%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. lift-*.f32 N/A

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

    7. lower-*.f32 97.8%

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

11. Applied rewrites 97.8%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. associate-*r/ N/A

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

    5. lift-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. associate-/l/ N/A

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

    9. lower-/.f32 N/A

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

13. Applied rewrites 97.7%

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

Alternative 5: 97.1% 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 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. 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 97.1%

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

Alternative 6: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 97.1%

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

Alternative 7: 96.2% accurate, 1.0× speedup

Math

\[\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(x \cdot x\right) \cdot 9.869604110717773}}{tau} \]

FPCore

(FPCore (x tau)
  :precision binary32
  (/
 (*
  (sin (* tau (* PI x)))
  (/ (sin (* PI x)) (* (* x x) 9.869604110717773)))
 tau))

C

float code(float x, float tau) {
	return (sinf((tau * (((float) M_PI) * x))) * (sinf((((float) M_PI) * x)) / ((x * x) * 9.869604110717773f))) / tau;
}

Julia

function code(x, tau)
	return Float32(Float32(sin(Float32(tau * Float32(Float32(pi) * x))) * Float32(sin(Float32(Float32(pi) * x)) / Float32(Float32(x * x) * Float32(9.869604110717773)))) / tau)
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((tau * (single(pi) * x))) * (sin((single(pi) * x)) / ((x * x) * single(9.869604110717773)))) / tau;
end

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

3. Applied rewrites 96.8%

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

4. Evaluated real constant 96.2%

   \[\leadsto \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{9.869604110717773}}}{tau} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f32 N/A

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

    2. lower-sin.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 85.2%

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

12. Applied rewrites 85.2%

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

Alternative 9: 71.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 71.2%

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

Alternative 10: 64.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = (((x * x) * single(-0.16666666666666666)) * single(pi)) * single(pi);
	tmp = single(1.0) * ((single(1.0) + (single(1.0) / t_1)) * 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. 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 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. sum-to-mult N/A

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

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

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

9. Applied rewrites 64.8%

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

Alternative 11: 64.9% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  (*
 1.0
 (fma (* x x) (* (exp (* (log PI) 2.0)) -0.16666666666666666) 1.0)))

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

    1. lift-*.f32 N/A

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

    2. pow2 N/A

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

    3. pow-to-exp N/A

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

    4. lower-unsound-exp.f32 N/A

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

    5. lift-PI.f32 N/A

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

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

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

    7. lift-PI.f32 N/A

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

    8. lower-unsound-log.f32 64.9%

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

11. Applied rewrites 64.9%

    \[\leadsto 1 \cdot \mathsf{fma}\left(x \cdot x, e^{\log \pi \cdot 2} \cdot -0.16666666666666666, 1\right) \]
12. Add Preprocessing

Alternative 12: 64.9% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 64.9%

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

   7. lift-pow.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 64.9%

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

   10. lift-pow.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

Alternative 13: 64.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. associate-*r* N/A

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

    8. lower-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. associate-*r* N/A

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

    11. *-commutative N/A

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

    12. lift-*.f32 N/A

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

    13. lower-*.f32 64.9%

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

    14. lift-*.f32 N/A

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

    15. *-commutative N/A

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

    16. lift-*.f32 64.9%

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

11. Applied rewrites 64.9%

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

Alternative 14: 64.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. lift-*.f32 N/A

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

    5. lower-fma.f32 64.9%

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

    6. lift-*.f32 N/A

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

    7. lift-*.f32 N/A

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

    8. lift-*.f32 N/A

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

    9. unswap-sqr N/A

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

    10. *-commutative N/A

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

    11. lift-*.f32 N/A

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

    12. associate-*r* N/A

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

    13. lower-*.f32 N/A

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

    14. lower-*.f32 64.9%

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

    15. lift-*.f32 N/A

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

    16. *-commutative N/A

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

    17. lift-*.f32 64.9%

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

11. Applied rewrites 64.9%

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

Alternative 15: 64.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-*.f32 64.9%

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

11. Applied rewrites 64.9%

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

Alternative 16: 64.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. lift-*.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. unswap-sqr N/A

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

    7. *-commutative N/A

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

    8. lift-*.f32 N/A

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

    9. *-commutative N/A

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

    10. lift-*.f32 N/A

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

    11. associate-*l* N/A

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

    12. lower-fma.f32 N/A

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

    13. lift-*.f32 N/A

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

    14. *-commutative N/A

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

    15. lift-*.f32 N/A

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

    16. lower-*.f32 64.9%

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

    17. lift-*.f32 N/A

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

    18. *-commutative N/A

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

    19. lift-*.f32 64.9%

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

11. Applied rewrites 64.9%

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

Alternative 17: 64.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

Alternative 18: 64.8% accurate, 8.1× speedup

Math

\[1 \cdot \mathsf{fma}\left(x \cdot x, -1.644934058189392, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  (* 1.0 (fma (* x x) -1.644934058189392 1.0)))

C

float code(float x, float tau) {
	return 1.0f * fmaf((x * x), -1.644934058189392f, 1.0f);
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) * fma(Float32(x * x), Float32(-1.644934058189392), Float32(1.0)))
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 64.8%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 64.9%

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

7. Applied rewrites 64.9%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-pow.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 64.9%

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 64.9%

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

9. Applied rewrites 64.9%

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

10. Evaluated real constant 64.9%

    \[\leadsto 1 \cdot \mathsf{fma}\left(x \cdot x, -1.644934058189392, 1\right) \]
11. Add Preprocessing

Alternative 19: 64.0% accurate, 20.3× speedup

Math

\[\frac{tau}{tau} \]

FPCore

(FPCore (x tau)
  :precision binary32
  (/ tau tau))

C

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

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 = tau / tau
end function

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = tau / tau;
end

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

3. Applied rewrites 96.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025212 
(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))))