Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 3.3s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 97.9

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-sin.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

7. Applied rewrites 97.9%

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

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := tau \cdot \left(\pi \cdot x\right)\\ \mathbf{if}\;x \leq 0.02500000037252903:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \cdot \sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(tau \cdot x, \pi, \pi\right)\right) \cdot \sin \left(\pi \cdot x\right)}{\left(\left(-x\right) \cdot \pi\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* tau (* PI x))))
   (if (<= x 0.02500000037252903)
     (/
      (*
       (fma (pow (* PI x) 2.0) -0.16666666666666666 1.0)
       (sin (* tau (* x PI))))
      t_1)
     (/ (* (sin (fma (* tau x) PI PI)) (sin (* PI x))) (* (* (- x) PI) t_1)))))

C

float code(float x, float tau) {
	float t_1 = tau * (((float) M_PI) * x);
	float tmp;
	if (x <= 0.02500000037252903f) {
		tmp = (fmaf(powf((((float) M_PI) * x), 2.0f), -0.16666666666666666f, 1.0f) * sinf((tau * (x * ((float) M_PI))))) / t_1;
	} else {
		tmp = (sinf(fmaf((tau * x), ((float) M_PI), ((float) M_PI))) * sinf((((float) M_PI) * x))) / ((-x * ((float) M_PI)) * t_1);
	}
	return tmp;
}

Julia

function code(x, tau)
	t_1 = Float32(tau * Float32(Float32(pi) * x))
	tmp = Float32(0.0)
	if (x <= Float32(0.02500000037252903))
		tmp = Float32(Float32(fma((Float32(Float32(pi) * x) ^ Float32(2.0)), Float32(-0.16666666666666666), Float32(1.0)) * sin(Float32(tau * Float32(x * Float32(pi))))) / t_1);
	else
		tmp = Float32(Float32(sin(fma(Float32(tau * x), Float32(pi), Float32(pi))) * sin(Float32(Float32(pi) * x))) / Float32(Float32(Float32(-x) * Float32(pi)) * t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0250000004

   1. Initial program 98.5%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f32 N/A

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 98.4

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 98.4

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 98.4

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.0250000004 < x

   1. Initial program 96.6%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f32 N/A

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

   3. Applied rewrites 96.6%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f32 N/A

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

      3. sin-+PI-rev N/A

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

      4. sin-+PI-rev N/A

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

      5. lower-sin.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-fma.f32 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 94.0

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

   10. Applied rewrites 94.0%

       \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{fma}\left(tau \cdot x, \pi, \pi\right)\right)} \cdot \sin \left(\pi \cdot x\right)}{\left(\left(-x\right) \cdot \pi\right) \cdot \left(tau \cdot \left(\pi \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03200000151991844:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\pi \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \cdot \sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau \cdot \left(\pi \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(tau, x, 1\right) \cdot \pi\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(\left(tau \cdot x\right) \cdot \pi\right) \cdot \left(\left(-\pi\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (if (<= x 0.03200000151991844)
   (/
    (*
     (fma (pow (* PI x) 2.0) -0.16666666666666666 1.0)
     (sin (* tau (* x PI))))
    (* tau (* PI x)))
   (*
    (sin (* (fma tau x 1.0) PI))
    (/ (sin (* PI x)) (* (* (* tau x) PI) (* (- PI) x))))))

C

float code(float x, float tau) {
	float tmp;
	if (x <= 0.03200000151991844f) {
		tmp = (fmaf(powf((((float) M_PI) * x), 2.0f), -0.16666666666666666f, 1.0f) * sinf((tau * (x * ((float) M_PI))))) / (tau * (((float) M_PI) * x));
	} else {
		tmp = sinf((fmaf(tau, x, 1.0f) * ((float) M_PI))) * (sinf((((float) M_PI) * x)) / (((tau * x) * ((float) M_PI)) * (-((float) M_PI) * x)));
	}
	return tmp;
}

Julia

function code(x, tau)
	tmp = Float32(0.0)
	if (x <= Float32(0.03200000151991844))
		tmp = Float32(Float32(fma((Float32(Float32(pi) * x) ^ Float32(2.0)), Float32(-0.16666666666666666), Float32(1.0)) * sin(Float32(tau * Float32(x * Float32(pi))))) / Float32(tau * Float32(Float32(pi) * x)));
	else
		tmp = Float32(sin(Float32(fma(tau, x, Float32(1.0)) * Float32(pi))) * Float32(sin(Float32(Float32(pi) * x)) / Float32(Float32(Float32(tau * x) * Float32(pi)) * Float32(Float32(-Float32(pi)) * x))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0320000015

   1. Initial program 98.5%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f32 N/A

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 98.4

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 98.4

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 98.4

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 98.1

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

   8. Applied rewrites 98.1%

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

   if 0.0320000015 < x

   1. Initial program 96.5%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f32 N/A

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

   3. Applied rewrites 96.4%

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

      1. remove-double-neg N/A

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

      2. lift-sin.f32 N/A

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

      3. sin-+PI-rev N/A

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

      4. sin-+PI-rev N/A

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

      5. lower-sin.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-fma.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-fma.f32 92.9

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

   5. Applied rewrites 92.9%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

   7. Applied rewrites 93.8%

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

   9. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(tau, x, 1\right) \cdot \pi\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(\left(tau \cdot x\right) \cdot \pi\right) \cdot \left(\left(-\pi\right) \cdot x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

3. Applied rewrites 97.7%

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

Alternative 6: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-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. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 97.8

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

3. Applied rewrites 97.8%

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

Alternative 7: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot \pi\\ t_2 := tau \cdot \left(\pi \cdot x\right)\\ \frac{\sin t\_1 \cdot \sin t\_2}{t\_1 \cdot t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (- x) PI)) (t_2 (* tau (* PI x))))
   (/ (* (sin t_1) (sin t_2)) (* t_1 t_2))))

C

float code(float x, float tau) {
	float t_1 = -x * ((float) M_PI);
	float t_2 = tau * (((float) M_PI) * x);
	return (sinf(t_1) * sinf(t_2)) / (t_1 * t_2);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(-x) * Float32(pi))
	t_2 = Float32(tau * Float32(Float32(pi) * x))
	return Float32(Float32(sin(t_1) * sin(t_2)) / Float32(t_1 * t_2))
end

MATLAB

function tmp = code(x, tau)
	t_1 = -x * single(pi);
	t_2 = tau * (single(pi) * x);
	tmp = (sin(t_1) * sin(t_2)) / (t_1 * t_2);
end

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. lift-sin.f32 N/A

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

   5. sin-+PI N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-fma.f32 N/A

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

   9. lift-sin.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

7. Applied rewrites 97.8%

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

Alternative 8: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 96.8%

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

Alternative 9: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 97.9

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 85.3

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

8. Applied rewrites 85.3%

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

Alternative 10: 78.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 97.9

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 78.9%

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

Alternative 11: 64.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 97.9

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 64.6

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

8. Applied rewrites 64.6%

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

Alternative 12: 63.9% accurate, 258.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

3. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 97.9

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 63.9%

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

Reproduce

herbie shell --seed 2025107 
(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))))