Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 15.6s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.0% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied egg-rr 98.2%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-sin.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. div-inv N/A

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

4. Applied egg-rr 97.8%

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

5. Final simplification 97.8%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. div-inv N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-sin.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

4. Applied egg-rr 97.7%

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

5. Final simplification 97.7%

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

Alternative 6: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. associate-/r* N/A

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

   14. associate-/l/ N/A

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

   15. lower-/.f32 N/A

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

5. Simplified 97.2%

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

6. Final simplification 97.2%

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

Alternative 7: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied egg-rr 97.9%

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

5. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-/.f32 N/A

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

   13. lower-sin.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. *-commutative N/A

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 8: 85.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 88.3

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

5. Simplified 88.3%

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

Alternative 9: 84.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(pi), fma(Float32(pi), Float32(Float32(-0.16666666666666666) * Float32(tau * tau)), Float32(Float32(pi) * Float32(-0.16666666666666666))), Float32(fma(Float32(tau * tau), fma(Float32(tau * tau), Float32(0.008333333333333333), Float32(0.027777777777777776)), Float32(0.008333333333333333)) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(x * x) * Float32(Float32(pi) * Float32(pi)))))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]

7. Applied egg-rr 88.1%

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

9. Applied egg-rr 88.1%

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

10. Final simplification 88.1%

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

Alternative 10: 84.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(x * x) * fma(Float32(pi), Float32(Float32(pi) * fma(tau, Float32(tau * Float32(-0.16666666666666666)), Float32(-0.16666666666666666))), Float32(Float32(x * x) * Float32(fma(Float32(tau * tau), fma(Float32(0.008333333333333333), Float32(tau * tau), Float32(0.027777777777777776)), Float32(0.008333333333333333)) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]

7. Applied egg-rr 88.1%

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

8. Final simplification 88.1%

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

Alternative 11: 84.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(Float32(x * Float32(pi)) * Float32(x * fma(Float32(tau * tau), fma(Float32(tau * tau), Float32(0.008333333333333333), Float32(0.027777777777777776)), Float32(0.008333333333333333)))), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * fma(tau, Float32(tau * Float32(-0.16666666666666666)), Float32(-0.16666666666666666)))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]

7. Applied egg-rr 88.1%

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

9. Applied egg-rr 88.1%

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

10. Final simplification 88.1%

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

Alternative 12: 79.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(Float32(-0.16666666666666666) * Float32(tau * tau)), Float32(Float32(pi) * Float32(pi)), fma(Float32(Float32(x * x) * fma(Float32(tau * tau), Float32(0.027777777777777776), Float32(0.008333333333333333))), Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(pi) * Float32(Float32(pi) * Float32(-0.16666666666666666))))), Float32(1.0))
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]

7. Applied egg-rr 88.1%

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

9. Applied egg-rr 88.1%

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

10. Taylor expanded in tau around 0

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\left(tau \cdot tau\right) \cdot \frac{-1}{6}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \mathsf{fma}\left(\mathsf{fma}\left(tau \cdot tau, \color{blue}{\frac{1}{36}}, \frac{1}{120}\right) \cdot \left(x \cdot x\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{-1}{6}\right)\right)\right), 1\right) \]
11. Step-by-step derivation

12. Simplified 83.1%

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

13. Final simplification 83.1%

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

Alternative 13: 78.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Simplified 88.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {\pi}^{4} \cdot \mathsf{fma}\left(0.008333333333333333, \left(tau \cdot tau\right) \cdot \left(tau \cdot tau\right), \mathsf{fma}\left(tau \cdot tau, 0.027777777777777776, 0.008333333333333333\right)\right), \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, tau \cdot tau, -0.16666666666666666\right)\right), 1\right)} \]

7. Applied egg-rr 88.1%

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

8. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 82.0

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

10. Simplified 82.0%

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

11. Final simplification 82.0%

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

Alternative 14: 78.5% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f32 N/A

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

7. Simplified 82.0%

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

8. Final simplification 82.0%

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

Alternative 15: 69.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied egg-rr 97.3%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 72.0

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

7. Simplified 72.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-PI.f32 71.9

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

10. Simplified 71.9%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-*.f32 N/A

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

12. Applied egg-rr 71.9%

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

13. Final simplification 71.9%

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

Alternative 16: 69.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sin.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

4. Applied egg-rr 97.3%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 72.0

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

7. Simplified 72.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-PI.f32 71.9

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

10. Simplified 71.9%

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

11. Final simplification 71.9%

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

Alternative 17: 63.3% accurate, 258.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in x around 0

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

5. Simplified 64.9%

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

Reproduce

herbie shell --seed 2024207 
(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))))