Lanczos kernel

Percentage Accurate: 97.9% → 97.8%

Time: 5.0s

Alternatives: 7

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. add-log-exp N/A

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

   4. log-pow-rev N/A

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

   5. lower-log.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lift-PI.f32 81.0

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

4. Applied rewrites 81.0%

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

6. Applied rewrites 97.9%

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

Alternative 2: 97.9% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

Alternative 3: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6, {\pi}^{4}, \mathsf{fma}\left(-4, {\pi}^{4}, \mathsf{fma}\left(13, {\pi}^{4}, -3 \cdot {\pi}^{4}\right)\right)\right) \cdot x, 0.041666666666666664, \mathsf{fma}\left(-3, {\pi}^{3}, 3 \cdot {\pi}^{3}\right) \cdot 0.16666666666666666\right), x, \left(0 \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.5\right), x, \pi\right) \cdot x\right)}{x \cdot \pi} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (*
    (/ (sin t_1) t_1)
    (/
     (sin
      (*
       (fma
        (fma
         (fma
          (*
           (fma
            -6.0
            (pow PI 4.0)
            (fma
             -4.0
             (pow PI 4.0)
             (fma 13.0 (pow PI 4.0) (* -3.0 (pow PI 4.0)))))
           x)
          0.041666666666666664
          (* (fma -3.0 (pow PI 3.0) (* 3.0 (pow PI 3.0))) 0.16666666666666666))
         x
         (* (* 0.0 (* PI PI)) 0.5))
        x
        PI)
       x))
     (* x PI)))))

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((fmaf(fmaf(fmaf((fmaf(-6.0f, powf(((float) M_PI), 4.0f), fmaf(-4.0f, powf(((float) M_PI), 4.0f), fmaf(13.0f, powf(((float) M_PI), 4.0f), (-3.0f * powf(((float) M_PI), 4.0f))))) * x), 0.041666666666666664f, (fmaf(-3.0f, powf(((float) M_PI), 3.0f), (3.0f * powf(((float) M_PI), 3.0f))) * 0.16666666666666666f)), x, ((0.0f * (((float) M_PI) * ((float) M_PI))) * 0.5f)), x, ((float) M_PI)) * x)) / (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(fma(fma(fma(Float32(fma(Float32(-6.0), (Float32(pi) ^ Float32(4.0)), fma(Float32(-4.0), (Float32(pi) ^ Float32(4.0)), fma(Float32(13.0), (Float32(pi) ^ Float32(4.0)), Float32(Float32(-3.0) * (Float32(pi) ^ Float32(4.0)))))) * x), Float32(0.041666666666666664), Float32(fma(Float32(-3.0), (Float32(pi) ^ Float32(3.0)), Float32(Float32(3.0) * (Float32(pi) ^ Float32(3.0)))) * Float32(0.16666666666666666))), x, Float32(Float32(Float32(0.0) * Float32(Float32(pi) * Float32(pi))) * Float32(0.5))), x, Float32(pi)) * x)) / Float32(x * Float32(pi))))
end

Derivation

1. Initial program 97.8%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. add-log-exp N/A

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

   4. log-pow-rev N/A

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

   5. lower-log.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lift-PI.f32 81.0

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

4. Applied rewrites 81.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) + x \cdot \left(\frac{1}{2} \cdot \left(-1 \cdot {\mathsf{PI}\left(\right)}^{2} + {\mathsf{PI}\left(\right)}^{2}\right) + x \cdot \left(\frac{1}{24} \cdot \left(x \cdot \left(-6 \cdot {\mathsf{PI}\left(\right)}^{4} + \left(-4 \cdot {\mathsf{PI}\left(\right)}^{4} + \left(-3 \cdot {\mathsf{PI}\left(\right)}^{4} + \left(12 \cdot {\mathsf{PI}\left(\right)}^{4} + {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right) + \frac{1}{6} \cdot \left(-3 \cdot {\mathsf{PI}\left(\right)}^{3} + \left(2 \cdot {\mathsf{PI}\left(\right)}^{3} + {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)}}{x \cdot \pi} \]

7. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6, {\pi}^{4}, \mathsf{fma}\left(-4, {\pi}^{4}, \mathsf{fma}\left(13, {\pi}^{4}, -3 \cdot {\pi}^{4}\right)\right)\right) \cdot x, 0.041666666666666664, \mathsf{fma}\left(-3, {\pi}^{3}, 3 \cdot {\pi}^{3}\right) \cdot 0.16666666666666666\right), x, \left(0 \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.5\right), x, \pi\right) \cdot x\right)}}{x \cdot \pi} \]
8. Add Preprocessing

Alternative 4: 97.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(\pi \cdot x\right) \cdot tau\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (sin (* (* PI x) tau)) (/ (sin (* PI x)) (* (pow (* PI x) 2.0) tau))))

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin(((single(pi) * x) * tau)) * (sin((single(pi) * x)) / (((single(pi) * x) ^ single(2.0)) * tau));
end

Derivation

1. Initial program 97.8%

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

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Applied rewrites 97.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 97.3

      \[\leadsto \sin \left(\left(\pi \cdot x\right) \cdot tau\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau} \]

7. Applied rewrites 97.3%

   \[\leadsto \sin \left(\left(\pi \cdot x\right) \cdot tau\right) \cdot \frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{{\left(\pi \cdot x\right)}^{2} \cdot tau} \]
8. Add Preprocessing

Alternative 5: 97.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(tau \cdot x\right) \cdot \pi\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (sin (* (* tau x) PI)) (/ (sin (* PI x)) (* (pow (* PI x) 2.0) tau))))

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin(((tau * x) * single(pi))) * (sin((single(pi) * x)) / (((single(pi) * x) ^ single(2.0)) * tau));
end

Derivation

1. Initial program 97.8%

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

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\sin \left(\left(tau \cdot x\right) \cdot \pi\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau}} \]
6. Add Preprocessing

Alternative 6: 97.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

5. Applied rewrites 97.0%

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

   1. lift-pow.f32 N/A

      \[\leadsto \sin \left(\left(tau \cdot x\right) \cdot \pi\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{{\left(\pi \cdot x\right)}^{2} \cdot tau} \]

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. unpow-prod-down N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 N/A

      \[\leadsto \sin \left(\left(tau \cdot x\right) \cdot \pi\right) \cdot \frac{\sin \left(\pi \cdot x\right)}{\left(\left(\pi \cdot \pi\right) \cdot {x}^{2}\right) \cdot tau} \]

   10. pow2 N/A

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

   11. lift-*.f32 96.7

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

7. Applied rewrites 96.7%

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

Alternative 7: 81.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. lift-/.f32 N/A

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

   9. lift-sin.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

4. Applied rewrites 80.4%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 80.4%

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

8. Final simplification 80.4%

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

Reproduce

herbie shell --seed 2025057 
(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))))