Lanczos kernel

Percentage Accurate: 98.0% → 97.9%

Time: 39.3s

Alternatives: 17

Speedup: N/A×

Specification

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (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

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* (* x PI) tau)))
  (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* PI (* x tau))))
  (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 97.9%

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

Alternative 2: 97.9% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* x (* tau PI))))
  (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 97.9%

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

Alternative 3: 97.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* (* x PI) tau))
 (/ (sin (* x PI)) (* (* (* x PI) (* x PI)) tau))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 96.8%

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

5. Applied rewrites 97.4%

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

Alternative 4: 97.2% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* x (* tau PI)))
 (/ (sin (* x PI)) (* (* x PI) (* (* x PI) tau)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 96.8%

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 97.2%

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

Alternative 5: 97.0% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* x PI))
 (/ (sin (* PI (* x tau))) (* (* (* x x) (* PI PI)) tau))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 96.8%

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

5. Applied rewrites 97.0%

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

Alternative 6: 96.2% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* x (* tau PI)))
 (/ (sin (* x PI)) (* (* (* x x) 9.869604110717773) tau))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 96.8%

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

4. Evaluated real constant 96.2%

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

6. Applied rewrites 96.2%

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

Alternative 7: 96.2% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* x PI))
 (/ (sin (* (* x PI) tau)) (* (* x (* x 9.869604110717773)) tau))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 96.8%

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

4. Evaluated real constant 96.2%

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

6. Applied rewrites 96.2%

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

Alternative 8: 85.0% accurate, 1.5× speedup

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

Math

\[\begin{array}{l} t_1 := x \cdot \left(tau \cdot \pi\right)\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), 9.869604110717773, 1\right) \end{array} \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (let* ((t_1 (* x (* tau PI))))
  (*
   (/ (sin t_1) t_1)
   (fma (* -0.16666666666666666 (* x x)) 9.869604110717773 1.0))))

C

float code(float x, float tau) {
	float t_1 = x * (tau * ((float) M_PI));
	return (sinf(t_1) / t_1) * fmaf((-0.16666666666666666f * (x * x)), 9.869604110717773f, 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(tau * Float32(pi)))
	return Float32(Float32(sin(t_1) / t_1) * fma(Float32(Float32(-0.16666666666666666) * Float32(x * x)), Float32(9.869604110717773), Float32(1.0)))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 85.0%

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

6. Applied rewrites 85.0%

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

8. Applied rewrites 85.0%

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

9. Evaluated real constant 85.0%

   \[\leadsto \frac{\sin \left(x \cdot \left(tau \cdot \pi\right)\right)}{x \cdot \left(tau \cdot \pi\right)} \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), 9.869604110717773, 1\right) \]
10. Add Preprocessing

Alternative 9: 79.2% accurate, 1.6× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (*
 (sin (* x PI))
 (fma (* -0.16666666666666666 (* tau tau)) (* x PI) (/ 1.0 (* x PI)))))

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

3. Applied rewrites 96.8%

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

4. Taylor expanded in tau around 0

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

6. Applied rewrites 79.2%

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

8. Applied rewrites 79.2%

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

Alternative 10: 78.7% accurate, 4.7× speedup

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

Math

\[1 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(\mathsf{fma}\left(tau, tau, 1\right) \cdot 9.869604110717773\right)\right)\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (+
 1.0
 (*
  x
  (*
   x
   (* -0.16666666666666666 (* (fma tau tau 1.0) 9.869604110717773))))))

C

float code(float x, float tau) {
	return 1.0f + (x * (x * (-0.16666666666666666f * (fmaf(tau, tau, 1.0f) * 9.869604110717773f))));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(x * Float32(x * Float32(Float32(-0.16666666666666666) * Float32(fma(tau, tau, Float32(1.0)) * Float32(9.869604110717773))))))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

8. Applied rewrites 78.7%

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

9. Evaluated real constant 78.7%

   \[\leadsto 1 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \left(\mathsf{fma}\left(tau, tau, 1\right) \cdot 9.869604110717773\right)\right)\right) \]
10. Add Preprocessing

Alternative 11: 78.7% accurate, 4.9× speedup

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

Math

\[\mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(tau \cdot tau, 9.869604110717773, 9.869604110717773\right), x \cdot x, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma
 (*
  -0.16666666666666666
  (fma (* tau tau) 9.869604110717773 9.869604110717773))
 (* x x)
 1.0))

C

float code(float x, float tau) {
	return fmaf((-0.16666666666666666f * fmaf((tau * tau), 9.869604110717773f, 9.869604110717773f)), (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * fma(Float32(tau * tau), Float32(9.869604110717773), Float32(9.869604110717773))), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

7. Evaluated real constant 78.7%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(tau \cdot tau, 9.869604110717773, 9.869604110717773\right), x \cdot x, 1\right) \]
8. Add Preprocessing

Alternative 12: 78.7% accurate, 4.9× speedup

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

Math

\[\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.16666666666666666, \left(tau \cdot tau\right) \cdot 9.869604110717773, -1.6449340184529622\right), 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma
 (* x x)
 (fma
  -0.16666666666666666
  (* (* tau tau) 9.869604110717773)
  -1.6449340184529622)
 1.0))

C

float code(float x, float tau) {
	return fmaf((x * x), fmaf(-0.16666666666666666f, ((tau * tau) * 9.869604110717773f), -1.6449340184529622f), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(x * x), fma(Float32(-0.16666666666666666), Float32(Float32(tau * tau) * Float32(9.869604110717773)), Float32(-1.6449340184529622)), Float32(1.0))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

5. Evaluated real constant 78.7%

   \[\leadsto 1 + {x}^{2} \cdot \mathsf{fma}\left(-0.16666666666666666, {tau}^{2} \cdot 9.869604110717773, -0.16666666666666666 \cdot 9.869604110717773\right) \]

7. Applied rewrites 78.7%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(-0.16666666666666666, \left(tau \cdot tau\right) \cdot 9.869604110717773, -1.6449340184529622\right), 1\right) \]
8. Add Preprocessing

Alternative 13: 69.7% accurate, 5.2× speedup

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

Math

\[1 + -0.16666666666666666 \cdot \left(\left(\left(tau \cdot tau\right) \cdot \left(x \cdot x\right)\right) \cdot 9.869604110717773\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (+
 1.0
 (*
  -0.16666666666666666
  (* (* (* tau tau) (* x x)) 9.869604110717773))))

C

float code(float x, float tau) {
	return 1.0f + (-0.16666666666666666f * (((tau * tau) * (x * x)) * 9.869604110717773f));
}

Fortran

real(4) function code(x, tau)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0 + ((-0.16666666666666666e0) * (((tau * tau) * (x * x)) * 9.869604110717773e0))
end function

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32(Float32(Float32(tau * tau) * Float32(x * x)) * Float32(9.869604110717773))))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * (((tau * tau) * (x * x)) * single(9.869604110717773)));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in tau around inf

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

7. Applied rewrites 69.7%

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

9. Applied rewrites 69.7%

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

10. Evaluated real constant 69.7%

    \[\leadsto 1 + -0.16666666666666666 \cdot \left(\left(\left(tau \cdot tau\right) \cdot \left(x \cdot x\right)\right) \cdot 9.869604110717773\right) \]
11. Add Preprocessing

Alternative 14: 64.5% accurate, 6.3× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (+ 1.0 (* (* x x) (* -0.16666666666666666 (* PI PI)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in tau around 0

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

7. Applied rewrites 64.5%

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

9. Applied rewrites 64.5%

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

Alternative 15: 64.5% accurate, 8.1× speedup

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

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (* 1.0 (fma x (* x -1.644934058189392) 1.0)))

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 85.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 64.5%

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

9. Applied rewrites 64.5%

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

10. Evaluated real constant 64.5%

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

Alternative 16: 64.5% accurate, 8.1× speedup

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

Math

\[\mathsf{fma}\left(-0.16666666666666666 \cdot 9.869604110717773, x \cdot x, 1\right) \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (fma (* -0.16666666666666666 9.869604110717773) (* x x) 1.0))

C

float code(float x, float tau) {
	return fmaf((-0.16666666666666666f * 9.869604110717773f), (x * x), 1.0f);
}

Julia

function code(x, tau)
	return fma(Float32(Float32(-0.16666666666666666) * Float32(9.869604110717773)), Float32(x * x), Float32(1.0))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.7%

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

7. Evaluated real constant 78.7%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \mathsf{fma}\left(tau \cdot tau, 9.869604110717773, 9.869604110717773\right), x \cdot x, 1\right) \]

8. Taylor expanded in tau around 0

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \frac{5174515}{524288}, x \cdot x, 1\right) \]

10. Applied rewrites 64.5%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot 9.869604110717773, x \cdot x, 1\right) \]
11. Add Preprocessing

Alternative 17: 64.5% accurate, 10.2× speedup

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

Math

\[1 + \left(x \cdot x\right) \cdot -1.644934058189392 \]

FPCore

(FPCore (x tau)
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0))
     (and (<= 1.0 tau) (<= tau 5.0)))
  (+ 1.0 (* (* x x) -1.644934058189392)))

C

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

Fortran

real(4) function code(x, tau)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0 + ((x * x) * (-1.644934058189392e0))
end function

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((x * x) * single(-1.644934058189392));
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in tau around 0

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

7. Applied rewrites 64.5%

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

9. Applied rewrites 64.5%

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

10. Evaluated real constant 64.5%

    \[\leadsto 1 + \left(x \cdot x\right) \cdot -1.644934058189392 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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))))