Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.6s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 77.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((single(pi) * x)) / (single(pi) * x)) * ((single(1.0) / x) / (tau * ((single(0.16666666666666666) * (tau * (x * (single(pi) ^ single(2.0))))) + (single(1.0) / (x * tau)))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 71.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 71.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * ((single(pi) ^ single(2.0)) * (x ^ single(2.0))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((single(pi) ^ single(2.0)) * (single(-0.16666666666666666) * (x ^ single(2.0))));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 64.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 63.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{x \cdot 0}{\pi} \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (+ 1.0 (/ (* x 0.0) PI)))

C

float code(float x, float tau) {
	return 1.0f + ((x * 0.0f) / ((float) M_PI));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) + Float32(Float32(x * Float32(0.0)) / Float32(pi)))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + ((x * single(0.0)) / single(pi));
end

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2024003 
(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))))