Lanczos kernel

Percentage Accurate: 97.9% → 97.9%
Time: 12.4s
Alternatives: 12
Speedup: 1.0×

Specification

?
\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))
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)));
}

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

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

\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}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 12 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 97.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (* (/ (sin t_1) t_1) (/ (sin (* x PI)) (* x PI)))))
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)));
}

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

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

\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}

Alternative 1: 97.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \end{array} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* PI (* x tau))))
   (* (/ (sin t_1) t_1) (/ (sin (* PI x)) (* PI x)))))
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));
}

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

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

\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}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 97.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \cdot \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (* (/ (sin (* PI x)) (* PI x)) (/ (sin (* tau (* PI x))) (* x (* PI tau)))))
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)));
}

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

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

\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}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 97.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \cdot \frac{\sin t_1}{t_1} \end{array} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (* (/ (sin (* PI x)) (* PI x)) (/ (sin t_1) t_1))))
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);
}

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

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

\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}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 79.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\sin \left(\pi \cdot x\right)}{\pi \cdot x} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(tau \cdot \left(\pi \cdot x\right)\right)}^{2}, 1\right) \end{array} \]
(FPCore (x tau)
 :precision binary32
 (*
  (/ (sin (* PI x)) (* PI x))
  (fma -0.16666666666666666 (pow (* tau (* PI x)) 2.0) 1.0)))
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);
}

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

\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}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 70.4% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \left(\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi} \cdot \frac{1}{x \cdot tau}\right) \cdot \frac{\pi \cdot x}{\pi \cdot x} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (* (* (/ (sin (* PI (* x tau))) PI) (/ 1.0 (* x tau))) (/ (* PI x) (* PI x))))
float code(float x, float tau) {
	return ((sinf((((float) M_PI) * (x * tau))) / ((float) M_PI)) * (1.0f / (x * tau))) * ((((float) M_PI) * x) / (((float) M_PI) * x));
}

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

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

\begin{array}{l}

\\
\left(\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\pi} \cdot \frac{1}{x \cdot tau}\right) \cdot \frac{\pi \cdot x}{\pi \cdot x}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 70.2% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\pi \cdot x}{\pi \cdot x} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (* (/ (sin (* tau (* PI x))) (* x (* PI tau))) (/ (* PI x) (* PI x))))
float code(float x, float tau) {
	return (sinf((tau * (((float) M_PI) * x))) / (x * (((float) M_PI) * tau))) * ((((float) M_PI) * x) / (((float) M_PI) * x));
}

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

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

\begin{array}{l}

\\
\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\pi \cdot x}{\pi \cdot x}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 63.8% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}^{2}, 1\right) \end{array} \]
(FPCore (x tau)
 :precision binary32
 (fma -0.16666666666666666 (pow (expm1 (log1p (* PI x))) 2.0) 1.0))
float code(float x, float tau) {
	return fmaf(-0.16666666666666666f, powf(expm1f(log1pf((((float) M_PI) * x))), 2.0f), 1.0f);
}

function code(x, tau)
	return fma(Float32(-0.16666666666666666), (expm1(log1p(Float32(Float32(pi) * x))) ^ Float32(2.0)), Float32(1.0))
end

\begin{array}{l}

\\
\mathsf{fma}\left(-0.16666666666666666, {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot x\right)\right)\right)}^{2}, 1\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 63.8% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, \pi \cdot \left(x \cdot \left(\pi \cdot x\right)\right), 1\right) \end{array} \]
(FPCore (x tau)
 :precision binary32
 (fma -0.16666666666666666 (* PI (* x (* PI x))) 1.0))
float code(float x, float tau) {
	return fmaf(-0.16666666666666666f, (((float) M_PI) * (x * (((float) M_PI) * x))), 1.0f);
}

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

\begin{array}{l}

\\
\mathsf{fma}\left(-0.16666666666666666, \pi \cdot \left(x \cdot \left(\pi \cdot x\right)\right), 1\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 63.8% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, {\left(\pi \cdot x\right)}^{2}, 1\right) \end{array} \]
(FPCore (x tau)
 :precision binary32
 (fma -0.16666666666666666 (pow (* PI x) 2.0) 1.0))
float code(float x, float tau) {
	return fmaf(-0.16666666666666666f, powf((((float) M_PI) * x), 2.0f), 1.0f);
}

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

\begin{array}{l}

\\
\mathsf{fma}\left(-0.16666666666666666, {\left(\pi \cdot x\right)}^{2}, 1\right)
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 10: 63.8% accurate, 3.0× speedup?

\[\begin{array}{l} \\ 1 + -0.16666666666666666 \cdot {\left(\pi \cdot x\right)}^{2} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (+ 1.0 (* -0.16666666666666666 (pow (* PI x) 2.0))))
float code(float x, float tau) {
	return 1.0f + (-0.16666666666666666f * powf((((float) M_PI) * x), 2.0f));
}

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

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

\begin{array}{l}

\\
1 + -0.16666666666666666 \cdot {\left(\pi \cdot x\right)}^{2}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 11: 4.7% accurate, 3.0× speedup?

\[\begin{array}{l} \\ -0.16666666666666666 \cdot {\left(\pi \cdot x\right)}^{2} \end{array} \]
(FPCore (x tau)
 :precision binary32
 (* -0.16666666666666666 (pow (* PI x) 2.0)))
float code(float x, float tau) {
	return -0.16666666666666666f * powf((((float) M_PI) * x), 2.0f);
}

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

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

\begin{array}{l}

\\
-0.16666666666666666 \cdot {\left(\pi \cdot x\right)}^{2}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 12: 62.9% accurate, 3.0× speedup?

\[\begin{array}{l} \\ \frac{\pi \cdot x}{\pi \cdot x} \end{array} \]
(FPCore (x tau) :precision binary32 (/ (* PI x) (* PI x)))
float code(float x, float tau) {
	return (((float) M_PI) * x) / (((float) M_PI) * x);
}

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

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

\begin{array}{l}

\\
\frac{\pi \cdot x}{\pi \cdot x}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023347 
(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))))