Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.1 → 1.3

Time: 4.3s

Precision: binary64

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\\ \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 6.67452746661305 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{{x}^{-1} \cdot \frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{\frac{1}{x}}{t_0}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (hypot 1.0 z) (sqrt y))))
   (if (<= (* y (+ 1.0 (* z z))) 6.67452746661305e-205)
     (/ (/ (* (pow x -1.0) (/ 1.0 y)) (hypot 1.0 z)) (hypot 1.0 z))
     (* (/ 1.0 t_0) (/ (/ 1.0 x) t_0)))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

double code(double x, double y, double z) {
	double t_0 = hypot(1.0, z) * sqrt(y);
	double tmp;
	if ((y * (1.0 + (z * z))) <= 6.67452746661305e-205) {
		tmp = ((pow(x, -1.0) * (1.0 / y)) / hypot(1.0, z)) / hypot(1.0, z);
	} else {
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(1.0, z) * Math.sqrt(y);
	double tmp;
	if ((y * (1.0 + (z * z))) <= 6.67452746661305e-205) {
		tmp = ((Math.pow(x, -1.0) * (1.0 / y)) / Math.hypot(1.0, z)) / Math.hypot(1.0, z);
	} else {
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

↓

def code(x, y, z):
	t_0 = math.hypot(1.0, z) * math.sqrt(y)
	tmp = 0
	if (y * (1.0 + (z * z))) <= 6.67452746661305e-205:
		tmp = ((math.pow(x, -1.0) * (1.0 / y)) / math.hypot(1.0, z)) / math.hypot(1.0, z)
	else:
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0)
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

↓

function code(x, y, z)
	t_0 = Float64(hypot(1.0, z) * sqrt(y))
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 6.67452746661305e-205)
		tmp = Float64(Float64(Float64((x ^ -1.0) * Float64(1.0 / y)) / hypot(1.0, z)) / hypot(1.0, z));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(1.0 / x) / t_0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = hypot(1.0, z) * sqrt(y);
	tmp = 0.0;
	if ((y * (1.0 + (z * z))) <= 6.67452746661305e-205)
		tmp = (((x ^ -1.0) * (1.0 / y)) / hypot(1.0, z)) / hypot(1.0, z);
	else
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.67452746661305e-205], N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	4.7
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 6.6745274666130502e-205

   1. Initial program 4.2

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   2. Applied egg-rr 6.6

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}} \]

   3. Applied egg-rr 3.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}} \]

   4. Applied egg-rr 3.4

      \[\leadsto \frac{\frac{\color{blue}{{x}^{-1} \cdot \frac{1}{y}}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)} \]

   if 6.6745274666130502e-205 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 6.9

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 6.67452746661305 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{{x}^{-1} \cdot \frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))