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

Average Error: 6.1 → 0.4

Time: 3.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}\\ \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5.786246149407601 \cdot 10^{+285}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{\frac{1}{x}}{t_0}\\ \end{array} \]

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))) 5.786246149407601e+285)
     (/ (* (/ 1.0 (hypot 1.0 z)) (/ (/ 1.0 y) (hypot 1.0 z))) x)
     (* (/ 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))) <= 5.786246149407601e+285) {
		tmp = ((1.0 / hypot(1.0, z)) * ((1.0 / y) / hypot(1.0, z))) / x;
	} 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))) <= 5.786246149407601e+285) {
		tmp = ((1.0 / Math.hypot(1.0, z)) * ((1.0 / y) / Math.hypot(1.0, z))) / x;
	} 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))) <= 5.786246149407601e+285:
		tmp = ((1.0 / math.hypot(1.0, z)) * ((1.0 / y) / math.hypot(1.0, z))) / x
	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))) <= 5.786246149407601e+285)
		tmp = Float64(Float64(Float64(1.0 / hypot(1.0, z)) * Float64(Float64(1.0 / y) / hypot(1.0, z))) / x);
	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))) <= 5.786246149407601e+285)
		tmp = ((1.0 / hypot(1.0, z)) * ((1.0 / y) / hypot(1.0, z))) / x;
	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], 5.786246149407601e+285], N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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	0.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5.7862461494076014e285

   1. Initial program 1.8

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

   2. Applied egg-rr 1.8

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

   3. Applied egg-rr 0.5

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

   if 5.7862461494076014e285 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 16.6

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

   2. Applied egg-rr 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5.786246149407601 \cdot 10^{+285}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\\ \end{array} \]

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)))))