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

Average Error: 6.1 → 1.7

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.47293109594615 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{1}{y}}{x} \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{z \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2.47293109594615e+270)
   (* (/ (/ 1.0 y) x) (/ 1.0 (fma z z 1.0)))
   (/ (/ (/ 1.0 y) z) (* z x))))

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 tmp;
	if ((z * z) <= 2.47293109594615e+270) {
		tmp = ((1.0 / y) / x) * (1.0 / fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / y) / z) / (z * x);
	}
	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)
	tmp = 0.0
	if (Float64(z * z) <= 2.47293109594615e+270)
		tmp = Float64(Float64(Float64(1.0 / y) / x) * Float64(1.0 / fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) / Float64(z * x));
	end
	return 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_] := If[LessEqual[N[(z * z), $MachinePrecision], 2.47293109594615e+270], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	5.5
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.47293109594615023e270

   1. Initial program 1.8

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

   2. Taylor expanded in x around 0 2.2

      \[\leadsto \color{blue}{\frac{1}{\left(y + y \cdot {z}^{2}\right) \cdot x}} \]

   3. Simplified 2.0

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

   4. Applied egg-rr 1.8

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

   if 2.47293109594615023e270 < (*.f64 z z)

   1. Initial program 16.1

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

   2. Taylor expanded in z around inf 16.3

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left({z}^{2} \cdot x\right)}} \]

   3. Simplified 1.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{z}}{z \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.47293109594615 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{1}{y}}{x} \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{z \cdot x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))))