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

Average Error: 6.5 → 0.9

Time: 4.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z, y \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1}{z \cdot z}\right) \cdot \frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 2e+304)
   (/ (/ 1.0 x) (fma z (* y z) y))
   (* (+ 1.0 (/ -1.0 (* z z))) (/ 1.0 (* z (* y (* 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 ((y * (1.0 + (z * z))) <= 2e+304) {
		tmp = (1.0 / x) / fma(z, (y * z), y);
	} else {
		tmp = (1.0 + (-1.0 / (z * z))) * (1.0 / (z * (y * (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(y * Float64(1.0 + Float64(z * z))) <= 2e+304)
		tmp = Float64(Float64(1.0 / x) / fma(z, Float64(y * z), y));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(z * z))) * Float64(1.0 / Float64(z * Float64(y * 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[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+304], N[(N[(1.0 / x), $MachinePrecision] / N[(z * N[(y * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.5
Target	5.2
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.9999999999999999e304

   1. Initial program 1.9

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

   2. Applied egg-rr 0.4

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

   if 1.9999999999999999e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 19.3

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

   2. Taylor expanded in z around inf 15.2

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left({z}^{2} \cdot x\right)} - \frac{1}{y \cdot \left({z}^{4} \cdot x\right)}} \]

   3. Simplified 2.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z, y \cdot z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-1}{z \cdot z}\right) \cdot \frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))