Average Error: 6.3 → 1.6
Time: 5.7s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\frac{\frac{1}{x} \cdot \left(\frac{1}{y} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)}\right)}{\mathsf{hypot}\left(1, z\right)} \]
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (/ (* (/ 1.0 x) (* (/ 1.0 y) (/ 1.0 (hypot 1.0 z)))) (hypot 1.0 z)))
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) {
	return ((1.0 / x) * ((1.0 / y) * (1.0 / hypot(1.0, z)))) / hypot(1.0, z);
}

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) {
	return ((1.0 / x) * ((1.0 / y) * (1.0 / Math.hypot(1.0, z)))) / Math.hypot(1.0, z);
}

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))
↓
def code(x, y, z):
	return ((1.0 / x) * ((1.0 / y) * (1.0 / math.hypot(1.0, z)))) / math.hypot(1.0, z)

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)
	return Float64(Float64(Float64(1.0 / x) * Float64(Float64(1.0 / y) * Float64(1.0 / hypot(1.0, z)))) / hypot(1.0, z))
end

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

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_] := N[(N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] * N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]

\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x} \cdot \left(\frac{1}{y} \cdot \frac{1}{\mathsf{hypot}\left(1, z\right)}\right)}{\mathsf{hypot}\left(1, z\right)}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.3
Target5.0
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 6.3

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

  2. Simplified6.3

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

  3. Applied egg-rr6.3

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

  4. Applied egg-rr3.7

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

  5. Applied egg-rr1.7

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

  6. Applied egg-rr1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2022133 
(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)))))