Average Error: 14.8 → 4.8
Time: 3.5s
Precision: binary64
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\frac{x \cdot \frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)} \]
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z) :precision binary64 (/ (* x (/ y z)) (fma z z z)))
double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}
↓
double code(double x, double y, double z) {
	return (x * (y / z)) / fma(z, z, z);
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original14.8
Target3.8
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Initial program 14.8

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

  2. Simplified4.9

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

  3. Applied egg-rr4.8

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

  4. Final simplification4.8

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

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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