Average Error: 14.7 → 2.6
Time: 5.5s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{z \cdot \frac{\frac{z}{y} \cdot \left(1 + z\right)}{x}}\\ \mathbf{if}\;x \cdot y \leq 4.586543263181661 \cdot 10^{-282}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 2.8102311057491402 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* z (/ (* (/ z y) (+ 1.0 z)) x)))))
   (if (<= (* x y) 4.586543263181661e-282)
     t_0
     (if (<= (* x y) 2.8102311057491402e+138)
       (/ (/ (* x y) (fma z z z)) z)
       t_0))))
double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}
↓
double code(double x, double y, double z) {
	double t_0 = 1.0 / (z * (((z / y) * (1.0 + z)) / x));
	double tmp;
	if ((x * y) <= 4.586543263181661e-282) {
		tmp = t_0;
	} else if ((x * y) <= 2.8102311057491402e+138) {
		tmp = ((x * y) / fma(z, z, z)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end
↓
function code(x, y, z)
	t_0 = Float64(1.0 / Float64(z * Float64(Float64(Float64(z / y) * Float64(1.0 + z)) / x)))
	tmp = 0.0
	if (Float64(x * y) <= 4.586543263181661e-282)
		tmp = t_0;
	elseif (Float64(x * y) <= 2.8102311057491402e+138)
		tmp = Float64(Float64(Float64(x * y) / fma(z, z, z)) / z);
	else
		tmp = t_0;
	end
	return tmp
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_] := Block[{t$95$0 = N[(1.0 / N[(z * N[(N[(N[(z / y), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], 4.586543263181661e-282], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 2.8102311057491402e+138], N[(N[(N[(x * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
t_0 := \frac{1}{z \cdot \frac{\frac{z}{y} \cdot \left(1 + z\right)}{x}}\\
\mathbf{if}\;x \cdot y \leq 4.586543263181661 \cdot 10^{-282}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \cdot y \leq 2.8102311057491402 \cdot 10^{+138}:\\
\;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original14.7
Target4.2
Herbie2.6
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 x y) < 4.5865432631816612e-282 or 2.8102311057491402e138 < (*.f64 x y)

    1. Initial program 18.6

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

    2. Simplified9.2

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

    3. Applied egg-rr6.3

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

    4. Applied egg-rr6.2

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

    5. Taylor expanded in z around 0 11.2

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

    6. Simplified3.7

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

    if 4.5865432631816612e-282 < (*.f64 x y) < 2.8102311057491402e138

    1. Initial program 5.5

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

    2. Simplified6.6

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

    3. Applied egg-rr2.1

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

    4. Applied egg-rr0.2

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 4.586543263181661 \cdot 10^{-282}:\\ \;\;\;\;\frac{1}{z \cdot \frac{\frac{z}{y} \cdot \left(1 + z\right)}{x}}\\ \mathbf{elif}\;x \cdot y \leq 2.8102311057491402 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \frac{\frac{z}{y} \cdot \left(1 + z\right)}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))