Average Error: 14.8 → 3.4
Time: 3.7s
Precision: binary64
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-28}:\\ \;\;\;\;{\left(\frac{z}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (/ (/ y z) (+ (/ z x) (/ z (/ x z))))
   (if (<= (* x y) -1e-28)
     (pow (/ z (/ (* x y) (fma z z z))) -1.0)
     (/ (/ y z) (/ (fma z z z) x)))))
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 tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = (y / z) / ((z / x) + (z / (x / z)));
	} else if ((x * y) <= -1e-28) {
		tmp = pow((z / ((x * y) / fma(z, z, z))), -1.0);
	} else {
		tmp = (y / z) / (fma(z, z, z) / x);
	}
	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)
	tmp = 0.0
	if (Float64(x * y) <= Float64(-Inf))
		tmp = Float64(Float64(y / z) / Float64(Float64(z / x) + Float64(z / Float64(x / z))));
	elseif (Float64(x * y) <= -1e-28)
		tmp = Float64(z / Float64(Float64(x * y) / fma(z, z, z))) ^ -1.0;
	else
		tmp = Float64(Float64(y / z) / Float64(fma(z, z, z) / x));
	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_] := If[LessEqual[N[(x * y), $MachinePrecision], (-Infinity)], N[(N[(y / z), $MachinePrecision] / N[(N[(z / x), $MachinePrecision] + N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -1e-28], N[Power[N[(z / N[(N[(x * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \leq -\infty:\\
\;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}\\

\mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-28}:\\
\;\;\;\;{\left(\frac{z}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}\right)}^{-1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\


\end{array}

Error

Target

Original14.8
Target4.0
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < -inf.0

    1. Initial program 64.0

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

    2. Simplified20.1

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

    3. Applied egg-rr20.1

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

    4. Taylor expanded in z around 0 20.1

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

    5. Simplified1.6

      \[\leadsto \frac{\frac{y}{z}}{\color{blue}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}} \]

    if -inf.0 < (*.f64 x y) < -9.99999999999999971e-29

    1. Initial program 7.0

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

    2. Simplified6.9

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

    3. Applied egg-rr2.1

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

    if -9.99999999999999971e-29 < (*.f64 x y)

    1. Initial program 14.7

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

    2. Simplified3.8

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

    3. Applied egg-rr3.7

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-28}:\\ \;\;\;\;{\left(\frac{z}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))