Average Error: 14.4 → 3.4
Time: 8.7s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(z, \sqrt{z}\right)\\ t_1 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{if}\;z \leq 5.615096020682109 \cdot 10^{-280}:\\ \;\;\;\;\frac{\left(x \cdot \frac{1}{t_1 \cdot t_1}\right) \cdot \frac{y}{t_1}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{y}{t_0} \cdot \frac{x}{t_0}}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot z (sqrt z))) (t_1 (cbrt (fma z z z))))
   (if (<= z 5.615096020682109e-280)
     (/ (* (* x (/ 1.0 (* t_1 t_1))) (/ y t_1)) z)
     (/ 1.0 (/ z (* (/ y t_0) (/ x 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 = hypot(z, sqrt(z));
	double t_1 = cbrt(fma(z, z, z));
	double tmp;
	if (z <= 5.615096020682109e-280) {
		tmp = ((x * (1.0 / (t_1 * t_1))) * (y / t_1)) / z;
	} else {
		tmp = 1.0 / (z / ((y / t_0) * (x / 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 = hypot(z, sqrt(z))
	t_1 = cbrt(fma(z, z, z))
	tmp = 0.0
	if (z <= 5.615096020682109e-280)
		tmp = Float64(Float64(Float64(x * Float64(1.0 / Float64(t_1 * t_1))) * Float64(y / t_1)) / z);
	else
		tmp = Float64(1.0 / Float64(z / Float64(Float64(y / t_0) * Float64(x / 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[Sqrt[z ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(z * z + z), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[z, 5.615096020682109e-280], N[(N[(N[(x * N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(1.0 / N[(z / N[(N[(y / t$95$0), $MachinePrecision] * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
t_0 := \mathsf{hypot}\left(z, \sqrt{z}\right)\\
t_1 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\
\mathbf{if}\;z \leq 5.615096020682109 \cdot 10^{-280}:\\
\;\;\;\;\frac{\left(x \cdot \frac{1}{t_1 \cdot t_1}\right) \cdot \frac{y}{t_1}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{z}{\frac{y}{t_0} \cdot \frac{x}{t_0}}}\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original14.4
Target3.8
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 2 regimes

  2. if z < 5.61509602068210881e-280

    1. Initial program 14.2

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

    2. Simplified8.8

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

    3. Applied associate-*r/_binary644.8

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

    4. Applied add-cube-cbrt_binary645.3

      \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}}{z} \]

    5. Applied *-un-lft-identity_binary645.3

      \[\leadsto \frac{x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

    6. Applied times-frac_binary645.3

      \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}} \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right)}}{z} \]

    7. Applied associate-*r*_binary644.4

      \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right) \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}}{z} \]

    if 5.61509602068210881e-280 < z

    1. Initial program 14.7

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

    2. Simplified8.7

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

    3. Applied associate-*r/_binary645.0

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

    4. Applied clear-num_binary645.1

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

    5. Simplified9.2

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

    6. Applied add-sqr-sqrt_binary649.2

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

    7. Applied times-frac_binary644.4

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

    8. Simplified4.4

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

    9. Simplified2.4

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.615096020682109 \cdot 10^{-280}:\\ \;\;\;\;\frac{\left(x \cdot \frac{1}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right) \cdot \frac{y}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{y}{\mathsf{hypot}\left(z, \sqrt{z}\right)} \cdot \frac{x}{\mathsf{hypot}\left(z, \sqrt{z}\right)}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))