Average Error: 14.6 → 3.2
Time: 4.0s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{elif}\;x \cdot y \leq 8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 5e-206)
   (/ (/ y z) (/ (fma z z z) x))
   (if (<= (* x y) 8e+81)
     (/ (/ (* x y) (fma z z z)) z)
     (/ (/ y z) (+ (/ z x) (/ z (/ x 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) {
	double tmp;
	if ((x * y) <= 5e-206) {
		tmp = (y / z) / (fma(z, z, z) / x);
	} else if ((x * y) <= 8e+81) {
		tmp = ((x * y) / fma(z, z, z)) / z;
	} else {
		tmp = (y / z) / ((z / x) + (z / (x / z)));
	}
	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) <= 5e-206)
		tmp = Float64(Float64(y / z) / Float64(fma(z, z, z) / x));
	elseif (Float64(x * y) <= 8e+81)
		tmp = Float64(Float64(Float64(x * y) / fma(z, z, z)) / z);
	else
		tmp = Float64(Float64(y / z) / Float64(Float64(z / x) + Float64(z / Float64(x / z))));
	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], 5e-206], N[(N[(y / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 8e+81], N[(N[(N[(x * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(N[(z / x), $MachinePrecision] + N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $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 5 \cdot 10^{-206}:\\
\;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\

\mathbf{elif}\;x \cdot y \leq 8 \cdot 10^{+81}:\\
\;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}\\


\end{array}

Error

Target

Original14.6
Target4.0
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < 5e-206

    1. Initial program 15.9

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

    2. Simplified4.0

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

    3. Applied egg-rr4.1

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

    4. Applied egg-rr3.9

      \[\leadsto \color{blue}{0 + \frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

    if 5e-206 < (*.f64 x y) < 7.99999999999999937e81

    1. Initial program 5.2

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

    2. Simplified5.9

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

    3. Applied egg-rr0.2

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

    if 7.99999999999999937e81 < (*.f64 x y)

    1. Initial program 23.7

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

    2. Simplified8.7

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

    3. Applied egg-rr8.8

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

    4. Taylor expanded in z around 0 10.3

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

    5. Simplified4.5

      \[\leadsto \frac{\frac{y}{z}}{\color{blue}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{elif}\;x \cdot y \leq 8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x} + \frac{z}{\frac{x}{z}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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))))