Average Error: 14.6 → 3.0
Time: 3.1s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{y}{z}}{\mathsf{fma}\left(\frac{z}{x}, z, \frac{z}{x}\right)}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -2e+205)
   (/ (/ y z) (fma (/ z x) z (/ z x)))
   (if (<= (* x y) 2e+127)
     (/ (/ x (/ z y)) (fma z z z))
     (/ (/ y z) (/ 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) <= -2e+205) {
		tmp = (y / z) / fma((z / x), z, (z / x));
	} else if ((x * y) <= 2e+127) {
		tmp = (x / (z / y)) / fma(z, z, z);
	} else {
		tmp = (y / z) / (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) <= -2e+205)
		tmp = Float64(Float64(y / z) / fma(Float64(z / x), z, Float64(z / x)));
	elseif (Float64(x * y) <= 2e+127)
		tmp = Float64(Float64(x / Float64(z / y)) / fma(z, z, z));
	else
		tmp = Float64(Float64(y / z) / 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], -2e+205], N[(N[(y / z), $MachinePrecision] / N[(N[(z / x), $MachinePrecision] * z + N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+127], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(z / N[(x / z), $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 -2 \cdot 10^{+205}:\\
\;\;\;\;\frac{\frac{y}{z}}{\mathsf{fma}\left(\frac{z}{x}, z, \frac{z}{x}\right)}\\

\mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+127}:\\
\;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{\mathsf{fma}\left(z, z, z\right)}\\

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


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < -2.00000000000000003e205

    1. Initial program 39.8

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

    2. Simplified13.7

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

    3. Applied egg-rr13.8

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

    4. Taylor expanded in z around 0 13.9

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

    5. Simplified2.2

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

    if -2.00000000000000003e205 < (*.f64 x y) < 1.99999999999999991e127

    1. Initial program 10.2

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

    2. Simplified3.6

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

    3. Applied egg-rr2.5

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

    4. Applied egg-rr2.4

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

    if 1.99999999999999991e127 < (*.f64 x y)

    1. Initial program 30.2

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

    2. Simplified11.6

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

    3. Applied egg-rr11.7

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

    4. Taylor expanded in z around inf 16.2

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

    5. Simplified7.8

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+205}:\\ \;\;\;\;\frac{\frac{y}{z}}{\mathsf{fma}\left(\frac{z}{x}, z, \frac{z}{x}\right)}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))