Average Error: 15.0 → 4.4
Time: 3.4s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z + 1}}{z}}{z}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le 0.0:\\
\;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\

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

\end{array}
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 temp;
	if (((x * y) <= 0.0)) {
		temp = (((x / z) / z) * (y / (z + 1.0)));
	} else {
		temp = (((x * (y / (z + 1.0))) / z) / z);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target4.2
Herbie4.4
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x y) < 0.0

    1. Initial program 16.3

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
    2. Using strategy rm

    3. Applied times-frac12.2

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}}\]
    4. Using strategy rm

    5. Applied associate-/r*6.1

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

    if 0.0 < (* x y)

    1. Initial program 13.4

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
    2. Using strategy rm

    3. Applied times-frac10.9

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt11.4

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

    6. Applied times-frac7.1

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

    7. Applied associate-*l*1.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}\]
    8. Using strategy rm

    9. Applied associate-*l/1.4

      \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \color{blue}{\frac{\sqrt[3]{x} \cdot \frac{y}{z + 1}}{z}}\]

    10. Applied associate-*r/1.6

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

    11. Simplified2.1

      \[\leadsto \frac{\color{blue}{\frac{x \cdot \frac{y}{z + 1}}{z}}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le 0.0:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z + 1}}{z}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))