Average Error: 14.8 → 2.4
Time: 7.2s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.16298716880188486 \cdot 10^{-137} \lor \neg \left(y \le 1.52173936496567621 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{z + 1}}{z}}{z}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.16298716880188486 \cdot 10^{-137} \lor \neg \left(y \le 1.52173936496567621 \cdot 10^{-187}\right):\\
\;\;\;\;\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)\\

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

\end{array}
double code(double x, double y, double z) {
	return (((double) (x * y)) / ((double) (((double) (z * z)) * ((double) (z + 1.0)))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((y <= -1.1629871688018849e-137) || !(y <= 1.5217393649656762e-187))) {
		VAR = ((double) ((1.0 / z) * ((double) ((x / z) * (y / ((double) (z + 1.0)))))));
	} else {
		VAR = ((double) (x * (((y / ((double) (z + 1.0))) / z) / z)));
	}
	return VAR;
}

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

Original14.8
Target4.3
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.16298716880188486e-137 or 1.52173936496567621e-187 < y

    1. Initial program 14.6

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

    3. Applied times-frac8.9

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

    5. Applied *-un-lft-identity8.9

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

    6. Applied times-frac3.5

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

    7. Applied associate-*l*1.4

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

    if -1.16298716880188486e-137 < y < 1.52173936496567621e-187

    1. Initial program 15.3

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

    3. Applied times-frac16.3

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

    5. Applied div-inv16.6

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

    6. Applied associate-*l*14.6

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

    7. Simplified5.0

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.16298716880188486 \cdot 10^{-137} \lor \neg \left(y \le 1.52173936496567621 \cdot 10^{-187}\right):\\ \;\;\;\;\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{z + 1}}{z}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))