Average Error: 15.2 → 3.9
Time: 2.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.21590082645987849 \cdot 10^{-178} \lor \neg \left(x \cdot y \le 6.23637277415903404 \cdot 10^{-305}\right):\\ \;\;\;\;\frac{\frac{\frac{x}{z} \cdot y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.21590082645987849 \cdot 10^{-178} \lor \neg \left(x \cdot y \le 6.23637277415903404 \cdot 10^{-305}\right):\\
\;\;\;\;\frac{\frac{\frac{x}{z} \cdot y}{z}}{z + 1}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (x * y)) / ((double) (((double) (z * z)) * ((double) (z + 1.0))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((((double) (x * y)) <= -1.2159008264598785e-178) || !(((double) (x * y)) <= 6.236372774159034e-305))) {
		VAR = ((double) (((double) (((double) (((double) (x / z)) * y)) / z)) / ((double) (z + 1.0))));
	} else {
		VAR = ((double) (((double) (((double) (x / z)) / z)) * ((double) (y / ((double) (z + 1.0))))));
	}
	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

Original15.2
Target4.0
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x y) < -1.2159008264598785e-178 or 6.236372774159034e-305 < (* x y)

    1. Initial program 13.1

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

    3. Applied times-frac9.5

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

    5. Applied *-un-lft-identity9.5

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

    6. Applied times-frac5.5

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

    7. Applied associate-*l*2.3

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

    9. Applied associate-*r/3.3

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

    11. Applied associate-*r/3.3

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

    12. Simplified3.3

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

    if -1.2159008264598785e-178 < (* x y) < 6.236372774159034e-305

    1. Initial program 21.4

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

    3. Applied times-frac15.0

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

    5. Applied associate-/r*5.6

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.21590082645987849 \cdot 10^{-178} \lor \neg \left(x \cdot y \le 6.23637277415903404 \cdot 10^{-305}\right):\\ \;\;\;\;\frac{\frac{\frac{x}{z} \cdot y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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))))