Average Error: 15.1 → 3.2
Time: 3.9s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{x \cdot \frac{\frac{y}{z}}{z + 1}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{x \cdot \frac{\frac{y}{z}}{z + 1}}{z}
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) {
	return ((double) (((double) (x * ((double) (((double) (y / z)) / ((double) (z + 1.0)))))) / z));
}

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

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

Derivation

  1. Initial program 15.1

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

  3. Applied associate-/r*13.3

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

  5. Applied *-un-lft-identity13.3

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

  6. Applied times-frac2.8

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

  7. Applied times-frac3.3

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

  8. Simplified3.3

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

  10. Applied associate-*l/3.2

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

  11. Final simplification3.2

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

Reproduce

herbie shell --seed 2020174 
(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))))