Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 15.1 → 2.4

Time: 5.5s

Precision: binary64

Math

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

↓

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

C

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) (1.0 / z)) * ((double) (((double) (x / z)) * ((double) (y / ((double) (z + 1.0))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.1
Target	3.9
Herbie	2.4

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

Derivation

1. Initial program 15.1

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

3. Applied times-frac 10.6

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

5. Applied *-un-lft-identity 10.6

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

6. Applied times-frac 5.5

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

7. Applied associate-*l* 2.4

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

8. Final simplification 2.4

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

Reproduce

herbie shell --seed 2020163 
(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))))