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

Average Error: 14.5 → 2.6

Time: 2.0s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((x * y) / ((z * z) * (z + 1.0)));
}

↓

double code(double x, double y, double z) {
	return (((x / z) * (y / z)) * (1.0 / (z + 1.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.5
Target	3.7
Herbie	2.6

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

Derivation

1. Initial program 14.5

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

3. Applied associate-*l* 14.5

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

4. Applied times-frac 4.7

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

6. Applied associate-/r* 3.0

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

7. Applied associate-*r/ 2.6

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

9. Applied div-inv 2.6

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

10. Final simplification 2.6

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

Reproduce

herbie shell --seed 2020071 +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))))