System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2

Average Error: 0.0 → 0.0

Time: 1.5s

Precision: 64

Math

\[0.5 \cdot \left(x \cdot x - y\right)\]

↓

\[\left(0.5 \cdot x\right) \cdot x + 0.5 \cdot \left(-y\right)\]

C

double code(double x, double y) {
	return (0.5 * ((x * x) - y));
}

↓

double code(double x, double y) {
	return (((0.5 * x) * x) + (0.5 * -y));
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.0

   \[0.5 \cdot \left(x \cdot x - y\right)\]
2. Using strategy rm

3. Applied *-un-lft-identity 0.0

   \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left(1 \cdot x\right)} - y\right)\]

4. Applied associate-*r* 0.0

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(x \cdot 1\right) \cdot x} - y\right)\]

5. Applied fma-neg 0.0

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x \cdot 1, x, -y\right)}\]
6. Using strategy rm

7. Applied fma-udef 0.0

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x \cdot 1\right) \cdot x + \left(-y\right)\right)}\]

8. Applied distribute-lft-in 0.0

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(x \cdot 1\right) \cdot x\right) + 0.5 \cdot \left(-y\right)}\]

9. Simplified 0.0

   \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot x} + 0.5 \cdot \left(-y\right)\]

10. Final simplification 0.0

    \[\leadsto \left(0.5 \cdot x\right) \cdot x + 0.5 \cdot \left(-y\right)\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y)
  :name "System.Random.MWC.Distributions:standard from mwc-random-0.13.3.2"
  :precision binary64
  (* 0.5 (- (* x x) y)))