a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 3.7s

Precision: binary64

\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]

Math

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]

↓

\[\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right) - 1 \cdot m\]

C

double code(double m, double v) {
	return ((double) (((double) (((double) (((double) (m * ((double) (1.0 - m)))) / v)) - 1.0)) * m));
}

↓

double code(double m, double v) {
	return ((double) (((double) (((double) (1.0 - m)) * ((double) (m * ((double) (m / v)))))) - ((double) (1.0 * m))));
}

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.2

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.4

   \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{\color{blue}{\sqrt{v} \cdot \sqrt{v}}} - 1\right) \cdot m\]

4. Applied associate-/r* 0.3

   \[\leadsto \left(\color{blue}{\frac{\frac{m \cdot \left(1 - m\right)}{\sqrt{v}}}{\sqrt{v}}} - 1\right) \cdot m\]

5. Simplified 0.3

   \[\leadsto \left(\frac{\color{blue}{\frac{m}{\sqrt{v}} \cdot \left(1 - m\right)}}{\sqrt{v}} - 1\right) \cdot m\]

6. Taylor expanded around 0 6.7

   \[\leadsto \color{blue}{1 \cdot \frac{{m}^{2}}{v} - \left(1 \cdot m + \frac{{m}^{3}}{v}\right)}\]

7. Simplified 0.2

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m}{v} \cdot m\right) - m \cdot 1}\]

8. Final simplification 0.2

   \[\leadsto \left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right) - 1 \cdot m\]

Reproduce

herbie shell --seed 2020184 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))