a parameter of renormalized beta distribution

Average Error: 0.2 → 0.4

Time: 4.7s

Precision: binary64

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

Math

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

↓

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

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) (((double) sqrt(m)) * ((double) (((double) sqrt(m)) * ((double) (((double) (1.0 - m)) / v)))))) - 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 *-un-lft-identity 0.2

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

4. Applied times-frac 0.2

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

5. Simplified 0.2

   \[\leadsto \left(\color{blue}{m} \cdot \frac{1 - m}{v} - 1\right) \cdot m\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.4

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

8. Applied associate-*l* 0.4

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

9. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020147 
(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))