a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 4.1s

Precision: 64

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

Math

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

↓

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

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) (((double) (m * ((double) (((double) sqrt(1.0)) + ((double) sqrt(m)))))) * ((double) (((double) sqrt(1.0)) - ((double) sqrt(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 add-sqr-sqrt 0.2

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied difference-of-squares 0.2

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

6. Applied associate-*r* 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020128 
(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))