a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 3.6s

Precision: 64

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

Math

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

↓

\[1 \cdot \left(\frac{m}{\frac{v}{m}} - m\right) - \frac{{m}^{3}}{v}\]

C

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

↓

double code(double m, double v) {
	return ((1.0 * ((m / (v / m)) - m)) - (pow(m, 3.0) / v));
}

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* 0.4

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

5. Taylor expanded around 0 7.1

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

6. Simplified 7.1

   \[\leadsto \color{blue}{1 \cdot \left(\frac{{m}^{2}}{v} - m\right) - \frac{{m}^{3}}{v}}\]
7. Using strategy rm

8. Applied unpow2 7.1

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

9. Applied associate-/l* 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020105 
(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 m)) v) 1) m))