b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 3.0s

Precision: 64

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

Math

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

↓

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

C

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

↓

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

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.1

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

3. Applied associate-/l* 0.1

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

5. Applied flip-- 0.1

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

6. Applied associate-/r/ 0.1

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

7. Applied associate-/r* 0.1

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

8. Taylor expanded around 0 0.1

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

9. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020071 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))