Error

Derivation

1. Initial program 0.1

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

2. Taylor expanded around 0 0.1

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

3. Simplified0.1

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

5. Applied *-un-lft-identity0.1

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

6. Applied associate-/l*0.2

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

7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))