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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 sub-neg0.1

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

6. Applied distribute-lft-in0.1

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

7. Simplified0.1

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

8. Taylor expanded around inf 0.1

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

9. Simplified0.1

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

10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019005 
(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)))

Details

Time bar (total: 31.7s)Debug log