Error

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 flip--0.1

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

4. Applied associate-*r/0.1

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

5. Applied associate-/l/0.1

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

6. Simplified0.1

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

8. Applied sub-neg0.1

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

9. Applied distribute-rgt-in0.1

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

10. Simplified0.1

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

11. Taylor expanded around 0 0.1

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

12. Simplified0.1

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

13. Final simplification0.1

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

Runtime

Time bar (total: 33.6s)Debug logProfile

herbie shell --seed 2018339 +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)))