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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. Taylor expanded around inf 0.1

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

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

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

5. Applied add-cube-cbrt0.6

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

6. Applied prod-diff0.6

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

7. Simplified0.1

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

8. Simplified0.1

   \[\leadsto \left(\left(\left(\frac{m}{v} - \frac{m}{v} \cdot m\right) + \color{blue}{0}\right) - 1\right) \cdot \left(1 - m\right)\]

9. Final simplification0.1

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

Runtime

Time bar (total: 52.7s)Debug logProfile

herbie shell --seed 2018290 +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)))