Average Error: 0.2 → 0.7
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[\left(\frac{\frac{\left(1 - {m}^{3}\right) \cdot m}{(m \cdot m + m)_* + 1}}{v} - 1\right) \cdot m\]

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.2

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

  3. Applied flip3--0.2

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

  4. Applied associate-*r/0.7

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

  5. Simplified0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) m))