Average Error: 0.1 → 0.1
Time: 42.9s
Precision: 64
Internal Precision: 128
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\frac{\left(\left(1 - m\right) \cdot \left(1 - m\right)\right) \cdot m}{v} - \left(1 - m\right)\]

Error

Bits error versus m

Bits error versus v

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

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

  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 pow10.1

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

  6. Applied pow10.1

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

  7. Applied pow-prod-down0.1

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

  8. Simplified0.1

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

  10. Applied associate-*r/0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019053 +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)))