Average Error: 0.1 → 0.1
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left((\left(\frac{m}{v}\right) \cdot \left(1 - m\right) + -1)_* \cdot \left(1 + \sqrt{m}\right)\right) \cdot \left(1 - \sqrt{m}\right)\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.1

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

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

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

  5. Applied difference-of-squares0.1

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

  6. Applied associate-*r*0.1

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

  8. Applied add-sqr-sqrt0.1

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

  9. Applied add-sqr-sqrt0.3

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

  10. Applied times-frac0.4

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

  11. Applied prod-diff0.4

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

  12. Simplified0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019090 +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)))