Average Error: 0.2 → 0.8
Time: 39.3s
Precision: 64
Internal Precision: 128
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m\]
↓
\[m \cdot \left(\frac{1}{(v \cdot \left((m \cdot m + m)_*\right) + v)_*} \cdot \left(m - {m}^{4}\right) - 1\right)\]

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 clear-num0.2

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

  5. Applied div-inv0.3

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

  6. Applied associate-/r*0.3

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

  8. Applied flip3--0.3

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

  9. Applied associate-*r/0.8

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

  10. Applied associate-/r/0.8

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

  11. Applied div-inv0.8

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

  12. Applied times-frac0.8

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

  13. Simplified0.8

    \[\leadsto \left(\color{blue}{\left(m - {m}^{4}\right)} \cdot \frac{\frac{1}{v}}{1 \cdot 1 + \left(m \cdot m + 1 \cdot m\right)} - 1\right) \cdot m\]

  14. Simplified0.8

    \[\leadsto \left(\left(m - {m}^{4}\right) \cdot \color{blue}{\frac{1}{(v \cdot \left((m \cdot m + m)_*\right) + v)_*}} - 1\right) \cdot m\]

  15. Final simplification0.8

    \[\leadsto m \cdot \left(\frac{1}{(v \cdot \left((m \cdot m + m)_*\right) + v)_*} \cdot \left(m - {m}^{4}\right) - 1\right)\]

Runtime

Time bar (total: 39.3s)Debug logProfile

herbie shell --seed 2018273 +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))