b parameter of renormalized beta distribution

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative99.9%

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

   2. sub-neg99.9%

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

   3. associate-/l*100.0%

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

   4. metadata-eval100.0%

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

3. Simplified100.0%

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

4. Final simplification100.0%

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

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative99.9%

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

   2. sub-neg99.9%

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

   3. associate-*l/100.0%

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

   4. metadata-eval100.0%

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

3. Simplified100.0%

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

4. Final simplification100.0%

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

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative99.9%

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

      2. sub-neg99.9%

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

      3. associate-*l/99.9%

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

      4. metadata-eval99.9%

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

   3. Simplified99.9%

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

   4. Taylor expanded in m around inf 98.4%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv98.3%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. cube-mult98.3%

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

      3. associate-*l*98.3%

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

      4. associate-*r*98.3%

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

      5. div-inv98.4%

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

   6. Applied egg-rr98.4%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{m}{v}\right)} \]
   7. Step-by-step derivation

      1. associate-*r*98.4%

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

      2. clear-num98.4%

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

      3. div-inv98.4%

         \[\leadsto \color{blue}{\frac{m \cdot m}{\frac{v}{m}}} \]

   8. Applied egg-rr98.4%

      \[\leadsto \color{blue}{\frac{m \cdot m}{\frac{v}{m}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

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

   if 1.6000000000000001 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative99.9%

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

      2. associate-*r/99.9%

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

      3. fma-neg99.9%

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

      4. metadata-eval99.9%

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

   3. Simplified99.9%

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
   4. Step-by-step derivation

      1. metadata-eval99.9%

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

      2. fma-neg99.9%

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

   5. Applied egg-rr99.9%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v} - 1\right)} \]
   6. Step-by-step derivation

      1. clear-num99.9%

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

      2. associate-/r/99.9%

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

   7. Applied egg-rr99.9%

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

   8. Taylor expanded in m around inf 29.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{{m}^{2}}{v} + \frac{{m}^{3}}{v}} \]
   9. Step-by-step derivation

      1. unpow229.5%

         \[\leadsto -2 \cdot \frac{\color{blue}{m \cdot m}}{v} + \frac{{m}^{3}}{v} \]

      2. associate-*r/29.5%

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

      3. unpow329.5%

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

      4. associate-*r/29.5%

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

      5. associate-*r*29.5%

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

      6. distribute-rgt-out99.2%

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

      7. associate-*r/99.2%

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

   10. Simplified99.2%

       \[\leadsto \color{blue}{\frac{m \cdot m}{v} \cdot \left(-2 + m\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v} \cdot \left(m + -2\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 0.38

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg98.0%

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

      2. distribute-lft-in98.0%

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

      3. *-commutative98.0%

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

      4. *-un-lft-identity98.0%

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

      5. sub-neg98.0%

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

      6. metadata-eval98.0%

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

      7. sub-neg98.0%

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

      8. metadata-eval98.0%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod97.8%

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

      11. sqr-neg97.8%

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

      12. sqrt-unprod97.8%

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

      13. add-sqr-sqrt97.8%

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

   4. Applied egg-rr97.8%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative97.8%

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

      2. distribute-rgt1-in97.8%

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

   6. Simplified97.8%

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

   7. Taylor expanded in m around 0 97.7%

      \[\leadsto \color{blue}{\left(\frac{1}{v} - 1\right) \cdot m - 1} \]
   8. Step-by-step derivation

      1. sub-neg97.7%

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

      2. *-commutative97.7%

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

      3. sub-neg97.7%

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

      4. metadata-eval97.7%

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

      5. distribute-rgt-in97.7%

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

      6. associate-*l/97.9%

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

      7. *-lft-identity97.9%

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

      8. neg-mul-197.9%

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

      9. sub-neg97.9%

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

      10. metadata-eval97.9%

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

   9. Simplified97.9%

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

   10. Taylor expanded in v around 0 97.9%

       \[\leadsto \color{blue}{\frac{m}{v}} + -1 \]

   if 0.38 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative99.9%

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

      2. sub-neg99.9%

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

      3. associate-*l/99.9%

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

      4. metadata-eval99.9%

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

   3. Simplified99.9%

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

   4. Taylor expanded in m around inf 98.4%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv98.3%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. cube-mult98.3%

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

      3. associate-*l*98.3%

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

      4. associate-*r*98.3%

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

      5. div-inv98.4%

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

   6. Applied egg-rr98.4%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{m}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 0.38

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg98.0%

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

      2. distribute-lft-in98.0%

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

      3. *-commutative98.0%

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

      4. *-un-lft-identity98.0%

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

      5. sub-neg98.0%

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

      6. metadata-eval98.0%

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

      7. sub-neg98.0%

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

      8. metadata-eval98.0%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod97.8%

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

      11. sqr-neg97.8%

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

      12. sqrt-unprod97.8%

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

      13. add-sqr-sqrt97.8%

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

   4. Applied egg-rr97.8%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative97.8%

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

      2. distribute-rgt1-in97.8%

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

   6. Simplified97.8%

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

   7. Taylor expanded in m around 0 97.7%

      \[\leadsto \color{blue}{\left(\frac{1}{v} - 1\right) \cdot m - 1} \]
   8. Step-by-step derivation

      1. sub-neg97.7%

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

      2. *-commutative97.7%

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

      3. sub-neg97.7%

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

      4. metadata-eval97.7%

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

      5. distribute-rgt-in97.7%

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

      6. associate-*l/97.9%

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

      7. *-lft-identity97.9%

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

      8. neg-mul-197.9%

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

      9. sub-neg97.9%

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

      10. metadata-eval97.9%

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

   9. Simplified97.9%

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

   10. Taylor expanded in v around 0 97.9%

       \[\leadsto \color{blue}{\frac{m}{v}} + -1 \]

   if 0.38 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative99.9%

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

      2. sub-neg99.9%

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

      3. associate-*l/99.9%

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

      4. metadata-eval99.9%

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

   3. Simplified99.9%

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

   4. Taylor expanded in m around inf 98.4%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv98.3%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. unpow398.3%

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

      3. associate-*l*98.3%

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

      4. div-inv98.4%

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

   6. Applied egg-rr98.4%

      \[\leadsto \color{blue}{\left(m \cdot m\right) \cdot \frac{m}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 0.38

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg98.0%

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

      2. distribute-lft-in98.0%

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

      3. *-commutative98.0%

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

      4. *-un-lft-identity98.0%

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

      5. sub-neg98.0%

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

      6. metadata-eval98.0%

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

      7. sub-neg98.0%

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

      8. metadata-eval98.0%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod97.8%

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

      11. sqr-neg97.8%

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

      12. sqrt-unprod97.8%

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

      13. add-sqr-sqrt97.8%

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

   4. Applied egg-rr97.8%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative97.8%

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

      2. distribute-rgt1-in97.8%

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

   6. Simplified97.8%

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

   7. Taylor expanded in m around 0 97.7%

      \[\leadsto \color{blue}{\left(\frac{1}{v} - 1\right) \cdot m - 1} \]
   8. Step-by-step derivation

      1. sub-neg97.7%

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

      2. *-commutative97.7%

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

      3. sub-neg97.7%

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

      4. metadata-eval97.7%

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

      5. distribute-rgt-in97.7%

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

      6. associate-*l/97.9%

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

      7. *-lft-identity97.9%

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

      8. neg-mul-197.9%

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

      9. sub-neg97.9%

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

      10. metadata-eval97.9%

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

   9. Simplified97.9%

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

   if 0.38 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative99.9%

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

      2. sub-neg99.9%

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

      3. associate-*l/99.9%

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

      4. metadata-eval99.9%

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

   3. Simplified99.9%

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

   4. Taylor expanded in m around inf 98.4%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv98.3%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. unpow398.3%

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

      3. associate-*l*98.3%

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

      4. div-inv98.4%

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

   6. Applied egg-rr98.4%

      \[\leadsto \color{blue}{\left(m \cdot m\right) \cdot \frac{m}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(\frac{m}{v} - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 0.38

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg98.0%

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

      2. distribute-lft-in98.0%

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

      3. *-commutative98.0%

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

      4. *-un-lft-identity98.0%

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

      5. sub-neg98.0%

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

      6. metadata-eval98.0%

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

      7. sub-neg98.0%

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

      8. metadata-eval98.0%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod97.8%

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

      11. sqr-neg97.8%

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

      12. sqrt-unprod97.8%

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

      13. add-sqr-sqrt97.8%

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

   4. Applied egg-rr97.8%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative97.8%

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

      2. distribute-rgt1-in97.8%

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

   6. Simplified97.8%

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

   7. Taylor expanded in m around 0 97.7%

      \[\leadsto \color{blue}{\left(\frac{1}{v} - 1\right) \cdot m - 1} \]
   8. Step-by-step derivation

      1. sub-neg97.7%

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

      2. *-commutative97.7%

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

      3. sub-neg97.7%

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

      4. metadata-eval97.7%

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

      5. distribute-rgt-in97.7%

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

      6. associate-*l/97.9%

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

      7. *-lft-identity97.9%

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

      8. neg-mul-197.9%

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

      9. sub-neg97.9%

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

      10. metadata-eval97.9%

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

   9. Simplified97.9%

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

   if 0.38 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative99.9%

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

      2. sub-neg99.9%

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

      3. associate-*l/99.9%

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

      4. metadata-eval99.9%

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

   3. Simplified99.9%

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

   4. Taylor expanded in m around inf 98.4%

      \[\leadsto \color{blue}{\frac{{m}^{3}}{v}} \]
   5. Step-by-step derivation

      1. div-inv98.3%

         \[\leadsto \color{blue}{{m}^{3} \cdot \frac{1}{v}} \]

      2. cube-mult98.3%

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

      3. associate-*l*98.3%

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

      4. associate-*r*98.3%

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

      5. div-inv98.4%

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

   6. Applied egg-rr98.4%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{m}{v}\right)} \]
   7. Step-by-step derivation

      1. associate-*r*98.4%

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

      2. clear-num98.4%

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

      3. div-inv98.4%

         \[\leadsto \color{blue}{\frac{m \cdot m}{\frac{v}{m}}} \]

   8. Applied egg-rr98.4%

      \[\leadsto \color{blue}{\frac{m \cdot m}{\frac{v}{m}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \left(\frac{m}{v} - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 1.6500000000000001e-28

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   2. Step-by-step derivation

      1. *-commutative100.0%

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

      2. sub-neg100.0%

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

      3. associate-*l/100.0%

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

      4. metadata-eval100.0%

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

   3. Simplified100.0%

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

   4. Taylor expanded in m around 0 47.6%

      \[\leadsto \color{blue}{-1} \]

   if 1.6500000000000001e-28 < m

   1. Initial program 99.8%

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

   2. Taylor expanded in m around 0 6.0%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg6.0%

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

      2. distribute-lft-in6.0%

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

      3. *-commutative6.0%

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

      4. *-un-lft-identity6.0%

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

      5. sub-neg6.0%

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

      6. metadata-eval6.0%

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

      7. sub-neg6.0%

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

      8. metadata-eval6.0%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod72.6%

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

      11. sqr-neg72.6%

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

      12. sqrt-unprod72.6%

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

      13. add-sqr-sqrt72.6%

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

   4. Applied egg-rr72.6%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative72.6%

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

      2. distribute-rgt1-in72.6%

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

   6. Simplified72.6%

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

   7. Taylor expanded in m around inf 67.6%

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   8. Step-by-step derivation

      1. unpow267.6%

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \]

      2. associate-*r/67.6%

         \[\leadsto \color{blue}{m \cdot \frac{m}{v}} \]

   9. Simplified67.6%

      \[\leadsto \color{blue}{m \cdot \frac{m}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.65 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

1. Split input into 2 regimes

2. if m < 0.27000000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 98.0%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg98.0%

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

      2. distribute-lft-in98.0%

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

      3. *-commutative98.0%

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

      4. *-un-lft-identity98.0%

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

      5. sub-neg98.0%

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

      6. metadata-eval98.0%

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

      7. sub-neg98.0%

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

      8. metadata-eval98.0%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod97.8%

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

      11. sqr-neg97.8%

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

      12. sqrt-unprod97.8%

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

      13. add-sqr-sqrt97.8%

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

   4. Applied egg-rr97.8%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative97.8%

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

      2. distribute-rgt1-in97.8%

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

   6. Simplified97.8%

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

   7. Taylor expanded in m around 0 97.7%

      \[\leadsto \color{blue}{\left(\frac{1}{v} - 1\right) \cdot m - 1} \]
   8. Step-by-step derivation

      1. sub-neg97.7%

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

      2. *-commutative97.7%

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

      3. sub-neg97.7%

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

      4. metadata-eval97.7%

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

      5. distribute-rgt-in97.7%

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

      6. associate-*l/97.9%

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

      7. *-lft-identity97.9%

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

      8. neg-mul-197.9%

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

      9. sub-neg97.9%

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

      10. metadata-eval97.9%

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

   9. Simplified97.9%

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

   10. Taylor expanded in v around 0 97.9%

       \[\leadsto \color{blue}{\frac{m}{v}} + -1 \]

   if 0.27000000000000002 < m

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 0.1%

      \[\leadsto \left(\frac{\color{blue}{m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   3. Step-by-step derivation

      1. sub-neg0.1%

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

      2. distribute-lft-in0.1%

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

      3. *-commutative0.1%

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

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

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

      5. sub-neg0.1%

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

      6. metadata-eval0.1%

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

      7. sub-neg0.1%

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

      8. metadata-eval0.1%

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

      9. add-sqr-sqrt0.0%

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

      10. sqrt-unprod72.4%

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

      11. sqr-neg72.4%

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

      12. sqrt-unprod72.4%

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

      13. add-sqr-sqrt72.4%

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

   4. Applied egg-rr72.4%

      \[\leadsto \color{blue}{\left(\frac{m}{v} + -1\right) + \left(\frac{m}{v} + -1\right) \cdot m} \]
   5. Step-by-step derivation

      1. *-commutative72.4%

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

      2. distribute-rgt1-in72.4%

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

   6. Simplified72.4%

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

   7. Taylor expanded in m around inf 72.4%

      \[\leadsto \color{blue}{\frac{{m}^{2}}{v}} \]
   8. Step-by-step derivation

      1. unpow272.4%

         \[\leadsto \frac{\color{blue}{m \cdot m}}{v} \]

      2. associate-*r/72.4%

         \[\leadsto \color{blue}{m \cdot \frac{m}{v}} \]

   9. Simplified72.4%

      \[\leadsto \color{blue}{m \cdot \frac{m}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.27:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative99.9%

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

   2. sub-neg99.9%

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

   3. associate-*l/100.0%

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

   4. metadata-eval100.0%

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

3. Simplified100.0%

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

4. Taylor expanded in v around inf 24.0%

   \[\leadsto \color{blue}{-1 \cdot \left(1 - m\right)} \]
5. Step-by-step derivation

   1. neg-mul-124.0%

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

   2. neg-sub024.0%

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

   3. associate--r-24.0%

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

   4. metadata-eval24.0%

      \[\leadsto \color{blue}{-1} + m \]

6. Simplified24.0%

   \[\leadsto \color{blue}{-1 + m} \]

7. Final simplification24.0%

   \[\leadsto m + -1 \]

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
2. Step-by-step derivation

   1. *-commutative99.9%

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

   2. sub-neg99.9%

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

   3. associate-*l/100.0%

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

   4. metadata-eval100.0%

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

3. Simplified100.0%

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

4. Taylor expanded in m around 0 21.4%

   \[\leadsto \color{blue}{-1} \]

5. Final simplification21.4%

   \[\leadsto -1 \]

herbie shell --seed 2023166 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))