1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Step-by-step derivation

   1. distribute-lft-in99.9%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

3. Applied egg-rr99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

4. Final simplification99.9%

   \[\leadsto x \cdot 0.5 + \left(y \cdot \left(1 - z\right) + y \cdot \log z\right) \]

1. Split input into 2 regimes

2. if y < -5.7999999999999999e63 or 4.5999999999999996e25 < y

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. sub-neg99.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(\color{blue}{\left(1 + \left(-z\right)\right)} + \log z\right) \]

      2. associate-+l+99.8%

         \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(1 + \left(\left(-z\right) + \log z\right)\right)} \]

      3. distribute-lft-in99.7%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot 1 + y \cdot \left(\left(-z\right) + \log z\right)\right)} \]

      4. *-rgt-identity99.7%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{y} + y \cdot \left(\left(-z\right) + \log z\right)\right) \]

      5. associate-+r+99.7%

         \[\leadsto \color{blue}{\left(x \cdot 0.5 + y\right) + y \cdot \left(\left(-z\right) + \log z\right)} \]

      6. fma-def99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y\right)} + y \cdot \left(\left(-z\right) + \log z\right) \]

      7. +-commutative99.7%

         \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + y \cdot \color{blue}{\left(\log z + \left(-z\right)\right)} \]

      8. unsub-neg99.7%

         \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + y \cdot \color{blue}{\left(\log z - z\right)} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y\right) + y \cdot \left(\log z - z\right)} \]

   4. Taylor expanded in x around 0 89.8%

      \[\leadsto \color{blue}{y + y \cdot \left(\log z - z\right)} \]
   5. Step-by-step derivation

      1. *-commutative89.8%

         \[\leadsto y + \color{blue}{\left(\log z - z\right) \cdot y} \]

      2. distribute-rgt1-in89.9%

         \[\leadsto \color{blue}{\left(\left(\log z - z\right) + 1\right) \cdot y} \]

      3. associate-+l-89.9%

         \[\leadsto \color{blue}{\left(\log z - \left(z - 1\right)\right)} \cdot y \]

   6. Applied egg-rr89.9%

      \[\leadsto \color{blue}{\left(\log z - \left(z - 1\right)\right) \cdot y} \]

   if -5.7999999999999999e63 < y < 4.5999999999999996e25

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in100.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   3. Applied egg-rr100.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Taylor expanded in z around inf 90.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   5. Step-by-step derivation

      1. associate-*r*90.3%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

      2. neg-mul-190.3%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right)} \cdot z \]

   6. Simplified90.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right) \cdot z} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out90.3%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y \cdot z\right)} \]

      2. unsub-neg90.3%

         \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]

   8. Applied egg-rr90.3%

      \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+63} \lor \neg \left(y \leq 4.6 \cdot 10^{+25}\right):\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

   2. Taylor expanded in z around 0 97.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(1 + \log z\right) \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative97.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(1 + \log z\right)} \]

      2. distribute-lft-in97.6%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot 1 + y \cdot \log z\right)} \]

      3. *-rgt-identity97.6%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{y} + y \cdot \log z\right) \]

   4. Simplified97.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + y \cdot \log z\right)} \]

   if 0.28000000000000003 < z

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in99.9%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   3. Applied egg-rr99.9%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

   4. Taylor expanded in z around inf 98.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   5. Step-by-step derivation

      1. associate-*r*98.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

      2. neg-mul-198.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right)} \cdot z \]

   6. Simplified98.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right) \cdot z} \]
   7. Step-by-step derivation

      1. distribute-lft-neg-out98.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y \cdot z\right)} \]

      2. unsub-neg98.0%

         \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]

   8. Applied egg-rr98.0%

      \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + \left(y + y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

2. Final simplification99.9%

   \[\leadsto x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Step-by-step derivation

   1. distribute-lft-in99.9%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

3. Applied egg-rr99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot \left(1 - z\right) + y \cdot \log z\right)} \]

4. Taylor expanded in z around inf 77.3%

   \[\leadsto x \cdot 0.5 + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
5. Step-by-step derivation

   1. associate-*r*77.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

   2. neg-mul-177.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right)} \cdot z \]

6. Simplified77.3%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y\right) \cdot z} \]
7. Step-by-step derivation

   1. distribute-lft-neg-out77.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(-y \cdot z\right)} \]

   2. unsub-neg77.3%

      \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]

8. Applied egg-rr77.3%

   \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]

9. Final simplification77.3%

   \[\leadsto x \cdot 0.5 - y \cdot z \]

1. Split input into 2 regimes

2. if z < 5.2e16

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

   2. Taylor expanded in x around inf 54.6%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 5.2e16 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. sub-neg100.0%

         \[\leadsto x \cdot 0.5 + y \cdot \left(\color{blue}{\left(1 + \left(-z\right)\right)} + \log z\right) \]

      2. associate-+l+100.0%

         \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(1 + \left(\left(-z\right) + \log z\right)\right)} \]

      3. distribute-lft-in100.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(y \cdot 1 + y \cdot \left(\left(-z\right) + \log z\right)\right)} \]

      4. *-rgt-identity100.0%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{y} + y \cdot \left(\left(-z\right) + \log z\right)\right) \]

      5. associate-+r+100.0%

         \[\leadsto \color{blue}{\left(x \cdot 0.5 + y\right) + y \cdot \left(\left(-z\right) + \log z\right)} \]

      6. fma-def100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y\right)} + y \cdot \left(\left(-z\right) + \log z\right) \]

      7. +-commutative100.0%

         \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + y \cdot \color{blue}{\left(\log z + \left(-z\right)\right)} \]

      8. unsub-neg100.0%

         \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + y \cdot \color{blue}{\left(\log z - z\right)} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y\right) + y \cdot \left(\log z - z\right)} \]

   4. Taylor expanded in y around -inf 75.1%

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

      1. mul-1-neg75.1%

         \[\leadsto \color{blue}{-y \cdot \left(-1 \cdot \left(\log z - z\right) - 1\right)} \]

      2. distribute-rgt-neg-in75.1%

         \[\leadsto \color{blue}{y \cdot \left(-\left(-1 \cdot \left(\log z - z\right) - 1\right)\right)} \]

      3. sub-neg75.1%

         \[\leadsto y \cdot \left(-\color{blue}{\left(-1 \cdot \left(\log z - z\right) + \left(-1\right)\right)}\right) \]

      4. mul-1-neg75.1%

         \[\leadsto y \cdot \left(-\left(\color{blue}{\left(-\left(\log z - z\right)\right)} + \left(-1\right)\right)\right) \]

      5. sub-neg75.1%

         \[\leadsto y \cdot \left(-\left(\left(-\color{blue}{\left(\log z + \left(-z\right)\right)}\right) + \left(-1\right)\right)\right) \]

      6. +-commutative75.1%

         \[\leadsto y \cdot \left(-\left(\left(-\color{blue}{\left(\left(-z\right) + \log z\right)}\right) + \left(-1\right)\right)\right) \]

      7. distribute-neg-in75.1%

         \[\leadsto y \cdot \left(-\left(\color{blue}{\left(\left(-\left(-z\right)\right) + \left(-\log z\right)\right)} + \left(-1\right)\right)\right) \]

      8. remove-double-neg75.1%

         \[\leadsto y \cdot \left(-\left(\left(\color{blue}{z} + \left(-\log z\right)\right) + \left(-1\right)\right)\right) \]

      9. sub-neg75.1%

         \[\leadsto y \cdot \left(-\left(\color{blue}{\left(z - \log z\right)} + \left(-1\right)\right)\right) \]

      10. metadata-eval75.1%

          \[\leadsto y \cdot \left(-\left(\left(z - \log z\right) + \color{blue}{-1}\right)\right) \]

      11. +-commutative75.1%

          \[\leadsto y \cdot \left(-\color{blue}{\left(-1 + \left(z - \log z\right)\right)}\right) \]

   6. Simplified75.1%

      \[\leadsto \color{blue}{y \cdot \left(-\left(-1 + \left(z - \log z\right)\right)\right)} \]

   7. Taylor expanded in z around inf 75.1%

      \[\leadsto y \cdot \left(-\color{blue}{z}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

2. Taylor expanded in x around inf 41.2%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

3. Final simplification41.2%

   \[\leadsto x \cdot 0.5 \]

herbie shell --seed 2023166 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))