1. Initial program 99.9%

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

   1. associate--l+99.9%

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

   2. sub-neg99.9%

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

   3. +-commutative99.9%

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

   4. associate-+l+99.9%

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

   5. *-commutative99.9%

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

   6. distribute-rgt-neg-in99.9%

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

   7. fma-def99.9%

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

   8. neg-sub099.9%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

   9. +-commutative99.9%

      \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

   10. associate--r+99.9%

       \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

   11. metadata-eval99.9%

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

   12. associate-+r-99.9%

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

   13. +-commutative99.9%

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

   14. associate-+r-99.9%

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

3. Simplified99.9%

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

4. Final simplification99.9%

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

1. Split input into 4 regimes

2. if x < -1.35e15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 89.3%

      \[\leadsto \color{blue}{x} - z \]

   if -1.35e15 < x < 3.79999999999999997e-222 or 1.85e-70 < x < 8.5e43

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 81.1%

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

      1. *-commutative81.1%

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

      2. log-rec81.1%

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

      3. cancel-sign-sub81.1%

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

      4. *-commutative81.1%

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

      5. mul-1-neg81.1%

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

      6. log-rec81.1%

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

      7. log-rec81.1%

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

      8. sub-neg81.1%

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

   4. Simplified81.1%

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

   if 3.79999999999999997e-222 < x < 1.85e-70

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 79.1%

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

   3. Taylor expanded in x around 0 79.1%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   4. Step-by-step derivation

      1. *-commutative79.1%

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

   5. Simplified79.1%

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

   if 8.5e43 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg99.9%

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

      2. distribute-rgt-neg-in99.9%

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

      3. log-rec99.9%

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

      4. remove-double-neg99.9%

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

   4. Simplified99.9%

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

      1. associate--l+99.9%

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

      2. add-cube-cbrt97.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}\right) \cdot \sqrt[3]{x - y \cdot \log y}} + \left(y - z\right) \]

      3. fma-def97.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

      4. pow297.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}}, \sqrt[3]{x - y \cdot \log y}, y - z\right) \]

   6. Applied egg-rr97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

   7. Taylor expanded in z around 0 92.3%

      \[\leadsto \color{blue}{y + {1}^{0.3333333333333333} \cdot \left(x - y \cdot \log y\right)} \]
   8. Step-by-step derivation

      1. pow-base-192.3%

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

      2. *-lft-identity92.3%

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

   9. Simplified92.3%

      \[\leadsto \color{blue}{y + \left(x - y \cdot \log y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+15}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-70}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+43}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if y < 3e15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.0%

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

   if 3e15 < y < 1.40000000000000001e74 or 5.80000000000000019e138 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg99.7%

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

      2. distribute-rgt-neg-in99.7%

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

      3. log-rec99.7%

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

      4. remove-double-neg99.7%

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

   4. Simplified99.7%

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

      1. associate--l+99.7%

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

      2. add-cube-cbrt98.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}\right) \cdot \sqrt[3]{x - y \cdot \log y}} + \left(y - z\right) \]

      3. fma-def98.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

      4. pow298.2%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}}, \sqrt[3]{x - y \cdot \log y}, y - z\right) \]

   6. Applied egg-rr98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

   7. Taylor expanded in z around 0 88.2%

      \[\leadsto \color{blue}{y + {1}^{0.3333333333333333} \cdot \left(x - y \cdot \log y\right)} \]
   8. Step-by-step derivation

      1. pow-base-188.2%

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

      2. *-lft-identity88.2%

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

   9. Simplified88.2%

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

   if 1.40000000000000001e74 < y < 5.80000000000000019e138

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 80.4%

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

      1. *-commutative80.4%

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

      2. log-rec80.4%

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

      3. cancel-sign-sub80.4%

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

      4. *-commutative80.4%

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

      5. mul-1-neg80.4%

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

      6. log-rec80.4%

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

      7. log-rec80.4%

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

      8. sub-neg80.4%

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

   4. Simplified80.4%

      \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} - z \]
3. Recombined 3 regimes into one program.

4. Final simplification93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+74} \lor \neg \left(y \leq 5.8 \cdot 10^{+138}\right):\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

1. Split input into 2 regimes

2. if z < -7.4999999999999995e75 or 4.15000000000000011e36 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 85.0%

      \[\leadsto \color{blue}{x} - z \]

   if -7.4999999999999995e75 < z < 4.15000000000000011e36

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 78.8%

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

      1. mul-1-neg78.8%

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

      2. distribute-rgt-neg-in78.8%

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

      3. log-rec78.8%

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

      4. remove-double-neg78.8%

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

   4. Simplified78.8%

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

      1. associate--l+78.8%

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

      2. add-cube-cbrt77.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}\right) \cdot \sqrt[3]{x - y \cdot \log y}} + \left(y - z\right) \]

      3. fma-def77.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

      4. pow277.5%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}}, \sqrt[3]{x - y \cdot \log y}, y - z\right) \]

   6. Applied egg-rr77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

   7. Taylor expanded in z around 0 77.5%

      \[\leadsto \color{blue}{y + {1}^{0.3333333333333333} \cdot \left(x - y \cdot \log y\right)} \]
   8. Step-by-step derivation

      1. pow-base-177.5%

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

      2. *-lft-identity77.5%

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

   9. Simplified77.5%

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

4. Final simplification80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+75}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 4.15 \cdot 10^{+36}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

   if 0.28000000000000003 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 99.4%

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

      1. mul-1-neg99.4%

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

      2. distribute-rgt-neg-in99.4%

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

      3. log-rec99.4%

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

      4. remove-double-neg99.4%

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

   4. Simplified99.4%

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

4. Final simplification99.2%

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

1. Initial program 99.9%

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

2. Final simplification99.9%

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

1. Initial program 99.9%

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

   1. +-commutative99.9%

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

   2. associate-+r-99.9%

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

3. Applied egg-rr99.9%

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

4. Final simplification99.9%

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

1. Split input into 2 regimes

2. if x < -44 or 1.3e15 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 84.7%

      \[\leadsto \color{blue}{x} - z \]

   if -44 < x < 1.3e15

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 66.3%

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

   3. Taylor expanded in x around 0 65.4%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   4. Step-by-step derivation

      1. *-commutative65.4%

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

   5. Simplified65.4%

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

4. Final simplification75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -44:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

1. Initial program 99.9%

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

2. Taylor expanded in x around inf 63.2%

   \[\leadsto \color{blue}{x} - z \]

3. Final simplification63.2%

   \[\leadsto x - z \]

1. Initial program 99.9%

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

   1. associate--l+99.9%

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

   2. sub-neg99.9%

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

   3. +-commutative99.9%

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

   4. associate-+l+99.9%

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

   5. *-commutative99.9%

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

   6. distribute-rgt-neg-in99.9%

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

   7. fma-def99.9%

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

   8. neg-sub099.9%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

   9. +-commutative99.9%

      \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

   10. associate--r+99.9%

       \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

   11. metadata-eval99.9%

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

   12. associate-+r-99.9%

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

   13. +-commutative99.9%

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

   14. associate-+r-99.9%

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

3. Simplified99.9%

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

4. Taylor expanded in z around inf 29.9%

   \[\leadsto \color{blue}{-1 \cdot z} \]
5. Step-by-step derivation

   1. mul-1-neg29.9%

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

6. Simplified29.9%

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

7. Final simplification29.9%

   \[\leadsto -z \]

1. Initial program 99.9%

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

2. Taylor expanded in y around inf 87.2%

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

   1. mul-1-neg87.2%

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

   2. distribute-rgt-neg-in87.2%

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

   3. log-rec87.2%

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

   4. remove-double-neg87.2%

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

4. Simplified87.2%

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

   1. associate--l+87.2%

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

   2. add-cube-cbrt86.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}\right) \cdot \sqrt[3]{x - y \cdot \log y}} + \left(y - z\right) \]

   3. fma-def86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x - y \cdot \log y} \cdot \sqrt[3]{x - y \cdot \log y}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

   4. pow286.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}}, \sqrt[3]{x - y \cdot \log y}, y - z\right) \]

6. Applied egg-rr86.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x - y \cdot \log y}\right)}^{2}, \sqrt[3]{x - y \cdot \log y}, y - z\right)} \]

7. Taylor expanded in y around inf 2.6%

   \[\leadsto \color{blue}{y} \]

8. Final simplification2.6%

   \[\leadsto y \]

herbie shell --seed 2023166 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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