math.sqrt on complex, imaginary part, im greater than 0 branch

1. Split input into 2 regimes

2. if re < 2.6999999999999997e27

   1. Initial program 48.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. hypot-def92.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right)} \]

   3. Simplified92.6%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   if 2.6999999999999997e27 < re

   1. Initial program 6.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around inf 52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow252.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   4. Simplified52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]

   5. Taylor expanded in im around 0 83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
   6. Step-by-step derivation

      1. *-commutative83.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. unpow1/283.2%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{\left(\frac{1}{re}\right)}^{0.5}}\right) \]

      3. unpow-183.2%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left({re}^{-1}\right)}}^{0.5}\right) \]

      4. exp-to-pow78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-178.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval78.0%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow83.2%

          \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]

   7. Simplified83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.7 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

1. Split input into 5 regimes

2. if re < -2.19999999999999987e68

   1. Initial program 24.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around -inf 81.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative81.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   4. Simplified81.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   if -2.19999999999999987e68 < re < 6.1000000000000001e-33

   1. Initial program 60.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around 0 80.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)} \]

   if 6.1000000000000001e-33 < re < 2.8e12

   1. Initial program 14.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around inf 43.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow243.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   4. Simplified43.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u43.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)}\right)\right)} \]

      2. expm1-udef5.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)}\right)} - 1\right)} \]

      3. associate-*r*5.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{im \cdot im}{re}}}\right)} - 1\right) \]

      4. metadata-eval5.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{1} \cdot \frac{im \cdot im}{re}}\right)} - 1\right) \]

      5. *-un-lft-identity5.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im \cdot im}{re}}}\right)} - 1\right) \]

      6. sqrt-div5.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{im \cdot im}}{\sqrt{re}}}\right)} - 1\right) \]

      7. sqrt-prod5.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{re}}\right)} - 1\right) \]

      8. add-sqr-sqrt5.3%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{re}}\right)} - 1\right) \]

   6. Applied egg-rr5.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def72.1%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p72.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified72.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   if 2.8e12 < re < 1.2200000000000001e27

   1. Initial program 4.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. *-commutative4.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2}} \]

      2. hypot-udef100.0%

         \[\leadsto 0.5 \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2} \]

      3. *-commutative100.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. add-sqr-sqrt99.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]

      5. pow299.1%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)}^{2}} \]

      6. pow1/299.1%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}}\right)}^{2} \]

      7. sqrt-pow199.1%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      8. metadata-eval99.1%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr99.1%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in re around 0 99.1%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

   5. Taylor expanded in im around 0 89.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + \log im\right)}\right)}^{2}} \]
   6. Step-by-step derivation

      1. log-prod89.9%

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)}}\right)}^{2} \]

      2. *-commutative89.9%

         \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{\log \left(2 \cdot im\right) \cdot 0.25}}\right)}^{2} \]

      3. exp-to-pow99.1%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

      4. unpow299.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left({\left(2 \cdot im\right)}^{0.25} \cdot {\left(2 \cdot im\right)}^{0.25}\right)} \]

      5. pow-sqr100.0%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot im\right)}^{\left(2 \cdot 0.25\right)}} \]

      6. metadata-eval100.0%

         \[\leadsto 0.5 \cdot {\left(2 \cdot im\right)}^{\color{blue}{0.5}} \]

      7. unpow1/2100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   7. Simplified100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   if 1.2200000000000001e27 < re

   1. Initial program 6.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around inf 52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow252.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   4. Simplified52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]

   5. Taylor expanded in im around 0 83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
   6. Step-by-step derivation

      1. *-commutative83.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. unpow1/283.2%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{\left(\frac{1}{re}\right)}^{0.5}}\right) \]

      3. unpow-183.2%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left({re}^{-1}\right)}}^{0.5}\right) \]

      4. exp-to-pow78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-178.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval78.0%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow83.2%

          \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]

   7. Simplified83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.1 \cdot 10^{-33}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2800000000000:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.22 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if re < -3.69999999999999998e68

   1. Initial program 24.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around -inf 81.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}} \]
   3. Step-by-step derivation

      1. *-commutative81.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   4. Simplified81.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re \cdot -2\right)}} \]

   if -3.69999999999999998e68 < re < 3.69999999999999988e26

   1. Initial program 55.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. *-commutative55.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2}} \]

      2. hypot-udef90.4%

         \[\leadsto 0.5 \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2} \]

      3. *-commutative90.4%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. add-sqr-sqrt89.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]

      5. pow289.7%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)}^{2}} \]

      6. pow1/289.7%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}}\right)}^{2} \]

      7. sqrt-pow189.7%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      8. metadata-eval89.7%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr89.7%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in re around 0 75.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

   5. Taylor expanded in im around 0 70.7%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + \log im\right)}\right)}^{2}} \]
   6. Step-by-step derivation

      1. log-prod71.0%

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)}}\right)}^{2} \]

      2. *-commutative71.0%

         \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{\log \left(2 \cdot im\right) \cdot 0.25}}\right)}^{2} \]

      3. exp-to-pow75.8%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

      4. unpow275.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left({\left(2 \cdot im\right)}^{0.25} \cdot {\left(2 \cdot im\right)}^{0.25}\right)} \]

      5. pow-sqr76.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot im\right)}^{\left(2 \cdot 0.25\right)}} \]

      6. metadata-eval76.4%

         \[\leadsto 0.5 \cdot {\left(2 \cdot im\right)}^{\color{blue}{0.5}} \]

      7. unpow1/276.4%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   7. Simplified76.4%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   if 3.69999999999999988e26 < re

   1. Initial program 6.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around inf 52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow252.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   4. Simplified52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]

   5. Taylor expanded in im around 0 83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
   6. Step-by-step derivation

      1. *-commutative83.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. unpow1/283.2%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{\left(\frac{1}{re}\right)}^{0.5}}\right) \]

      3. unpow-183.2%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left({re}^{-1}\right)}}^{0.5}\right) \]

      4. exp-to-pow78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-178.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval78.0%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow83.2%

          \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]

   7. Simplified83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.7 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if re < 1.19999999999999999e27

   1. Initial program 48.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. *-commutative48.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2}} \]

      2. hypot-udef92.6%

         \[\leadsto 0.5 \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2} \]

      3. *-commutative92.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. add-sqr-sqrt91.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]

      5. pow291.9%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)}^{2}} \]

      6. pow1/291.9%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}}\right)}^{2} \]

      7. sqrt-pow191.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      8. metadata-eval91.9%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr91.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in re around 0 63.7%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

   5. Taylor expanded in im around 0 59.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + \log im\right)}\right)}^{2}} \]
   6. Step-by-step derivation

      1. log-prod59.7%

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)}}\right)}^{2} \]

      2. *-commutative59.7%

         \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{\log \left(2 \cdot im\right) \cdot 0.25}}\right)}^{2} \]

      3. exp-to-pow63.7%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

      4. unpow263.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left({\left(2 \cdot im\right)}^{0.25} \cdot {\left(2 \cdot im\right)}^{0.25}\right)} \]

      5. pow-sqr64.1%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot im\right)}^{\left(2 \cdot 0.25\right)}} \]

      6. metadata-eval64.1%

         \[\leadsto 0.5 \cdot {\left(2 \cdot im\right)}^{\color{blue}{0.5}} \]

      7. unpow1/264.1%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   7. Simplified64.1%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   if 1.19999999999999999e27 < re

   1. Initial program 6.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around inf 52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow252.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   4. Simplified52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]

   5. Taylor expanded in im around 0 83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
   6. Step-by-step derivation

      1. *-commutative83.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]

      2. unpow1/283.2%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{\left(\frac{1}{re}\right)}^{0.5}}\right) \]

      3. unpow-183.2%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left({re}^{-1}\right)}}^{0.5}\right) \]

      4. exp-to-pow78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\color{blue}{\left(e^{\log re \cdot -1}\right)}}^{0.5}\right) \]

      5. *-commutative78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-1 \cdot \log re}}\right)}^{0.5}\right) \]

      6. neg-mul-178.0%

         \[\leadsto 0.5 \cdot \left(im \cdot {\left(e^{\color{blue}{-\log re}}\right)}^{0.5}\right) \]

      7. exp-prod78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{e^{\left(-\log re\right) \cdot 0.5}}\right) \]

      8. distribute-lft-neg-out78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{-\log re \cdot 0.5}}\right) \]

      9. distribute-rgt-neg-in78.0%

         \[\leadsto 0.5 \cdot \left(im \cdot e^{\color{blue}{\log re \cdot \left(-0.5\right)}}\right) \]

      10. metadata-eval78.0%

          \[\leadsto 0.5 \cdot \left(im \cdot e^{\log re \cdot \color{blue}{-0.5}}\right) \]

      11. exp-to-pow83.2%

          \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{{re}^{-0.5}}\right) \]

   7. Simplified83.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.2 \cdot 10^{+27}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if re < 6.2000000000000001e28

   1. Initial program 48.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   2. Step-by-step derivation

      1. *-commutative48.2%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2}} \]

      2. hypot-udef92.6%

         \[\leadsto 0.5 \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2} \]

      3. *-commutative92.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

      4. add-sqr-sqrt91.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]

      5. pow291.9%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)}^{2}} \]

      6. pow1/291.9%

         \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}}\right)}^{2} \]

      7. sqrt-pow191.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      8. metadata-eval91.9%

         \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   3. Applied egg-rr91.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in re around 0 63.7%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

   5. Taylor expanded in im around 0 59.4%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + \log im\right)}\right)}^{2}} \]
   6. Step-by-step derivation

      1. log-prod59.7%

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)}}\right)}^{2} \]

      2. *-commutative59.7%

         \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{\log \left(2 \cdot im\right) \cdot 0.25}}\right)}^{2} \]

      3. exp-to-pow63.7%

         \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

      4. unpow263.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left({\left(2 \cdot im\right)}^{0.25} \cdot {\left(2 \cdot im\right)}^{0.25}\right)} \]

      5. pow-sqr64.1%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot im\right)}^{\left(2 \cdot 0.25\right)}} \]

      6. metadata-eval64.1%

         \[\leadsto 0.5 \cdot {\left(2 \cdot im\right)}^{\color{blue}{0.5}} \]

      7. unpow1/264.1%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   7. Simplified64.1%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

   if 6.2000000000000001e28 < re

   1. Initial program 6.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]

   2. Taylor expanded in re around inf 52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{{im}^{2}}{re}\right)}} \]
   3. Step-by-step derivation

      1. unpow252.7%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{im \cdot im}}{re}\right)} \]

   4. Simplified52.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(0.5 \cdot \frac{im \cdot im}{re}\right)}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u52.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)}\right)\right)} \]

      2. expm1-udef26.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(0.5 \cdot \frac{im \cdot im}{re}\right)}\right)} - 1\right)} \]

      3. associate-*r*26.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(2 \cdot 0.5\right) \cdot \frac{im \cdot im}{re}}}\right)} - 1\right) \]

      4. metadata-eval26.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{1} \cdot \frac{im \cdot im}{re}}\right)} - 1\right) \]

      5. *-un-lft-identity26.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im \cdot im}{re}}}\right)} - 1\right) \]

      6. sqrt-div26.4%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{im \cdot im}}{\sqrt{re}}}\right)} - 1\right) \]

      7. sqrt-prod35.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{re}}\right)} - 1\right) \]

      8. add-sqr-sqrt35.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{im}}{\sqrt{re}}\right)} - 1\right) \]

   6. Applied egg-rr35.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def82.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p83.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]

   8. Simplified83.1%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\sqrt{re}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

1. Initial program 39.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
2. Step-by-step derivation

   1. *-commutative39.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} - re\right) \cdot 2}} \]

   2. hypot-udef81.6%

      \[\leadsto 0.5 \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(re, im\right)} - re\right) \cdot 2} \]

   3. *-commutative81.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \]

   4. add-sqr-sqrt81.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}} \cdot \sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)} \]

   5. pow281.0%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}}\right)}^{2}} \]

   6. pow1/281.0%

      \[\leadsto 0.5 \cdot {\left(\sqrt{\color{blue}{{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.5}}}\right)}^{2} \]

   7. sqrt-pow181.0%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

   8. metadata-eval81.0%

      \[\leadsto 0.5 \cdot {\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \]

3. Applied egg-rr81.0%

   \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

4. Taylor expanded in re around 0 54.6%

   \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

5. Taylor expanded in im around 0 50.9%

   \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + \log im\right)}\right)}^{2}} \]
6. Step-by-step derivation

   1. log-prod51.2%

      \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)}}\right)}^{2} \]

   2. *-commutative51.2%

      \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{\log \left(2 \cdot im\right) \cdot 0.25}}\right)}^{2} \]

   3. exp-to-pow54.6%

      \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(2 \cdot im\right)}^{0.25}\right)}}^{2} \]

   4. unpow254.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left({\left(2 \cdot im\right)}^{0.25} \cdot {\left(2 \cdot im\right)}^{0.25}\right)} \]

   5. pow-sqr55.0%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(2 \cdot im\right)}^{\left(2 \cdot 0.25\right)}} \]

   6. metadata-eval55.0%

      \[\leadsto 0.5 \cdot {\left(2 \cdot im\right)}^{\color{blue}{0.5}} \]

   7. unpow1/255.0%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

7. Simplified55.0%

   \[\leadsto 0.5 \cdot \color{blue}{\sqrt{2 \cdot im}} \]

8. Final simplification55.0%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]

herbie shell --seed 2023166 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))