\[\sqrt{re \cdot re + im \cdot im}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.836812280349736 \cdot 10^{+127}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le -5.6974351952720775 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 1.0892602907625438 \cdot 10^{-235}:\\ \;\;\;\;im\\ \mathbf{if}\;re \le 2.8288165587341483 \cdot 10^{-187}:\\ \;\;\;\;{\left(\frac{-1}{im}\right)}^{-1}\\ \mathbf{if}\;re \le 1.23303634520905 \cdot 10^{-159}:\\ \;\;\;\;im\\ \mathbf{if}\;re \le 1.2688591977550717 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Derivation

Split input into 5 regimes
if re < -3.836812280349736e+127
1. Initial program 52.9
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around -inf 8.8
  \[\leadsto \color{blue}{-1 \cdot re}\]
3. Applied simplify8.8
  \[\leadsto \color{blue}{-re}\]
if -3.836812280349736e+127 < re < -5.6974351952720775e-213 or 1.23303634520905e-159 < re < 1.2688591977550717e+154
1. Initial program 16.6
  \[\sqrt{re \cdot re + im \cdot im}\]
if -5.6974351952720775e-213 < re < 1.0892602907625438e-235 or 2.8288165587341483e-187 < re < 1.23303634520905e-159
1. Initial program 28.2
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around 0 34.0
  \[\leadsto \color{blue}{im}\]
if 1.0892602907625438e-235 < re < 2.8288165587341483e-187
1. Initial program 30.6
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Using strategy rm
3. Applied add-cbrt-cube42.3
  \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}\right) \cdot \sqrt{re \cdot re + im \cdot im}}}\]
4. Applied simplify42.3
  \[\leadsto \sqrt[3]{\color{blue}{\left(im \cdot im + re \cdot re\right) \cdot \sqrt{im \cdot im + re \cdot re}}}\]
5. Taylor expanded around -inf 38.0
  \[\leadsto \color{blue}{{\left(\frac{-1}{im}\right)}^{-1}}\]
if 1.2688591977550717e+154 < re
1. Initial program 59.4
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around inf 7.6
  \[\leadsto \color{blue}{re}\]
Recombined 5 regimes into one program.

Error

Derivation

`if re < -3.836812280349736e+127`

`if -3.836812280349736e+127 < re < -5.6974351952720775e-213 or 1.23303634520905e-159 < re < 1.2688591977550717e+154`

`if -5.6974351952720775e-213 < re < 1.0892602907625438e-235 or 2.8288165587341483e-187 < re < 1.23303634520905e-159`

`if 1.0892602907625438e-235 < re < 2.8288165587341483e-187`

`if 1.2688591977550717e+154 < re`

Runtime