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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.9133903609356435 \cdot 10^{+137}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le -1.4143330855213813 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 1.5363993963951212 \cdot 10^{-139}:\\ \;\;\;\;im\\ \mathbf{if}\;re \le 4.594816212135136 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Derivation

Split input into 4 regimes
if re < -1.9133903609356435e+137
1. Initial program 55.1
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around -inf 9.0
  \[\leadsto \color{blue}{-1 \cdot re}\]
3. Applied simplify9.0
  \[\leadsto \color{blue}{-re}\]
if -1.9133903609356435e+137 < re < -1.4143330855213813e-180 or 1.5363993963951212e-139 < re < 4.594816212135136e+128
1. Initial program 14.9
  \[\sqrt{re \cdot re + im \cdot im}\]
if -1.4143330855213813e-180 < re < 1.5363993963951212e-139
1. Initial program 30.2
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around 0 35.3
  \[\leadsto \color{blue}{im}\]
if 4.594816212135136e+128 < re
1. Initial program 53.6
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around inf 9.3
  \[\leadsto \color{blue}{re}\]
Recombined 4 regimes into one program.

Runtime

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))

Error

Derivation

`if re < -1.9133903609356435e+137`

`if -1.9133903609356435e+137 < re < -1.4143330855213813e-180 or 1.5363993963951212e-139 < re < 4.594816212135136e+128`

`if -1.4143330855213813e-180 < re < 1.5363993963951212e-139`

`if 4.594816212135136e+128 < re`

Runtime