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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.333951939884602 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{if}\;re \le -1.1350134350066445 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re \le 1.0354870292778926 \cdot 10^{-272}:\\ \;\;\;\;im\\ \mathbf{if}\;re \le 4.236891465196376 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Derivation

Split input into 4 regimes
if re < -1.333951939884602e+154
1. Initial program 59.4
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around -inf 7.2
  \[\leadsto \color{blue}{-1 \cdot re}\]
3. Applied simplify7.2
  \[\leadsto \color{blue}{-re}\]
if -1.333951939884602e+154 < re < -1.1350134350066445e-200 or 1.0354870292778926e-272 < re < 4.236891465196376e+104
1. Initial program 18.0
  \[\sqrt{re \cdot re + im \cdot im}\]
if -1.1350134350066445e-200 < re < 1.0354870292778926e-272
1. Initial program 29.0
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around 0 32.2
  \[\leadsto \color{blue}{im}\]
if 4.236891465196376e+104 < re
1. Initial program 47.6
  \[\sqrt{re \cdot re + im \cdot im}\]
2. Taylor expanded around inf 9.3
  \[\leadsto \color{blue}{re}\]
Recombined 4 regimes into one program.

Runtime

herbie shell --seed 2018201 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))

Error

Try it out

Derivation

`if re < -1.333951939884602e+154`

`if -1.333951939884602e+154 < re < -1.1350134350066445e-200 or 1.0354870292778926e-272 < re < 4.236891465196376e+104`

`if -1.1350134350066445e-200 < re < 1.0354870292778926e-272`

`if 4.236891465196376e+104 < re`

Runtime

In
Out