\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.1 s
Input Error: 30.2
Output Error: 13.2
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -1.0880485702476334 \cdot 10^{+152} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le -1.5608708420993077 \cdot 10^{-235} \\ im & \text{when } re \le -2.3665634646026902 \cdot 10^{-293} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 2.168102476723288 \cdot 10^{+152} \\ re & \text{otherwise} \end{cases}\)

    if re < -1.0880485702476334e+152

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      58.9
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      58.9
    3. Applied taylor to get
      \[\sqrt{{re}^2 + im \cdot im} \leadsto -1 \cdot re\]
      0
    4. Taylor expanded around -inf to get
      \[\color{red}{-1 \cdot re} \leadsto \color{blue}{-1 \cdot re}\]
      0
    5. Applied simplify to get
      \[\color{red}{-1 \cdot re} \leadsto \color{blue}{-re}\]
      0

    if -1.0880485702476334e+152 < re < -1.5608708420993077e-235 or -2.3665634646026902e-293 < re < 2.168102476723288e+152

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      18.8
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      18.8

    if -1.5608708420993077e-235 < re < -2.3665634646026902e-293

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      46.1
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      46.1
    3. Applied taylor to get
      \[\sqrt{{re}^2 + im \cdot im} \leadsto im\]
      0
    4. Taylor expanded around 0 to get
      \[\color{red}{im} \leadsto \color{blue}{im}\]
      0

    if 2.168102476723288e+152 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      59.2
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      59.2
    3. Applied taylor to get
      \[\sqrt{{re}^2 + im \cdot im} \leadsto re\]
      0
    4. Taylor expanded around inf to get
      \[\color{red}{re} \leadsto \color{blue}{re}\]
      0
  1. Removed slow pow expressions

Original test:


(lambda ((re default) (im default))
  #:name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))