\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.1 s
Input Error: 33.0
Output Error: 10.1
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -3.2136220183709947 \cdot 10^{+127} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 9.995586505648043 \cdot 10^{-300} \\ im & \text{when } re \le 1.6515950423484695 \cdot 10^{-133} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 1.5261071406778172 \cdot 10^{+112} \\ re & \text{otherwise} \end{cases}\)

    if re < -3.2136220183709947e+127

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      53.7
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      53.7
    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 -3.2136220183709947e+127 < re < 9.995586505648043e-300 or 1.6515950423484695e-133 < re < 1.5261071406778172e+112

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      18.5
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      18.5

    if 9.995586505648043e-300 < re < 1.6515950423484695e-133

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      46.4
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      46.4
    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 1.5261071406778172e+112 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      51.0
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      51.0
    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))))