\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 3.8 s
Input Error: 31.2
Output Error: 10.4
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -1.560410170116619 \cdot 10^{+149} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le -8.68041930647669 \cdot 10^{-252} \\ im & \text{when } re \le 1.9727183986347288 \cdot 10^{-223} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 7.79868197508907 \cdot 10^{+130} \\ re & \text{otherwise} \end{cases}\)

    if re < -1.560410170116619e+149

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      58.4
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      58.4
    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 -1.560410170116619e+149 < re < -8.68041930647669e-252 or 1.9727183986347288e-223 < re < 7.79868197508907e+130

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      17.2
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      17.2

    if -8.68041930647669e-252 < re < 1.9727183986347288e-223

    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 7.79868197508907e+130 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      53.6
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      53.6
    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))))