\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 3.8 s
Input Error: 32.0
Output Error: 10.5
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -4.282075037384891 \cdot 10^{+142} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le -4.379674431420248 \cdot 10^{-198} \\ im & \text{when } re \le 4.185543182523104 \cdot 10^{-238} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 1.5757889061495052 \cdot 10^{+162} \\ re & \text{otherwise} \end{cases}\)

    if re < -4.282075037384891e+142

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      56.7
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      56.7
    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 -4.282075037384891e+142 < re < -4.379674431420248e-198 or 4.185543182523104e-238 < re < 1.5757889061495052e+162

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      17.7
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      17.7

    if -4.379674431420248e-198 < re < 4.185543182523104e-238

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      45.1
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      45.1
    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 1.5757889061495052e+162 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      59.4
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      59.4
    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))))