\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 3.5 s
Input Error: 29.6
Output Error: 13.6
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -8.533487505443716 \cdot 10^{+145} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 3.751610550051807 \cdot 10^{+91} \\ re & \text{otherwise} \end{cases}\)

    if re < -8.533487505443716e+145

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      57.1
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      57.1
    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 -8.533487505443716e+145 < re < 3.751610550051807e+91

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      19.6
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      19.6

    if 3.751610550051807e+91 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      48.3
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      48.3
    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))))