\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.5 s
Input Error: 13.5
Output Error: 6.3
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -3.924884f+16 \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 1.9490856f+19 \\ re & \text{otherwise} \end{cases}\)

    if re < -3.924884f+16

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      25.4
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      25.4
    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 -3.924884f+16 < re < 1.9490856f+19

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      8.7
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      8.7

    if 1.9490856f+19 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      27.5
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      27.5
    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))))