\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 3.3 s
Input Error: 31.2
Output Error: 12.8
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -9.965948042809399 \cdot 10^{+153} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le -1.0527568012471275 \cdot 10^{-305} \\ im & \text{when } re \le 2.743020608759423 \cdot 10^{-240} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 1.1106623338517268 \cdot 10^{+105} \\ re & \text{otherwise} \end{cases}\)

    if re < -9.965948042809399e+153

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      59.6
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      59.6
    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 -9.965948042809399e+153 < re < -1.0527568012471275e-305 or 2.743020608759423e-240 < re < 1.1106623338517268e+105

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      19.7
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      19.7

    if -1.0527568012471275e-305 < re < 2.743020608759423e-240

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      45.4
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      45.4
    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 1.1106623338517268e+105 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      49.8
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      49.8
    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))))