\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
Test:
math.log/1 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.4 s
Input Error: 31.8
Output Error: 8.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(-re\right) & \text{when } re \le -3.4790326343865455 \cdot 10^{+38} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le -4.379674431420248 \cdot 10^{-198} \\ \log im & \text{when } re \le 4.185543182523104 \cdot 10^{-238} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 1.8039960838892567 \cdot 10^{+48} \\ \log re & \text{otherwise} \end{cases}\)

    if re < -3.4790326343865455e+38

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      45.1
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      45.1
    3. Applied taylor to get
      \[\log \left(\sqrt{{re}^2 + im \cdot im}\right) \leadsto \log \left(-1 \cdot re\right)\]
      0
    4. Taylor expanded around -inf to get
      \[\log \color{red}{\left(-1 \cdot re\right)} \leadsto \log \color{blue}{\left(-1 \cdot re\right)}\]
      0
    5. Applied simplify to get
      \[\color{red}{\log \left(-1 \cdot re\right)} \leadsto \color{blue}{\log \left(-re\right)}\]
      0

    if -3.4790326343865455e+38 < re < -4.379674431420248e-198 or 4.185543182523104e-238 < re < 1.8039960838892567e+48

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      19.3
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      19.3

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

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      30.2
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      30.2
    3. Applied taylor to get
      \[\log \left(\sqrt{{re}^2 + im \cdot im}\right) \leadsto \log im\]
      0
    4. Taylor expanded around 0 to get
      \[\log \color{red}{im} \leadsto \log \color{blue}{im}\]
      0

    if 1.8039960838892567e+48 < re

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      43.9
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      43.9
    3. Applied taylor to get
      \[\log \left(\sqrt{{re}^2 + im \cdot im}\right) \leadsto \log re\]
      0
    4. Taylor expanded around inf to get
      \[\log \color{red}{re} \leadsto \log \color{blue}{re}\]
      0
  1. Removed slow pow expressions

Original test:


(lambda ((re default) (im default))
  #:name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))