\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
Test:
math.log/1 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.2 s
Input Error: 30.7
Output Error: 10.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(-re\right) & \text{when } re \le -1.4760992303283469 \cdot 10^{+119} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le -8.68041930647669 \cdot 10^{-252} \\ \log im & \text{when } re \le 1.9727183986347288 \cdot 10^{-223} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 1.6728927718931055 \cdot 10^{+97} \\ \log re & \text{otherwise} \end{cases}\)

    if re < -1.4760992303283469e+119

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      54.3
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      54.3
    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 -1.4760992303283469e+119 < re < -8.68041930647669e-252 or 1.9727183986347288e-223 < re < 1.6728927718931055e+97

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      18.2
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      18.2

    if -8.68041930647669e-252 < re < 1.9727183986347288e-223

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      31.6
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      31.6
    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 1.6728927718931055e+97 < re

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      49.6
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      49.6
    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)))))