\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
Test:
math.log/1 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.1 s
Input Error: 31.5
Output Error: 12.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(-re\right) & \text{when } re \le -5.001969231040948 \cdot 10^{+151} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 6.526828139904173 \cdot 10^{-268} \\ \log im & \text{when } re \le 3.51720187277729 \cdot 10^{-183} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 3.0634872044683813 \cdot 10^{+39} \\ \log re & \text{otherwise} \end{cases}\)

    if re < -5.001969231040948e+151

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      61.1
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      61.1
    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 -5.001969231040948e+151 < re < 6.526828139904173e-268 or 3.51720187277729e-183 < re < 3.0634872044683813e+39

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      20.4
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      20.4

    if 6.526828139904173e-268 < re < 3.51720187277729e-183

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      31.0
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      31.0
    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 3.0634872044683813e+39 < re

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      44.4
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      44.4
    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)))))