\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
Test:
math.log/1 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 5.2 s
Input Error: 31.8
Output Error: 11.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(-re\right) & \text{when } re \le -3.999543611763718 \cdot 10^{+108} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 9.995586505648043 \cdot 10^{-300} \\ \log im & \text{when } re \le 1.6515950423484695 \cdot 10^{-133} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 1.5261071406778172 \cdot 10^{+112} \\ \log re & \text{otherwise} \end{cases}\)

    if re < -3.999543611763718e+108

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      52.4
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      52.4
    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 -3.999543611763718e+108 < re < 9.995586505648043e-300 or 1.6515950423484695e-133 < re < 1.5261071406778172e+112

    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 9.995586505648043e-300 < re < 1.6515950423484695e-133

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      29.0
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      29.0
    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 1.5261071406778172e+112 < re

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      53.0
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      53.0
    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)))))