\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
Test:
math.log/1 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 4.5 s
Input Error: 30.7
Output Error: 12.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(-re\right) & \text{when } re \le -2.2370754592070436 \cdot 10^{+105} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le -1.5608708420993077 \cdot 10^{-235} \\ \log im & \text{when } re \le -2.3665634646026902 \cdot 10^{-293} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 2.56976209479494 \cdot 10^{+109} \\ \log re & \text{otherwise} \end{cases}\)

    if re < -2.2370754592070436e+105

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      50.6
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      50.6
    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 -2.2370754592070436e+105 < re < -1.5608708420993077e-235 or -2.3665634646026902e-293 < re < 2.56976209479494e+109

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      20.0
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      20.0

    if -1.5608708420993077e-235 < re < -2.3665634646026902e-293

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      30.7
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      30.7
    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 2.56976209479494e+109 < re

    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
      \[\log \left(\sqrt{{re}^2 + im \cdot im}\right) \leadsto \log re\]
      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)))))