\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
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: 31.2
Output Error: 13.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \log \left(-re\right) & \text{when } re \le -9.965948042809399 \cdot 10^{+153} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le -1.0527568012471275 \cdot 10^{-305} \\ \log im & \text{when } re \le 2.743020608759423 \cdot 10^{-240} \\ \log \left(\sqrt{{re}^2 + im \cdot im}\right) & \text{when } re \le 7.720496126429738 \cdot 10^{+93} \\ \log re & \text{otherwise} \end{cases}\)

    if re < -9.965948042809399e+153

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      62.0
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      62.0
    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 -9.965948042809399e+153 < re < -1.0527568012471275e-305 or 2.743020608759423e-240 < re < 7.720496126429738e+93

    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 -1.0527568012471275e-305 < re < 2.743020608759423e-240

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      29.3
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      29.3
    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 7.720496126429738e+93 < re

    1. Started with
      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
      50.1
    2. Applied simplify to get
      \[\color{red}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \leadsto \color{blue}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}\]
      50.1
    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)))))