\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 9.2 s
Input Error: 31.5
Output Error: 14.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-re\right)}{\log 10} & \text{when } re \le -9.965948042809399 \cdot 10^{+153} \\ \frac{\log \left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}}\right)}^3\right)}^3\right)}{\log 10} & \text{when } re \le -1.0527568012471275 \cdot 10^{-305} \\ \frac{\log im}{\log 10} & \text{when } re \le 2.743020608759423 \cdot 10^{-240} \\ \frac{\log \left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}}\right)}^3\right)}^3\right)}{\log 10} & \text{when } re \le 1.0475622453733284 \cdot 10^{+105} \\ \frac{\log re}{\log 10} & \text{otherwise} \end{cases}\)

    if re < -9.965948042809399e+153

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      62.0
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      62.0
    3. Applied taylor to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log \left(-1 \cdot re\right)}{\log 10}\]
      0.6
    4. Taylor expanded around -inf to get
      \[\frac{\log \color{red}{\left(-1 \cdot re\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]
      0.6
    5. Applied simplify to get
      \[\color{red}{\frac{\log \left(-1 \cdot re\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(-re\right)}{\log 10}}\]
      0.6

    if -9.965948042809399e+153 < re < -1.0527568012471275e-305 or 2.743020608759423e-240 < re < 1.0475622453733284e+105

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      20.7
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      20.7
    3. Using strategy rm
      20.7
    4. Applied add-cube-cbrt to get
      \[\frac{\log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)}}{\log 10} \leadsto \frac{\log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)}}{\log 10}\]
      20.7
    5. Using strategy rm
      20.7
    6. Applied add-cube-cbrt to get
      \[\frac{\log \left({\color{red}{\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}}^3\right)}{\log 10} \leadsto \frac{\log \left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}}\right)}^3\right)}}^3\right)}{\log 10}\]
      20.7

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

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      29.6
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      29.6
    3. Applied taylor to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log im}{\log 10}\]
      0.6
    4. Taylor expanded around 0 to get
      \[\frac{\log \color{red}{im}}{\log 10} \leadsto \frac{\log \color{blue}{im}}{\log 10}\]
      0.6

    if 1.0475622453733284e+105 < re

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      51.6
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      51.6
    3. Applied taylor to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log re}{\log 10}\]
      0.6
    4. Taylor expanded around inf to get
      \[\frac{\log \color{red}{re}}{\log 10} \leadsto \frac{\log \color{blue}{re}}{\log 10}\]
      0.6
  1. Removed slow pow expressions

Original test:


(lambda ((re default) (im default))
  #:name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))