\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
Test:
math.log10 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 11.7 s
Input Error: 31.7
Output Error: 12.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-re\right)}{\log 10} & \text{when } re \le -5.001969231040948 \cdot 10^{+151} \\ \frac{1}{\frac{\log 10}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}} & \text{when } re \le 4.633757336962986 \cdot 10^{-280} \\ \frac{\log im}{\log 10} & \text{when } re \le 3.51720187277729 \cdot 10^{-183} \\ \frac{1}{\frac{\log 10}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}} & \text{when } re \le 3.0634872044683813 \cdot 10^{+39} \\ \frac{1}{\frac{\log 10}{\log re}} & \text{otherwise} \end{cases}\)

    if re < -5.001969231040948e+151

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      61.2
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      61.2
    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 -5.001969231040948e+151 < re < 4.633757336962986e-280 or 3.51720187277729e-183 < re < 3.0634872044683813e+39

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      20.5
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      20.5
    3. Using strategy rm
      20.5
    4. Applied clear-num to get
      \[\color{red}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}}\]
      20.5

    if 4.633757336962986e-280 < re < 3.51720187277729e-183

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      32.2
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      32.2
    3. Applied taylor to get
      \[\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10} \leadsto \frac{\log im}{\log 10}\]
      0.5
    4. Taylor expanded around 0 to get
      \[\frac{\log \color{red}{im}}{\log 10} \leadsto \frac{\log \color{blue}{im}}{\log 10}\]
      0.5

    if 3.0634872044683813e+39 < re

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
      44.5
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}}\]
      44.5
    3. Using strategy rm
      44.5
    4. Applied clear-num to get
      \[\color{red}{\frac{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}{\log 10}} \leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}}\]
      44.5
    5. Using strategy rm
      44.5
    6. Applied add-cbrt-cube to get
      \[\frac{1}{\frac{\log 10}{\color{red}{\log \left(\sqrt{{re}^2 + im \cdot im}\right)}}} \leadsto \frac{1}{\frac{\log 10}{\color{blue}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}}\]
      44.5
    7. Applied add-cbrt-cube to get
      \[\frac{1}{\frac{\color{red}{\log 10}}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}} \leadsto \frac{1}{\frac{\color{blue}{\sqrt[3]{{\left(\log 10\right)}^3}}}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}\]
      44.7
    8. Applied cbrt-undiv to get
      \[\frac{1}{\color{red}{\frac{\sqrt[3]{{\left(\log 10\right)}^3}}{\sqrt[3]{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}} \leadsto \frac{1}{\color{blue}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}}}\]
      44.5
    9. Applied taylor to get
      \[\frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log \left(\sqrt{{re}^2 + im \cdot im}\right)\right)}^3}}} \leadsto \frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log re\right)}^3}}}\]
      0.7
    10. Taylor expanded around inf to get
      \[\frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log \color{red}{re}\right)}^3}}} \leadsto \frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log \color{blue}{re}\right)}^3}}}\]
      0.7
    11. Applied simplify to get
      \[\frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log re\right)}^3}}} \leadsto \frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log re\right)}^3}}}\]
      0.7
    12. Applied final simplification
    13. Applied simplify to get
      \[\color{red}{\frac{1}{\sqrt[3]{\frac{{\left(\log 10\right)}^3}{{\left(\log re\right)}^3}}}} \leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log re}}}\]
      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)))