\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Test:
math.log/2 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Bits error versus base
Time: 19.5 s
Input Error: 14.6
Output Error: 6.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-im\right)}{\log base} & \text{when } im \le -2.2638177f+14 \\ \sqrt[3]{\frac{{\left(\log \left(\sqrt{{im}^2 + re \cdot re}\right)\right)}^3}{{\left(\log base\right)}^3}} & \text{when } im \le 5.397122f+09 \\ \frac{\log im \cdot \log base}{{\left(\log base\right)}^{\left(1 + 1\right)}} & \text{otherwise} \end{cases}\)

    if im < -2.2638177f+14

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      25.7
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      25.7
    3. Using strategy rm
      25.7
    4. Applied add-cbrt-cube to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{\sqrt[3]{{\left(\log base\right)}^3}}}\]
      25.7
    5. Applied add-cbrt-cube to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot \sqrt[3]{{\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3}} \cdot \sqrt[3]{{\left(\log base\right)}^3}}\]
      25.7
    6. Applied cbrt-unprod to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\sqrt[3]{{\left(\log base\right)}^3} \cdot \sqrt[3]{{\left(\log base\right)}^3}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
      25.7
    7. Applied add-cbrt-cube to get
      \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}\]
      25.7
    8. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
      25.7
    9. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{\frac{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^3}{{\left(\log base\right)}^3}}}\]
      25.7
    10. Applied taylor to get
      \[\sqrt[3]{\frac{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^3}{{\left(\log base\right)}^3}} \leadsto \sqrt[3]{\frac{{\left(\log \left(-1 \cdot im\right)\right)}^3}{{\left(\log base\right)}^3}}\]
      0.4
    11. Taylor expanded around -inf to get
      \[\sqrt[3]{\frac{{\left(\log \color{red}{\left(-1 \cdot im\right)}\right)}^3}{{\left(\log base\right)}^3}} \leadsto \sqrt[3]{\frac{{\left(\log \color{blue}{\left(-1 \cdot im\right)}\right)}^3}{{\left(\log base\right)}^3}}\]
      0.4
    12. Applied simplify to get
      \[\color{red}{\sqrt[3]{\frac{{\left(\log \left(-1 \cdot im\right)\right)}^3}{{\left(\log base\right)}^3}}} \leadsto \color{blue}{\frac{\log \left(-im\right)}{\log base}}\]
      0.4

    if -2.2638177f+14 < im < 5.397122f+09

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      9.7
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      9.7
    3. Using strategy rm
      9.7
    4. Applied add-cbrt-cube to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{\sqrt[3]{{\left(\log base\right)}^3}}}\]
      9.7
    5. Applied add-cbrt-cube to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot \sqrt[3]{{\left(\log base\right)}^3}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3}} \cdot \sqrt[3]{{\left(\log base\right)}^3}}\]
      9.7
    6. Applied cbrt-unprod to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\sqrt[3]{{\left(\log base\right)}^3} \cdot \sqrt[3]{{\left(\log base\right)}^3}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
      9.7
    7. Applied add-cbrt-cube to get
      \[\frac{\color{red}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}\]
      9.8
    8. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}}{\sqrt[3]{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}}\]
      9.7
    9. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0\right)}^3}{{\left(\log base\right)}^3 \cdot {\left(\log base\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{\frac{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^3}{{\left(\log base\right)}^3}}}\]
      9.7
    10. Applied simplify to get
      \[\sqrt[3]{\frac{\color{red}{{\left(\log \left(\sqrt{im \cdot im + re \cdot re}\right)\right)}^3}}{{\left(\log base\right)}^3}} \leadsto \sqrt[3]{\frac{\color{blue}{{\left(\log \left(\sqrt{{im}^2 + re \cdot re}\right)\right)}^3}}{{\left(\log base\right)}^3}}\]
      9.7

    if 5.397122f+09 < im

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      22.8
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      22.8
    3. Using strategy rm
      22.8
    4. Applied pow1 to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{red}{\log base}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \color{blue}{{\left(\log base\right)}^{1}}}\]
      22.8
    5. Applied pow1 to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{\log base} \cdot {\left(\log base\right)}^{1}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\log base\right)}^{1}} \cdot {\left(\log base\right)}^{1}}\]
      22.8
    6. Applied pow-prod-up to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{red}{{\left(\log base\right)}^{1} \cdot {\left(\log base\right)}^{1}}} \leadsto \frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\color{blue}{{\left(\log base\right)}^{\left(1 + 1\right)}}}\]
      22.8
    7. Applied taylor to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{{\left(\log base\right)}^{\left(1 + 1\right)}} \leadsto \frac{\log base \cdot \log im + 0}{{\left(\log base\right)}^{\left(1 + 1\right)}}\]
      0.4
    8. Taylor expanded around 0 to get
      \[\frac{\log base \cdot \log \color{red}{im} + 0}{{\left(\log base\right)}^{\left(1 + 1\right)}} \leadsto \frac{\log base \cdot \log \color{blue}{im} + 0}{{\left(\log base\right)}^{\left(1 + 1\right)}}\]
      0.4
    9. Applied simplify to get
      \[\color{red}{\frac{\log base \cdot \log im + 0}{{\left(\log base\right)}^{\left(1 + 1\right)}}} \leadsto \color{blue}{\frac{\log im \cdot \log base}{{\left(\log base\right)}^{\left(1 + 1\right)}}}\]
      0.4
  1. Removed slow pow expressions

Original test:


(lambda ((re default) (im default) (base default))
  #:name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))