\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Test:
math.log/2 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Bits error versus base
Time: 21.8 s
Input Error: 14.9
Output Error: 6.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-im\right)}{\log base} & \text{when } im \le -7.1582915f+11 \\ \frac{\log base \cdot \log \left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right) + 0}{\log base \cdot \log base} & \text{when } im \le 1.566448f+13 \\ \frac{\log im}{\log base} + \frac{\frac{1}{2}}{\log base} \cdot {\left(\frac{re}{im}\right)}^2 & \text{otherwise} \end{cases}\)

    if im < -7.1582915f+11

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      24.2
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      24.2
    3. Using strategy rm
      24.2
    4. Applied add-cube-cbrt to get
      \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}^3}\]
      24.3
    5. Applied simplify to get
      \[{\color{red}{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}}^3\]
      24.3
    6. Applied taylor to get
      \[{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)}^3\]
      0.8
    7. Taylor expanded around -inf to get
      \[{\left(\sqrt[3]{\frac{\log \color{red}{\left(-1 \cdot im\right)}}{\log base}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{\log \color{blue}{\left(-1 \cdot im\right)}}{\log base}}\right)}^3\]
      0.8
    8. Applied simplify to get
      \[{\left(\sqrt[3]{\frac{\log \left(-1 \cdot im\right)}{\log base}}\right)}^3 \leadsto \frac{\log \left(-im\right)}{\log base}\]
      0.4
    9. Applied final simplification

    if -7.1582915f+11 < im < 1.566448f+13

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      9.9
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      9.9
    3. Using strategy rm
      9.9
    4. Applied add-cube-cbrt to get
      \[\frac{\log base \cdot \log \color{red}{\left(\sqrt{{re}^2 + im \cdot im}\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left({\left(\sqrt[3]{\sqrt{{re}^2 + im \cdot im}}\right)}^3\right)} + 0}{\log base \cdot \log base}\]
      9.9

    if 1.566448f+13 < im

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      24.6
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      24.6
    3. Using strategy rm
      24.6
    4. Applied add-cube-cbrt to get
      \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}^3}\]
      24.7
    5. Applied simplify to get
      \[{\color{red}{\left(\sqrt[3]{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}}^3\]
      24.7
    6. Applied taylor to get
      \[{\left(\sqrt[3]{\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}}\right)}^3 \leadsto \frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base}\]
      7.1
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base}}\]
      7.1
    8. Applied simplify to get
      \[\frac{1}{2} \cdot \frac{{re}^2}{{im}^2 \cdot \log base} + \frac{\log im}{\log base} \leadsto \frac{\log im}{\log base} + \frac{\frac{1}{2}}{\log base} \cdot {\left(\frac{re}{im}\right)}^2\]
      0.5
    9. Applied final simplification
  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))))