\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Test:
math.log/2 on complex, real part
Bits:
128 bits
Bits error versus re
Bits error versus im
Bits error versus base
Time: 14.5 s
Input Error: 31.4
Output Error: 15.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log \left(-re\right)}{\log base} & \text{when } re \le -5.230879555317119 \cdot 10^{+121} \\ \frac{1}{\log base \cdot \frac{1}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} & \text{when } re \le 3.0929452776661224 \cdot 10^{+125} \\ \frac{1}{\frac{\log base}{\log re}} & \text{otherwise} \end{cases}\)

    if re < -5.230879555317119e+121

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      53.9
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      53.9
    3. Applied taylor to get
      \[\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}\]
      0.5
    4. Taylor expanded around -inf to get
      \[\frac{\log base \cdot \log \color{red}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base} \leadsto \frac{\log base \cdot \log \color{blue}{\left(-1 \cdot re\right)} + 0}{\log base \cdot \log base}\]
      0.5
    5. Applied simplify to get
      \[\color{red}{\frac{\log base \cdot \log \left(-1 \cdot re\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{\log \left(-re\right)}{\log base}}\]
      0.4

    if -5.230879555317119e+121 < re < 3.0929452776661224e+125

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      21.6
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      21.6
    3. Using strategy rm
      21.6
    4. Applied clear-num to get
      \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{1}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}}\]
      21.6
    5. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}} \leadsto \frac{1}{\color{blue}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
      21.5
    6. Using strategy rm
      21.5
    7. Applied div-inv to get
      \[\frac{1}{\color{red}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \leadsto \frac{1}{\color{blue}{\log base \cdot \frac{1}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
      21.6

    if 3.0929452776661224e+125 < re

    1. Started with
      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
      55.1
    2. Applied simplify to get
      \[\color{red}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}} \leadsto \color{blue}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}}\]
      55.1
    3. Using strategy rm
      55.1
    4. Applied clear-num to get
      \[\color{red}{\frac{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}{\log base \cdot \log base}} \leadsto \color{blue}{\frac{1}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}}\]
      55.1
    5. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{\log base \cdot \log base}{\log base \cdot \log \left(\sqrt{{re}^2 + im \cdot im}\right) + 0}}} \leadsto \frac{1}{\color{blue}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
      55.1
    6. Using strategy rm
      55.1
    7. Applied div-inv to get
      \[\frac{1}{\color{red}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}} \leadsto \frac{1}{\color{blue}{\log base \cdot \frac{1}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
      55.1
    8. Applied taylor to get
      \[\frac{1}{\log base \cdot \frac{1}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} \leadsto \frac{1}{\log base \cdot \frac{1}{\log re}}\]
      0.5
    9. Taylor expanded around inf to get
      \[\frac{1}{\log base \cdot \frac{1}{\log \color{red}{re}}} \leadsto \frac{1}{\log base \cdot \frac{1}{\log \color{blue}{re}}}\]
      0.5
    10. Applied simplify to get
      \[\color{red}{\frac{1}{\log base \cdot \frac{1}{\log re}}} \leadsto \color{blue}{\frac{1}{\frac{\log base}{\log re}}}\]
      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))))