Error

Derivation

1. Split input into 4 regimes

2. if (- re) < -1.335204076576571e+85

   1. Initial program 47.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Taylor expanded around inf 10.2

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]

   3. Applied simplify10.2

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]

   if -1.335204076576571e+85 < (- re) < -5.33950021962182e-183 or 2.6755888641379018e-211 < (- re) < 1.3645053119469467e+49

   1. Initial program 17.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt17.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\sqrt{\log base \cdot \log base + 0 \cdot 0} \cdot \sqrt{\log base \cdot \log base + 0 \cdot 0}}}\]

   4. Applied *-un-lft-identity17.5

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right)}}{\sqrt{\log base \cdot \log base + 0 \cdot 0} \cdot \sqrt{\log base \cdot \log base + 0 \cdot 0}}\]

   5. Applied times-frac17.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0 \cdot 0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{\log base \cdot \log base + 0 \cdot 0}}}\]

   6. Applied simplify17.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{\log base \cdot \log base + 0 \cdot 0}}\]

   7. Applied simplify17.5

      \[\leadsto \frac{1}{\sqrt{\log base \cdot \log base}} \cdot \color{blue}{\frac{\log base \cdot \log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\sqrt{\log base \cdot \log base}}}\]

   if -5.33950021962182e-183 < (- re) < 2.6755888641379018e-211

   1. Initial program 30.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Taylor expanded around 0 33.2

      \[\leadsto \frac{\log \color{blue}{im} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   3. Applied simplify33.2

      \[\leadsto \color{blue}{\frac{\log base \cdot \log im}{\log base \cdot \log base}}\]

   if 1.3645053119469467e+49 < (- re)

   1. Initial program 44.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Taylor expanded around -inf 12.3

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   3. Applied simplify12.2

      \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\log base}}\]
3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (re im base)
  :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))))