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Derivation

1. Split input into 2 regimes

2. if re < -7.317903954721519e-279

   1. Initial program 57.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

   2. Initial simplification57.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

   3. Taylor expanded around -inf 55.6

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log 10}\]

   4. Simplified55.6

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log 10}\]

   if -7.317903954721519e-279 < re

   1. Initial program 57.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

   2. Initial simplification57.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

   3. Taylor expanded around inf 56.1

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\frac{1}{re}\right)}{\log 10}}\]

   4. Simplified56.1

      \[\leadsto \color{blue}{\frac{\log re}{\log 10}}\]
3. Recombined 2 regimes into one program.

4. Final simplification55.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.317903954721519 \cdot 10^{-279}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Runtime

Time bar (total: 22.5s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))