Error

Derivation

1. Split input into 4 regimes

2. if re < -1.219198831026942e+122

   1. Initial program 54.2

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

   2. Taylor expanded around -inf 7.9

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

   3. Simplified7.9

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

   if -1.219198831026942e+122 < re < -1.3987143113439338e-265 or 9.615181704877145e-268 < re < 1.5569157992329785e+79

   1. Initial program 19.9

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

   if -1.3987143113439338e-265 < re < 9.615181704877145e-268

   1. Initial program 30.9

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

   2. Taylor expanded around 0 30.2

      \[\leadsto \log \color{blue}{im}\]

   if 1.5569157992329785e+79 < re

   1. Initial program 46.4

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

   2. Taylor expanded around inf 9.7

      \[\leadsto \log \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.219198831026942 \cdot 10^{+122}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le -1.3987143113439338 \cdot 10^{-265}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 9.615181704877145 \cdot 10^{-268}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.5569157992329785 \cdot 10^{+79}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))