Error

Derivation

1. Split input into 4 regimes

2. if re < -2.610120343778248e+40

   1. Initial program 42.3

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

   2. Taylor expanded around -inf 11.2

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

   3. Simplified11.2

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

   if -2.610120343778248e+40 < re < 6.339057535698283e-246 or 1.4621303669194703e-218 < re < 1.5215402511937908e+94

   1. Initial program 20.8

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

   if 6.339057535698283e-246 < re < 1.4621303669194703e-218

   1. Initial program 33.2

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

   2. Taylor expanded around 0 35.6

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

   if 1.5215402511937908e+94 < re

   1. Initial program 49.5

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

   2. Taylor expanded around inf 9.0

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

4. Final simplification17.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.610120343778248 \cdot 10^{+40}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \le 6.339057535698283 \cdot 10^{-246}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{elif}\;re \le 1.4621303669194703 \cdot 10^{-218}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.5215402511937908 \cdot 10^{+94}:\\ \;\;\;\;\log \left(\sqrt{im \cdot im + re \cdot re}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))