put in writing vs write down

put in writing

write down

verb
  • Used other than figuratively or idiomatically: see write, down. 

  • To produce or set (something) down in writing; to record something. 

  • To write (something) in a simple or condescending style. 

  • To make a downward adjustment in the value of an asset. 

  • To condemn in writing; to document the faults, offenses, or wrongdoing of. 

How often have the words put in writing and write down occurred in a corpus of books? (source: Google Ngram Viewer )