Approximation of the following distribution (source):

e5b3bbf9636ac0c05e3655920b67420a.png

t :: temperature [degrees C]
p :: precipitation [cm/year]

temperature < 0                => Tundra
temperature < 8
  p < 30                       => Cold desert
  else                         => Boreal forest
temperature < 20
  p < 30 + (t - 8) * 20 / 12   => Cold desert
  p < 50 + (t - 8) * 70 / 12   => Woodland/shrubland
  p < 180 + (t - 8) * 60 / 12  => Temperate forest
  else                         => Temperate rainforest
else
  p < 50 + (t - 20) * 50 / 10  => Subtropical desert
  p < 240 + (t - 20) * 40 / 10 => Savanna
  else                         => Tropical rainforest