In a long sum, how can we find how many terms are preceded by the plus (or minus) sign

Question

I have a module that computes a very long expression. I need to count the number of terms in the expression and how many are preceded by a minus sign.

Example: in a + bc - e + df - gh we have 5 terms and two are preceded by a minus sign.

I was looking at the Cases command but I could not find any similar examples.

Answer 1

Generalizing a bit to expressions whose coefficients are not just ±1, one has the following:

expr = a + bc - 2 e + 3 df - gh;
Length@expr
Count[expr, _?Internal`SyntacticNegativeQ]
(*
  5
  2
*)

Count[expr, _?Negative * Optional[__]]
(*  2  *)

They are not strictly equivalent. Internal`SyntacticNegativeQ is true if the term is typeset with a minus sign. The following shows the difference:

expr2 = Cos[1] x y + Cos[3] y;
Count[expr2, _?Negative * Optional[__]]
Count[expr2, _?Internal`SyntacticNegativeQ]
(*
  1
  0
*)

The use of Optional[__] is in case there is a term that is simply a number and not a product of a number and an expression (note that y-x is Plus[Times[-1, x], y] internally):

expr3 = expr - 3
Count[expr3, _?Negative*Optional[__]]
Count[expr3, _?Internal`SyntacticNegativeQ]
(*
  -3 + a + bc + 3 df - 2 e - gh
  3
  3
*)

Answer 2

If your expression is a sum, as in your example, you can do the following:

expr = a + bc - e + df - gh

make a list:

lexpr = List @@ expr

Count the expressions with a preceding minus sign

Count[lexpr, -x_]
(* 2 *)

Answer 3

expr = a + bc - 2 e + 3 df - gh + Sin[ij] - Cos[klm];

neg = Length @ Select[CoefficientRules @ expr, #[[2]] < 0 &]
pos = Length @ expr - neg

3

4

Answer 4

expr = a + bc - e + df - gh
If[StringMatchQ[ToString@#, "-" ~~ ___], "-", "+"] & /@ 
  List @@ expr // Tally

{{"+", 3}, {"-", 2}}

note this breaks if you have ratios within terms, eg expr = a - b/c the -b/c becomes Times[-1, b, Power[c, -1]] and appears to not have a lead -. Some of the other answers have the same issue, ( I didn't try all.. )