Fix Integer.extended_gcd/2 returning negative GCD for zero base cases (#15163)

When one argument is 0 and the other is negative, `extended_gcd/2`
was returning a negative GCD, violating both the @SPEC (which
declares `non_neg_integer`) and consistency with `gcd/2`.

Author: pnezis

lib/elixir/lib/integer.ex

@@ -532,17 +532,23 @@ defmodule Integer do
 
       iex> Integer.extended_gcd(10, 0)
       {10, 1, 0}
+      iex> Integer.extended_gcd(-10, 0)
+      {10, -1, 0}
       iex> Integer.extended_gcd(0, 10)
       {10, 0, 1}
+      iex> Integer.extended_gcd(0, -10)
+      {10, 0, -1}
       iex> Integer.extended_gcd(0, 0)
       {0, 0, 0}
 
   """
   @doc since: "1.12.0"
   @spec extended_gcd(integer, integer) :: {non_neg_integer, integer, integer}
   def extended_gcd(0, 0), do: {0, 0, 0}
-  def extended_gcd(0, b), do: {b, 0, 1}
-  def extended_gcd(a, 0), do: {a, 1, 0}
+  def extended_gcd(0, b) when b > 0, do: {b, 0, 1}
+  def extended_gcd(0, b) when b < 0, do: {-b, 0, -1}
+  def extended_gcd(a, 0) when a > 0, do: {a, 1, 0}
+  def extended_gcd(a, 0) when a < 0, do: {-a, -1, 0}
 
   def extended_gcd(integer1, integer2) when is_integer(integer1) and is_integer(integer2) do
     extended_gcd(integer2, integer1, 0, 1, 1, 0)

lib/elixir/test/elixir/integer_test.exs

@@ -282,13 +282,20 @@ defmodule IntegerTest do
   end
 
   test "extended_gcd" do
-    # Poor's man properby based testing
+    # Poor's man property based testing
     for _ <- 1..100 do
       left = :rand.uniform(1000)
       right = :rand.uniform(1000)
       {gcd, m, n} = Integer.extended_gcd(left, right)
       assert Integer.gcd(left, right) == gcd
       assert m * left + n * right == gcd
     end
+
+    # zero cases
+    for {left, right} <- [{10, 0}, {0, 10}, {0, -10}, {-10, 0}] do
+      {gcd, m, n} = Integer.extended_gcd(left, right)
+      assert Integer.gcd(left, right) == gcd
+      assert m * left + n * right == gcd
+    end
   end
 end