Implement Base.cmp

This operation is useful for implementing `:(==)`, `:(<)`, and `:(<=)`.
The more efficient `cmp` is, the more efficient these comparisons are.

src/equals.jl

@@ -1,45 +1,64 @@
-# Equality
+_sign(x::BigInt) = Int(sign(x))
+_sign(x::Decimal) = x.s ? -1 : 1
 
-# equals() now depends on == instead
-# of the other way round.
-function Base.:(==)(x::Decimal, y::Decimal)
-    # return early on zero
-    x_is_zero = iszero(x)
-    y_is_zero = iszero(y)
-    if x_is_zero || y_is_zero
-        return x_is_zero === y_is_zero
+function Base.cmp(x::Decimal, y::Decimal)
+    # We try to avoid computing x - y because it allocates a new BigInt
+
+    if iszero(x) && iszero(y)
+        return 0
+    elseif iszero(x) # && !iszero(y)
+        return -_sign(y)
+    elseif iszero(y) # && !iszero(x)
+        return _sign(x)
     end
 
-    a = normalize(x)
-    b = normalize(y)
-    a.c == b.c && a.q == b.q && a.s == b.s
-end
+    # Neither x nor y is zero here
 
-function Base.:(<)(x::Decimal, y::Decimal)
-    # return early on zero
-    if iszero(x) && iszero(y)
-        return false
+    if x.s != y.s
+        # x and y have different signs, so
+        #  if x < 0, then return -1 (because y is positive)
+        #  if x > 0, then return +1 (because y is negative)
+        return _sign(x)
     end
 
-    # avoid normalization if possible
-    if x.q == y.q
-        return isless(x.s == 0 ? x.c : -x.c, y.s == 0 ? y.c : -y.c)
+    cmp_c = cmp(x.c, y.c)
+    cmp_q = cmp(x.q, y.q)
+
+    # If both x.c and x.q is greater (or equal, or less) than y.c and y.q,
+    # then x is greater (or equal, or less) than y
+    if cmp_c == cmp_q
+        return cmp_c
     end
 
-    diff = y - x
+    # Let x = a * 10^p, y = b * 10^q.
+    #
+    # If p ≥ q:
+    #
+    #   sign(x - y)
+    # = sign(a * 10^p - b * 10^q)
+    # = sign((a * 10^(p - q) - b) * 10^q)
+    # = sign(a * 10^(p - q) - b)
+    #
+    # If p < q:
+    #
+    #   sign(x - y)
+    # = a - b * 10^(q - p)
 
-    farther_from_0 = diff.c > 0 || (iszero(diff.c) && diff.q > 0)
+    xcoef = (-1)^x.s * x.c
+    ycoef = (-1)^y.s * y.c
 
-    if diff.s == 1
-        return !farther_from_0
+    if x.q ≥ y.q
+        q = x.q - y.q
+        return _sign(xcoef * big(10) ^ q - ycoef)
     else
-        return farther_from_0
+        q = y.q - x.q
+        return _sign(xcoef  - ycoef * big(10) ^ q)
     end
 end
 
-function Base.:(<=)(x::Decimal, y::Decimal)
-    return x < y || x == y
-end
+Base.:(==)(x::Decimal, y::Decimal) = iszero(cmp(x, y))
+Base.:(<)(x::Decimal, y::Decimal) = cmp(x, y) < 0
+Base.:(<=)(x::Decimal, y::Decimal) = cmp(x, y) <= 0
 
 # Special case equality with AbstractFloat to allow comparison against Inf/Nan
 # which are not representable in Decimal

test/test_equals.jl

@@ -3,6 +3,34 @@ using Test
 
 @testset "Equality" begin
 
+@testset "cmp" begin
+    @test cmp(parse(Decimal, "-2" ), parse(Decimal, "-2")) == 0
+    @test cmp(parse(Decimal, "-2" ), parse(Decimal, "-1")) == -1
+    @test cmp(parse(Decimal, "-2" ), parse(Decimal, "0")) == -1
+    @test cmp(parse(Decimal, "-2" ), parse(Decimal, "1")) == -1
+    @test cmp(parse(Decimal, "-2" ), parse(Decimal, "2")) == -1
+    @test cmp(parse(Decimal, "-1" ), parse(Decimal, "-2")) == 1
+    @test cmp(parse(Decimal, "-1" ), parse(Decimal, "-1")) == 0
+    @test cmp(parse(Decimal, "-1" ), parse(Decimal, "0")) == -1
+    @test cmp(parse(Decimal, "-1" ), parse(Decimal, "1")) == -1
+    @test cmp(parse(Decimal, "-1" ), parse(Decimal, "2")) == -1
+    @test cmp(parse(Decimal, "0" ), parse(Decimal, "-2")) == 1
+    @test cmp(parse(Decimal, "0" ), parse(Decimal, "-1")) == 1
+    @test cmp(parse(Decimal, "0" ), parse(Decimal, "0")) == 0
+    @test cmp(parse(Decimal, "0" ), parse(Decimal, "1")) == -1
+    @test cmp(parse(Decimal, "0" ), parse(Decimal, "2")) == -1
+    @test cmp(parse(Decimal, "1" ), parse(Decimal, "-2")) == 1
+    @test cmp(parse(Decimal, "1" ), parse(Decimal, "-1")) == 1
+    @test cmp(parse(Decimal, "1" ), parse(Decimal, "0")) == 1
+    @test cmp(parse(Decimal, "1" ), parse(Decimal, "1")) == 0
+    @test cmp(parse(Decimal, "1" ), parse(Decimal, "2")) == -1
+    @test cmp(parse(Decimal, "2" ), parse(Decimal, "-2")) == 1
+    @test cmp(parse(Decimal, "2" ), parse(Decimal, "-1")) == 1
+    @test cmp(parse(Decimal, "2" ), parse(Decimal, "0")) == 1
+    @test cmp(parse(Decimal, "2" ), parse(Decimal, "1")) == 1
+    @test cmp(parse(Decimal, "2" ), parse(Decimal, "2")) == 0
+end
+
 @testset "isequal" begin
     @test isequal(Decimal(false, 2, -3), Decimal(false, 2, -3))
     @test !isequal(Decimal(false, 2, -3), Decimal(false, 2, 3))