Clean up mod1

Implement fld1 and fldmod1.
Add test cases.
Deprecate rem1.

Author: eschnett

base/deprecated.jl

@@ -873,6 +873,15 @@ end
 @deprecate chol(A::Number, ::Type{Val{:L}})         ctranspose(chol(A))
 @deprecate chol(A::AbstractMatrix, ::Type{Val{:L}}) ctranspose(chol(A))
 
+# Number updates
+
+# rem1 is inconsistent for x==0: The result should both have the same
+# sign as x, and should be non-zero.
+function rem1{T<:Real}(x::T, y::T)
+    depwarn("`rem1(x,y)` is discontinued, as it cannot be defined consistently for `x==0`. Rewrite the expression using `mod1` instead.", :rem1)
+    rem(x-1,y)+1
+end
+
 # Filesystem module updates
 
 @deprecate_binding FS Filesystem

base/exports.jl

@@ -357,7 +357,9 @@ export
     factor,
     factorial,
     fld,
+    fld1,
     fldmod,
+    fldmod1,
     flipsign,
     float,
     tryparse,

base/operators.jl

@@ -149,10 +149,11 @@ const % = rem
 .%(x::Real, y::Real) = x%y
 const ÷ = div
 
-# mod returns in [0,y) whereas mod1 returns in (0,y]
+# mod returns in [0,y) or (y,0] (for negative y),
+# whereas mod1 returns in (0,y] or [y,0)
 mod1{T<:Real}(x::T, y::T) = (m=mod(x,y); ifelse(m==0, y, m))
-rem1{T<:Real}(x::T, y::T) = rem(x-1,y)+1
-fld1{T<:Real}(x::T, y::T) = fld(x-1,y)+1
+fld1{T<:Real}(x::T, y::T) = fld(x - mod1(x,y), y)
+fldmod1{T<:Real}(x::T, y::T) = (m=mod1(x,y); (fld(x-m, y), m))
 
 # transpose
 transpose(x) = x

test/numbers.jl

@@ -2530,6 +2530,12 @@ end
 # test second branch, after all small primes in list have been searched
 @test factor(10009 * Int128(1000000000000037)) == Dict(10009=>1,1000000000000037=>1)
 
+@test all(x -> (m=mod1(x,3); 0<m<=3), -5:+5)
+@test all(x -> x == fld1(x,3)*3 + mod1(x,3), -5:+5)
+@test all(x -> fldmod1(x,3) == (fld1(x,3), mod1(x,3)))
+@test all(x -> (m=mod1(x,-3); -3<=m<0), -5:+5)
+@test all(x -> x == fld1(x,-3)*-3 + mod1(x,-3), -5:+5)
+@test all(x -> fldmod1(x,-3) == (fld1(x,-3), mod1(x,-3)))
 #Issue #5570
 @test map(x -> Int(mod1(UInt(x),UInt(5))), 0:15) == [5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5]