Add mul_hi function for bit integers

Move the _mul_high function from base/multinverses.jl to base/int.jl.
Rename it to mul_hi.

Author: sumiya11

base/int.jl

@@ -96,6 +96,44 @@ inv(x::Integer) = float(one(x)) / float(x)
 # skip promotion for system integer types
 (/)(x::BitInteger, y::BitInteger) = float(x) / float(y)
 
+
+"""
+    mul_hi(a::T, b::T) where {T<:Integer}
+
+Returns the higher half of the product of `a` and `b`.
+
+# Examples
+```jldoctest
+julia> mul_hi(12345678987654321, 123456789)
+82624
+
+julia> (widen(12345678987654321) * 123456789) >> 64
+82624
+
+julia> mul_hi(0xff, 0xff)
+0xfe
+```
+"""
+function mul_hi(a::T, b::T) where {T<:BitInteger}
+    ((widen(a)*b) >>> (sizeof(a)*8)) % T
+end
+
+function mul_hi(a::UInt128, b::UInt128)
+    shift = sizeof(a)*4
+    mask = typemax(UInt128) >> shift
+    a1, a2 = a >>> shift, a & mask
+    b1, b2 = b >>> shift, b & mask
+    a1b1, a1b2, a2b1, a2b2 = a1*b1, a1*b2, a2*b1, a2*b2
+    carry = ((a1b2 & mask) + (a2b1 & mask) + (a2b2 >>> shift)) >>> shift
+    a1b1 + (a1b2 >>> shift) + (a2b1 >>> shift) + carry
+end
+
+function mul_hi(a::Int128, b::Int128)
+    shift = sizeof(a)*8 - 1
+    t1, t2 = (a >> shift) & b % UInt128, (b >> shift) & a % UInt128
+    (mul_hi(a % UInt128, b % UInt128) - t1 - t2) % Int128
+end
+
 """
     isodd(x::Number) -> Bool
 

base/multinverses.jl

@@ -2,7 +2,7 @@
 
 module MultiplicativeInverses
 
-import Base: div, divrem, rem, unsigned
+import Base: div, divrem, mul_hi, rem, unsigned
 using  Base: IndexLinear, IndexCartesian, tail
 export multiplicativeinverse
 
@@ -134,33 +134,13 @@ struct UnsignedMultiplicativeInverse{T<:Unsigned} <: MultiplicativeInverse{T}
 end
 UnsignedMultiplicativeInverse(x::Unsigned) = UnsignedMultiplicativeInverse{typeof(x)}(x)
 
-# Returns the higher half of the product a*b
-function _mul_high(a::T, b::T) where {T<:Union{Signed, Unsigned}}
-    ((widen(a)*b) >>> (sizeof(a)*8)) % T
-end
-
-function _mul_high(a::UInt128, b::UInt128)
-    shift = sizeof(a)*4
-    mask = typemax(UInt128) >> shift
-    a1, a2 = a >>> shift, a & mask
-    b1, b2 = b >>> shift, b & mask
-    a1b1, a1b2, a2b1, a2b2 = a1*b1, a1*b2, a2*b1, a2*b2
-    carry = ((a1b2 & mask) + (a2b1 & mask) + (a2b2 >>> shift)) >>> shift
-    a1b1 + (a1b2 >>> shift) + (a2b1 >>> shift) + carry
-end
-function _mul_high(a::Int128, b::Int128)
-    shift = sizeof(a)*8 - 1
-    t1, t2 = (a >> shift) & b % UInt128, (b >> shift) & a % UInt128
-    (_mul_high(a % UInt128, b % UInt128) - t1 - t2) % Int128
-end
-
 function div(a::T, b::SignedMultiplicativeInverse{T}) where T
-    x = _mul_high(a, b.multiplier)
+    x = mul_hi(a, b.multiplier)
     x += (a*b.addmul) % T
     ifelse(abs(b.divisor) == 1, a*b.divisor, (signbit(x) + (x >> b.shift)) % T)
 end
 function div(a::T, b::UnsignedMultiplicativeInverse{T}) where T
-    x = _mul_high(a, b.multiplier)
+    x = mul_hi(a, b.multiplier)
     x = ifelse(b.add, convert(T, convert(T, (convert(T, a - x) >>> 1)) + x), x)
     ifelse(b.divisor == 1, a, x >>> b.shift)
 end