From 8fbc64ecd2c31138a75e34ff9bc799672f07a2b3 Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 13:12:53 +0900
Subject: [PATCH 1/8] Add docstrings and Update comments

---
 src/FixedPointNumbers.jl | 10 ++++++++--
 src/fixed.jl             | 17 ++++++++++++++---
 src/normed.jl            | 17 +++++++++++++----
 3 files changed, 35 insertions(+), 9 deletions(-)

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index 60ec762d..bf86cd9a 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -10,8 +10,14 @@ import Base: ==, <, <=, -, +, *, /, ~, isapprox,
 
 using Base: @pure
 
-# T => BaseType
-# f => Number of bits reserved for fractional part
+"""
+    FixedPoint{T <: Integer, f} <: Real
+
+Supertype of the two fixed-point number types: `Fixed{T, f}` and `Normed{T, f}`.
+
+The parameter `T` is the underlying machine representation and `f` is the number
+of fraction bits.
+"""
 abstract type FixedPoint{T <: Integer, f} <: Real end
 
 
diff --git a/src/fixed.jl b/src/fixed.jl
index 3576a610..82a43114 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -1,5 +1,16 @@
-# 32-bit fixed point; parameter `f` is the number of fraction bits
-struct Fixed{T <: Signed,f} <: FixedPoint{T,  f}
+"""
+    Fixed{T <: Signed, f} <: FixedPoint{T, f}
+
+`Fixed{T,f}` maps `Signed` integers from `-2^f` to `2^f` to the range
+[-1.0, 1.0]. For example, `Fixed{Int8,7}` maps `-128` to `-1.0` and `127` to
+`127/128 ≈ 0.992`.
+
+There are the typealiases for `Fixed` in the `QXfY` notation, where `Y` is
+the number of fractional bits (i.e. `f`), and `X+Y+1` equals the number of
+underlying bits used (`+1` means the sign bit). For example, `Q0f7` is aliased
+to `Fixed{Int8,7}` and `Q3f12` is aliased to `Fixed{Int16,12}`.
+"""
+struct Fixed{T <: Signed, f} <: FixedPoint{T, f}
     i::T
 
     # constructor for manipulating the representation;
@@ -37,7 +48,7 @@ abs(x::Fixed{T,f}) where {T,f} = Fixed{T,f}(abs(x.i),0)
 -(x::Fixed{T,f}, y::Fixed{T,f}) where {T,f} = Fixed{T,f}(x.i-y.i,0)
 
 # with truncation:
-#*{f}(x::Fixed32{f}, y::Fixed32{f}) = Fixed32{f}(Base.widemul(x.i,y.i)>>f,0)
+#*(x::Fixed{T,f}, y::Fixed{T,f}) = Fixed{T,f}(Base.widemul(x.i,y.i)>>f,0)
 # with rounding up:
 *(x::Fixed{T,f}, y::Fixed{T,f}) where {T,f} = Fixed{T,f}((Base.widemul(x.i,y.i) + (one(widen(T)) << (f-1)))>>f,0)
 
diff --git a/src/normed.jl b/src/normed.jl
index f0fa8aad..40d55553 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -1,7 +1,16 @@
-# Normed{T,f} maps UInts from 0 to 2^f-1 to the range [0.0, 1.0]
-# For example, Normed{UInt8,8} == N0f8 maps 0x00 to 0.0 and 0xff to 1.0
-
-struct Normed{T<:Unsigned,f} <: FixedPoint{T,f}
+"""
+    Normed{T <: Unsigned, f} <: FixedPoint{T, f}
+
+`Normed{T,f}` maps `Unsigned` integers from `0` to `2^f-1` to the range
+[0.0, 1.0]. For example, `Normed{UInt8,8}` maps `0x00` to `0.0` and `0xff` to
+`1.0`.
+
+There are the typealiases for `Normed` in the `NXfY` notation, where `Y` is
+the number of fractional bits (i.e. `f`), and `X+Y` equals the number of
+underlying bits used. For example, `N0f8` is aliased to `Normed{UInt8,8}` and
+`N4f12` is aliased to `Normed{UInt16,12}`.
+"""
+struct Normed{T <: Unsigned, f} <: FixedPoint{T, f}
     i::T
 
     Normed{T, f}(i::Integer,_) where {T,f} = new{T, f}(i%T)   # for setting by raw representation

From dfc366947dcabdfff59cb69371a7caa722dc722c Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 13:24:01 +0900
Subject: [PATCH 2/8] Modify scope of variables

---
 src/FixedPointNumbers.jl | 7 ++-----
 src/fixed.jl             | 3 ++-
 src/normed.jl            | 3 ++-
 3 files changed, 6 insertions(+), 7 deletions(-)

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index bf86cd9a..4fdd67f1 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -119,9 +119,6 @@ rawtype(::Type{FixedPoint{T,f}}) where {T <: Integer,f} = T
 rawtype(::Type{F}) where {F <: FixedPoint} = rawtype(supertype(F))
 rawtype(x::FixedPoint) = rawtype(typeof(x))
 
-# This IOBuffer is used during module definition to generate typealias names
-_iotypealias = IOBuffer()
-
 # Printing. These are used to generate type-symbols, so we need them
 # before we include any files.
 function showtype(io::IO, ::Type{X}) where {X <: FixedPoint}
@@ -132,10 +129,10 @@ function showtype(io::IO, ::Type{X}) where {X <: FixedPoint}
     io
 end
 function show(io::IO, x::FixedPoint{T,f}) where {T,f}
-    show(io, round(convert(Float64,x), digits=ceil(Int,f/_log2_10)))
+    log10_2 = 0.3010299956639812
+    show(io, round(convert(Float64,x), digits=ceil(Int, f * log10_2)))
     get(io, :compact, false) || showtype(io, typeof(x))
 end
-const _log2_10 = 3.321928094887362
 
 function Base.showarg(io::IO, a::Array{T}, toplevel) where {T<:FixedPoint}
     toplevel || print(io, "::")
diff --git a/src/fixed.jl b/src/fixed.jl
index 82a43114..edee8b8c 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -31,8 +31,9 @@ typechar(::Type{X}) where {X <: Fixed} = 'Q'
 signbits(::Type{X}) where {X <: Fixed} = 1
 
 for T in (Int8, Int16, Int32, Int64)
+    io = IOBuffer()
     for f in 0:sizeof(T)*8-1
-        sym = Symbol(String(take!(showtype(_iotypealias, Fixed{T,f}))))
+        sym = Symbol(String(take!(showtype(io, Fixed{T,f}))))
         @eval begin
             const $sym = Fixed{$T,$f}
             export $sym
diff --git a/src/normed.jl b/src/normed.jl
index 40d55553..34ba405d 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -25,8 +25,9 @@ typechar(::Type{X}) where {X <: Normed} = 'N'
 signbits(::Type{X}) where {X <: Normed} = 0
 
 for T in (UInt8, UInt16, UInt32, UInt64)
+    io = IOBuffer()
     for f in 1:sizeof(T)*8
-        sym = Symbol(String(take!(showtype(_iotypealias, Normed{T,f}))))
+        sym = Symbol(String(take!(showtype(io, Normed{T,f}))))
         @eval begin
             const $sym = Normed{$T,$f}
             export $sym

From 745ce91914b7d6c5cafc09eeb568739c1ba58c2c Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 17:06:39 +0900
Subject: [PATCH 3/8] Put utility functions and macros into a new file
 "utilities.jl"

---
 src/FixedPointNumbers.jl | 31 +++++++-------------
 src/fixed.jl             |  8 +++---
 src/normed.jl            | 62 +++++++++++++++++-----------------------
 src/utilities.jl         | 31 ++++++++++++++++++++
 test/fixed.jl            |  3 +-
 test/normed.jl           |  9 +++---
 6 files changed, 78 insertions(+), 66 deletions(-)
 create mode 100644 src/utilities.jl

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index 4fdd67f1..1bf00714 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -31,6 +31,9 @@ export
 # Functions
     scaledual
 
+include("utilities.jl")
+
+# reinterpretation
 reinterpret(x::FixedPoint) = x.i
 reinterpret(::Type{T}, x::FixedPoint{T,f}) where {T,f} = x.i
 
@@ -64,20 +67,6 @@ typemin(::Type{T}) where {T <: FixedPoint} = T(typemin(rawtype(T)), 0)
 floatmin(::Type{T}) where {T <: FixedPoint} = eps(T)
 floatmax(::Type{T}) where {T <: FixedPoint} = typemax(T)
 
-widen1(::Type{Int8})   = Int16
-widen1(::Type{UInt8})  = UInt16
-widen1(::Type{Int16})  = Int32
-widen1(::Type{UInt16}) = UInt32
-widen1(::Type{Int32})  = Int64
-widen1(::Type{UInt32}) = UInt64
-widen1(::Type{Int64})  = Int128
-widen1(::Type{UInt64}) = UInt128
-widen1(::Type{Int128}) = Int128
-widen1(::Type{UInt128}) = UInt128
-widen1(x::Integer) = x % widen1(typeof(x))
-
-const ShortInts = Union{Int8,UInt8,Int16,UInt16}
-const LongInts = Union{UInt64, UInt128, Int64, Int128, BigInt}
 
 """
     floattype(::Type{T})
@@ -124,7 +113,7 @@ rawtype(x::FixedPoint) = rawtype(typeof(x))
 function showtype(io::IO, ::Type{X}) where {X <: FixedPoint}
     print(io, typechar(X))
     f = nbitsfrac(X)
-    m = sizeof(X)*8-f-signbits(X)
+    m = bitwidth(X)-f-signbits(X)
     print(io, m, 'f', f)
     io
 end
@@ -188,13 +177,13 @@ scaledual(::Type{Tdual}, x::FixedPoint) where Tdual = convert(Tdual, 1/rawone(x)
 scaledual(::Type{Tdual}, x::AbstractArray{T}) where {Tdual, T <: FixedPoint} =
     convert(Tdual, 1/rawone(T)), reinterpret(rawtype(T), x)
 
-@noinline function throw_converterror(::Type{T}, x) where {T <: FixedPoint}
-    n = 2^(8*sizeof(T))
-    bitstring = sizeof(T) == 1 ? "an 8-bit" : "a $(8*sizeof(T))-bit"
+@noinline function throw_converterror(::Type{X}, x) where {X <: FixedPoint}
+    n = 2^bitwidth(X)
+    bitstring = bitwidth(X) == 8 ? "an 8-bit" : "a $(bitwidth(X))-bit"
     io = IOBuffer()
-    show(IOContext(io, :compact=>true), typemin(T)); Tmin = String(take!(io))
-    show(IOContext(io, :compact=>true), typemax(T)); Tmax = String(take!(io))
-    throw(ArgumentError("$T is $bitstring type representing $n values from $Tmin to $Tmax; cannot represent $x"))
+    show(IOContext(io, :compact=>true), typemin(X)); Xmin = String(take!(io))
+    show(IOContext(io, :compact=>true), typemax(X)); Xmax = String(take!(io))
+    throw(ArgumentError("$X is $bitstring type representing $n values from $Xmin to $Xmax; cannot represent $x"))
 end
 
 rand(::Type{T}) where {T <: FixedPoint} = reinterpret(T, rand(rawtype(T)))
diff --git a/src/fixed.jl b/src/fixed.jl
index edee8b8c..f2e97d9d 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -32,7 +32,7 @@ signbits(::Type{X}) where {X <: Fixed} = 1
 
 for T in (Int8, Int16, Int32, Int64)
     io = IOBuffer()
-    for f in 0:sizeof(T)*8-1
+    for f in 0:bitwidth(T)-1
         sym = Symbol(String(take!(showtype(io, Fixed{T,f}))))
         @eval begin
             const $sym = Fixed{$T,$f}
@@ -96,13 +96,13 @@ promote_rule(::Type{Fixed{T,f}}, ::Type{Rational{TR}}) where {T,f,TR} = Rational
     f = max(f1, f2)  # ensure we have enough precision
     T = promote_type(T1, T2)
     # make sure we have enough integer bits
-    i1, i2 = 8*sizeof(T1)-f1, 8*sizeof(T2)-f2  # number of integer bits for each
-    i = 8*sizeof(T)-f
+    i1, i2 = bitwidth(T1)-f1, bitwidth(T2)-f2  # number of integer bits for each
+    i = bitwidth(T)-f
     while i < max(i1, i2)
         Tw = widen1(T)
         T == Tw && break
         T = Tw
-        i = 8*sizeof(T)-f
+        i = bitwidth(T)-f
     end
     :(Fixed{$T,$f})
 end
diff --git a/src/normed.jl b/src/normed.jl
index 34ba405d..6711da68 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -26,7 +26,7 @@ signbits(::Type{X}) where {X <: Normed} = 0
 
 for T in (UInt8, UInt16, UInt32, UInt64)
     io = IOBuffer()
-    for f in 1:sizeof(T)*8
+    for f in 1:bitwidth(T)
         sym = Symbol(String(take!(showtype(io, Normed{T,f}))))
         @eval begin
             const $sym = Normed{$T,$f}
@@ -39,7 +39,7 @@ reinterpret(::Type{Normed{T,f}}, x::T) where {T <: Unsigned,f} = Normed{T,f}(x,
 
 zero(::Type{Normed{T,f}}) where {T,f} = Normed{T,f}(zero(T),0)
 function oneunit(::Type{T}) where {T <: Normed}
-    T(typemax(rawtype(T)) >> (8*sizeof(T)-nbitsfrac(T)), 0)
+    T(typemax(rawtype(T)) >> (bitwidth(T)-nbitsfrac(T)), 0)
 end
 one(::Type{T}) where {T <: Normed} = oneunit(T)
 zero(x::Normed) = zero(typeof(x))
@@ -76,36 +76,34 @@ function _convert(::Type{U}, x::Float16) where {T, f, U <: Normed{T,f}}
     end
     return _convert(U, Float32(x))
 end
-function _convert(::Type{U}, x::Tf) where {T, f, U <: Normed{T,f}, Tf <: Union{Float32, Float64}}
-    if T == UInt128 && f == 53
-        0 <= x <= Tf(3.777893186295717e22) || throw_converterror(U, x)
+function _convert(::Type{N}, x::Tf) where {T, f, N <: Normed{T,f}, Tf <: Union{Float32, Float64}}
+    if T === UInt128 && f == 53
+        0 <= x <= Tf(3.777893186295717e22) || throw_converterror(N, x)
     else
-        0 <= x <= Tf((typemax(T)-rawone(U))/rawone(U)+1) || throw_converterror(U, x)
+        0 <= x <= Tf((typemax(T)-rawone(N))/rawone(N)+1) || throw_converterror(N, x)
     end
 
-    significand_bits = Tf == Float64 ? 52 : 23
-    if f <= (significand_bits + 1) && sizeof(T) * 8 < significand_bits
-        return reinterpret(U, unsafe_trunc(T, round(rawone(U) * x)))
+    if f <= (significand_bits(Tf) + 1) && bitwidth(T) < significand_bits(Tf)
+        return reinterpret(N, unsafe_trunc(T, round(rawone(N) * x)))
     end
     # cf. the implementation of `frexp`
-    Tw = f < sizeof(T) * 8 ? T : widen1(T)
-    bits = sizeof(Tw) * 8 - 1
-    xu = reinterpret(Tf == Float64 ? UInt64 : UInt32, x)
-    k = Int(xu >> significand_bits)
-    k == 0 && return zero(U) # neglect subnormal numbers
-    significand = xu | (one(xu) << significand_bits)
-    yh = unsafe_trunc(Tw, significand) << (bits - significand_bits)
-    exponent_bias = Tf == Float64 ? 1023 : 127
-    ex = exponent_bias - k + bits - f
+    Tw = f < bitwidth(T) ? T : widen1(T)
+    bits = bitwidth(Tw) - 1
+    xu = reinterpret(Unsigned, x)
+    k = Int(xu >> significand_bits(Tf))
+    k == 0 && return zero(N) # neglect subnormal numbers
+    significand = xu | (oneunit(xu) << significand_bits(Tf))
+    yh = unsafe_trunc(Tw, significand) << (bits - significand_bits(Tf))
+    ex = exponent_bias(Tf) - k + bits - f
     yi = bits >= f ? yh - (yh >> f) : yh
     if ex <= 0
-        ex == 0 && return reinterpret(U, unsafe_trunc(T, yi))
-        ex != -1 || signbit(signed(yi)) && return typemax(U)
-        return reinterpret(U, unsafe_trunc(T, yi + yi))
+        ex == 0 && return reinterpret(N, unsafe_trunc(T, yi))
+        ex != -1 || signbit(signed(yi)) && return typemax(N)
+        return reinterpret(N, unsafe_trunc(T, yi + yi))
     end
-    ex > bits && return reinterpret(U, ex == bits + 1 ? one(T) : zero(T))
-    yi += one(Tw)<<((ex - 1) & bits) # RoundNearestTiesUp
-    return reinterpret(U, unsafe_trunc(T, yi >> (ex & bits)))
+    ex > bits && return reinterpret(N, ex == bits + 1 ? oneunit(T) : zero(T))
+    yi += oneunit(Tw)<<((ex - 1) & bits) # RoundNearestTiesUp
+    return reinterpret(N, unsafe_trunc(T, yi >> (ex & bits)))
 end
 
 rem(x::T, ::Type{T}) where {T <: Normed} = x
@@ -115,14 +113,6 @@ rem(x::Float16, ::Type{T}) where {T <: Normed} = rem(Float32(x), T)  # avoid ove
 
 float(x::Normed) = convert(floattype(x), x)
 
-macro f32(x::Float64) # just for hexadecimal floating-point literals
-    :(Float32($x))
-end
-macro exp2(n)
-     :(_exp2(Val($(esc(n)))))
-end
-_exp2(::Val{N}) where {N} = exp2(N)
-
 # for Julia v1.0, which does not fold `div_float` before inlining
 inv_rawone(x) = (@generated) ? (y = 1.0 / rawone(x); :($y)) : 1.0 / rawone(x)
 
@@ -275,7 +265,7 @@ function round(x::Normed{T,f}) where {T,f}
             Normed{T,f}(y+oneunit(Normed{T,f})) : y
 end
 function ceil(x::Normed{T,f}) where {T,f}
-    k = 8*sizeof(T)-f
+    k = bitwidth(T)-f
     mask = (typemax(T)<<k)>>k
     y = trunc(x)
     return convert(T, reinterpret(x)-reinterpret(y)) & (mask)>0 ?
@@ -324,13 +314,13 @@ end
     f = max(f1, f2)  # ensure we have enough precision
     T = promote_type(T1, T2)
     # make sure we have enough integer bits
-    i1, i2 = 8*sizeof(T1)-f1, 8*sizeof(T2)-f2  # number of integer bits for each
-    i = 8*sizeof(T)-f
+    i1, i2 = bitwidth(T1)-f1, bitwidth(T2)-f2  # number of integer bits for each
+    i = bitwidth(T)-f
     while i < max(i1, i2)
         Tw = widen1(T)
         T == Tw && break
         T = Tw
-        i = 8*sizeof(T)-f
+        i = bitwidth(T)-f
     end
     :(Normed{$T,$f})
 end
diff --git a/src/utilities.jl b/src/utilities.jl
new file mode 100644
index 00000000..5bc17388
--- /dev/null
+++ b/src/utilities.jl
@@ -0,0 +1,31 @@
+# utility functions and macros, which are independent of `FixedPoint`
+bitwidth(T::Type) = 8sizeof(T)
+
+widen1(::Type{Int8})    = Int16
+widen1(::Type{UInt8})   = UInt16
+widen1(::Type{Int16})   = Int32
+widen1(::Type{UInt16})  = UInt32
+widen1(::Type{Int32})   = Int64
+widen1(::Type{UInt32})  = UInt64
+widen1(::Type{Int64})   = Int128
+widen1(::Type{UInt64})  = UInt128
+widen1(::Type{Int128})  = Int128
+widen1(::Type{UInt128}) = UInt128
+widen1(x::Integer) = x % widen1(typeof(x))
+
+const ShortInts = Union{Int8, UInt8, Int16, UInt16}
+const LongInts = Union{Int64, UInt64, Int128, UInt128, BigInt}
+
+macro f32(x::Float64) # just for hexadecimal floating-point literals
+    :(Float32($x))
+end
+macro exp2(n)
+     :(_exp2(Val($(esc(n)))))
+end
+_exp2(::Val{N}) where {N} = exp2(N)
+
+# these are defined in julia/float.jl or julia/math.jl, but not exported
+significand_bits(::Type{Float32}) = 23
+significand_bits(::Type{Float64}) = 52
+exponent_bias(::Type{Float32}) = 127
+exponent_bias(::Type{Float64}) = 1023
diff --git a/test/fixed.jl b/test/fixed.jl
index cc8ea449..9fd26bbb 100644
--- a/test/fixed.jl
+++ b/test/fixed.jl
@@ -1,4 +1,5 @@
 using FixedPointNumbers, Test
+using FixedPointNumbers: bitwidth
 
 function test_op(fun::F, ::Type{T}, fx, fy, fxf, fyf, tol) where {F,T}
     # Make sure that the result is representable
@@ -164,7 +165,7 @@ end
                  (Int64, 63))
         tmax = typemax(Fixed{T, f})
         @test tmax == BigInt(typemax(T)) / BigInt(2)^f
-        tol = (tmax + BigFloat(1.0)) / (sizeof(T) * 8)
+        tol = (tmax + BigFloat(1.0)) / bitwidth(T)
         for x in range(-1, stop=BigFloat(tmax)-tol, length=50)
             @test abs(Fixed{T, f}(x) - x) <= tol
         end
diff --git a/test/normed.jl b/test/normed.jl
index 6fb4d125..c663aaef 100644
--- a/test/normed.jl
+++ b/test/normed.jl
@@ -1,4 +1,5 @@
 using FixedPointNumbers, Test
+using FixedPointNumbers: bitwidth
 
 @testset "reinterpret" begin
     @test reinterpret(N0f8, 0xa2).i  === 0xa2
@@ -106,7 +107,7 @@ end
     # issue 102
     for T in (UInt8, UInt16, UInt32, UInt64, UInt128)
         for Tf in (Float16, Float32, Float64)
-            @testset "Normed{$T,$f}(::$Tf)" for f = 1:sizeof(T)*8
+            @testset "Normed{$T,$f}(::$Tf)" for f = 1:bitwidth(T)
                 U = Normed{T,f}
                 r = FixedPointNumbers.rawone(U)
 
@@ -123,7 +124,7 @@ end
                 isinf(input_upper) && continue # for Julia v0.7
                 @test reinterpret(U(input_upper)) == T(min(round(BigFloat(input_upper) * r), typemax(T)))
 
-                input_exp2 = Tf(exp2(sizeof(T) * 8 - f))
+                input_exp2 = Tf(exp2(bitwidth(T) - f))
                 isinf(input_exp2) && continue
                 @test reinterpret(U(input_exp2)) == T(input_exp2) * r
             end
@@ -143,7 +144,7 @@ end
 
     for Tf in (Float16, Float32, Float64)
         @testset "$Tf(::Normed{$Ti})" for Ti in (UInt8, UInt16)
-            @testset "$Tf(::Normed{$Ti,$f})" for f = 1:(sizeof(Ti)*8)
+            @testset "$Tf(::Normed{$Ti,$f})" for f = 1:bitwidth(Ti)
                 T = Normed{Ti,f}
                 float_err = 0.0
                 for i = typemin(Ti):typemax(Ti)
@@ -156,7 +157,7 @@ end
             end
         end
         @testset "$Tf(::Normed{$Ti})" for Ti in (UInt32, UInt64, UInt128)
-            @testset "$Tf(::Normed{$Ti,$f})" for f = 1:(sizeof(Ti)*8)
+            @testset "$Tf(::Normed{$Ti,$f})" for f = 1:bitwidth(Ti)
                 T = Normed{Ti,f}
                 error_count = 0
                 for i in vcat(Ti(0x00):Ti(0xFF), (typemax(Ti)-0xFF):typemax(Ti))

From 91554633dccaa6a3bb8919ffabc5547468803232 Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 20:58:26 +0900
Subject: [PATCH 4/8] Clean traits and identities up

This changes the exception type of `one(Q0f7)` etc. from `InexactError` to `ArgumentError`.
---
 src/FixedPointNumbers.jl | 58 ++++++++++++++++++++++------------------
 src/fixed.jl             |  5 ++++
 src/normed.jl            | 12 ++-------
 test/fixed.jl            |  8 ++++++
 4 files changed, 47 insertions(+), 36 deletions(-)

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index 1bf00714..f320c777 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -37,6 +37,10 @@ include("utilities.jl")
 reinterpret(x::FixedPoint) = x.i
 reinterpret(::Type{T}, x::FixedPoint{T,f}) where {T,f} = x.i
 
+# static parameters
+nbitsfrac(::Type{X}) where {T, f, X <: FixedPoint{T,f}} = f
+rawtype(::Type{X}) where {T, X <: FixedPoint{T}} = T
+
 # construction using the (approximate) intended value, i.e., N0f8
 *(x::Real, ::Type{X}) where {X<:FixedPoint} = X(x)
 
@@ -61,7 +65,17 @@ end
 # predicates
 isinteger(x::FixedPoint{T,f}) where {T,f} = (x.i&(1<<f-1)) == 0
 
+# identities
+zero(::Type{X}) where {X <: FixedPoint} = X(zero(rawtype(X)), 0)
+oneunit(::Type{X}) where {X <: FixedPoint} = X(rawone(X), 0)
+one(::Type{X}) where {X <: FixedPoint} = oneunit(X)
+
+# for Julia v1.0, which does not fold `div_float` before inlining
+inv_rawone(x) = (@generated) ? (y = 1.0 / rawone(x); :($y)) : 1.0 / rawone(x)
+
 # traits
+sizeof(::Type{X}) where {X <: FixedPoint} = sizeof(rawtype(X))
+eps(::Type{X}) where {X <: FixedPoint} = X(oneunit(rawtype(X)), 0)
 typemax(::Type{T}) where {T <: FixedPoint} = T(typemax(rawtype(T)), 0)
 typemin(::Type{T}) where {T <: FixedPoint} = T(typemin(rawtype(T)), 0)
 floatmin(::Type{T}) where {T <: FixedPoint} = eps(T)
@@ -95,18 +109,25 @@ floattype(::Type{T}) where {T <: Real} = T # fallback
 floattype(::Type{T}) where {T <: Union{ShortInts, Bool}} = Float32
 floattype(::Type{T}) where {T <: Integer} = Float64
 floattype(::Type{T}) where {T <: LongInts} = BigFloat
-floattype(::Type{FixedPoint{T,f}}) where {T <: ShortInts,f} = Float32
-floattype(::Type{FixedPoint{T,f}}) where {T <: Integer,f} = Float64
-floattype(::Type{FixedPoint{T,f}}) where {T <: LongInts,f} = BigFloat
-floattype(::Type{F}) where {F <: FixedPoint} = floattype(supertype(F))
-floattype(x::FixedPoint) = floattype(typeof(x))
-
-nbitsfrac(::Type{FixedPoint{T,f}}) where {T <: Integer,f} = f
-nbitsfrac(::Type{F}) where {F <: FixedPoint} = nbitsfrac(supertype(F))
+floattype(::Type{X}) where {T <: ShortInts, X <: FixedPoint{T}} = Float32
+floattype(::Type{X}) where {T <: Integer, X <: FixedPoint{T}} = Float64
+floattype(::Type{X}) where {T <: LongInts, X <: FixedPoint{T}} = BigFloat
 
-rawtype(::Type{FixedPoint{T,f}}) where {T <: Integer,f} = T
-rawtype(::Type{F}) where {F <: FixedPoint} = rawtype(supertype(F))
-rawtype(x::FixedPoint) = rawtype(typeof(x))
+for f in (:zero, :oneunit, :one, :eps, :rawone, :rawtype, :floattype)
+    @eval begin
+        $f(x::FixedPoint) = $f(typeof(x))
+    end
+end
+for f in (:div, :fld, :fld1)
+    @eval begin
+        $f(x::X, y::X) where {X <: FixedPoint} = $f(x.i, y.i)
+    end
+end
+for f in (:rem, :mod, :mod1, :min, :max)
+    @eval begin
+        $f(x::X, y::X) where {X <: FixedPoint} = X($f(x.i, y.i), 0)
+    end
+end
 
 # Printing. These are used to generate type-symbols, so we need them
 # before we include any files.
@@ -136,10 +157,6 @@ include("normed.jl")
 include("deprecations.jl")
 const UF = (N0f8, N6f10, N4f12, N2f14, N0f16)
 
-eps(::Type{T}) where {T <: FixedPoint} = T(oneunit(rawtype(T)),0)
-eps(::T) where {T <: FixedPoint} = eps(T)
-sizeof(::Type{T}) where {T <: FixedPoint} = sizeof(rawtype(T))
-
 # Promotions for reductions
 const Treduce = Float64
 Base.add_sum(x::FixedPoint, y::FixedPoint) = Treduce(x) + Treduce(y)
@@ -150,17 +167,6 @@ Base.reduce_empty(::typeof(Base.mul_prod), ::Type{F}) where {F<:FixedPoint} = on
 Base.reduce_first(::typeof(Base.mul_prod), x::FixedPoint)  = Treduce(x)
 
 
-for f in (:div, :fld, :fld1)
-    @eval begin
-        $f(x::T, y::T) where {T <: FixedPoint} = $f(reinterpret(x),reinterpret(y))
-    end
-end
-for f in (:rem, :mod, :mod1, :min, :max)
-    @eval begin
-        $f(x::T, y::T) where {T <: FixedPoint} = T($f(reinterpret(x),reinterpret(y)),0)
-    end
-end
-
 """
     sd, ad = scaledual(s::Number, a)
 
diff --git a/src/fixed.jl b/src/fixed.jl
index f2e97d9d..5333b899 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -41,6 +41,11 @@ for T in (Int8, Int16, Int32, Int64)
     end
 end
 
+function rawone(::Type{Fixed{T,f}}) where {T, f}
+    f >= bitwidth(T)-1 && throw_converterror(Fixed{T,f}, 1)
+    oneunit(T) << f
+end
+
 # basic operators
 -(x::Fixed{T,f}) where {T,f} = Fixed{T,f}(-x.i,0)
 abs(x::Fixed{T,f}) where {T,f} = Fixed{T,f}(abs(x.i),0)
diff --git a/src/normed.jl b/src/normed.jl
index 6711da68..5d019507 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -37,15 +37,9 @@ end
 
 reinterpret(::Type{Normed{T,f}}, x::T) where {T <: Unsigned,f} = Normed{T,f}(x, 0)
 
-zero(::Type{Normed{T,f}}) where {T,f} = Normed{T,f}(zero(T),0)
-function oneunit(::Type{T}) where {T <: Normed}
-    T(typemax(rawtype(T)) >> (bitwidth(T)-nbitsfrac(T)), 0)
+function rawone(::Type{Normed{T,f}}) where {T <: Unsigned, f}
+    typemax(T) >> (bitwidth(T) - f)
 end
-one(::Type{T}) where {T <: Normed} = oneunit(T)
-zero(x::Normed) = zero(typeof(x))
-oneunit(x::Normed) =  one(typeof(x))
-one(x::Normed) = oneunit(x)
-rawone(v) = reinterpret(one(v))
 
 # Conversions
 function Normed{T,f}(x::Normed{T2}) where {T <: Unsigned,T2 <: Unsigned,f}
@@ -113,8 +107,6 @@ rem(x::Float16, ::Type{T}) where {T <: Normed} = rem(Float32(x), T)  # avoid ove
 
 float(x::Normed) = convert(floattype(x), x)
 
-# for Julia v1.0, which does not fold `div_float` before inlining
-inv_rawone(x) = (@generated) ? (y = 1.0 / rawone(x); :($y)) : 1.0 / rawone(x)
 
 function (::Type{T})(x::Normed) where {T <: AbstractFloat}
     # The following optimization for constant division may cause rounding errors.
diff --git a/test/fixed.jl b/test/fixed.jl
index 9fd26bbb..290c04e0 100644
--- a/test/fixed.jl
+++ b/test/fixed.jl
@@ -50,6 +50,14 @@ function test_fixed(::Type{T}, f) where {T}
     end
 end
 
+@testset "inexactness" begin
+    @test_throws InexactError Q0f7(-2)
+    # TODO: change back to InexactError when it allows message strings
+    @test_throws ArgumentError one(Q0f15)
+    @test_throws ArgumentError oneunit(Q0f31)
+    @test_throws ArgumentError one(Fixed{Int8,8})
+end
+
 @testset "conversion" begin
     @test isapprox(convert(Fixed{Int8,7}, 0.8), 0.797, atol=0.001)
     @test isapprox(convert(Fixed{Int8,7}, 0.9), 0.898, atol=0.001)

From e5be990052b807d5ac704d828fbd228970b5c714 Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 21:14:45 +0900
Subject: [PATCH 5/8] Commonize `reinterpret` and `float` between `Fixed` and
 `Normed`

---
 src/FixedPointNumbers.jl |  3 +++
 src/fixed.jl             |  3 ---
 src/normed.jl            |  4 ----
 test/fixed.jl            | 10 ++++++++++
 test/normed.jl           |  2 ++
 5 files changed, 15 insertions(+), 7 deletions(-)

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index f320c777..0042f4e1 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -36,6 +36,7 @@ include("utilities.jl")
 # reinterpretation
 reinterpret(x::FixedPoint) = x.i
 reinterpret(::Type{T}, x::FixedPoint{T,f}) where {T,f} = x.i
+reinterpret(::Type{X}, x::T) where {T <: Integer, X <: FixedPoint{T}} = X(x, 0)
 
 # static parameters
 nbitsfrac(::Type{X}) where {T, f, X <: FixedPoint{T,f}} = f
@@ -113,6 +114,8 @@ floattype(::Type{X}) where {T <: ShortInts, X <: FixedPoint{T}} = Float32
 floattype(::Type{X}) where {T <: Integer, X <: FixedPoint{T}} = Float64
 floattype(::Type{X}) where {T <: LongInts, X <: FixedPoint{T}} = BigFloat
 
+float(x::FixedPoint) = convert(floattype(x), x)
+
 for f in (:zero, :oneunit, :one, :eps, :rawone, :rawtype, :floattype)
     @eval begin
         $f(x::FixedPoint) = $f(typeof(x))
diff --git a/src/fixed.jl b/src/fixed.jl
index 5333b899..60c97618 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -25,8 +25,6 @@ Fixed{T,f}(x::Integer) where {T,f} = Fixed{T,f}(round(T, convert(widen1(T),x)<<f
 Fixed{T,f}(x::AbstractFloat) where {T,f} = Fixed{T,f}(round(T, trunc(widen1(T),x)<<f + rem(x,1)*(one(widen1(T))<<f)),0)
 Fixed{T,f}(x::Rational) where {T,f} = Fixed{T,f}(x.num)/Fixed{T,f}(x.den)
 
-reinterpret(::Type{Fixed{T,f}}, x::T) where {T <: Signed,f} = Fixed{T,f}(x, 0)
-
 typechar(::Type{X}) where {X <: Fixed} = 'Q'
 signbits(::Type{X}) where {X <: Fixed} = 1
 
@@ -73,7 +71,6 @@ end
 rem(x::Integer, ::Type{Fixed{T,f}}) where {T,f} = Fixed{T,f}(rem(x,T)<<f,0)
 rem(x::Real,    ::Type{Fixed{T,f}}) where {T,f} = Fixed{T,f}(rem(Integer(trunc(x)),T)<<f + rem(Integer(round(rem(x,1)*(one(widen1(T))<<f))),T),0)
 
-float(x::Fixed) = convert(floattype(x), x)
 
 Base.BigFloat(x::Fixed{T,f}) where {T,f} =
     BigFloat(x.i>>f) + BigFloat(x.i&(one(widen1(T))<<f - 1))/BigFloat(one(widen1(T))<<f)
diff --git a/src/normed.jl b/src/normed.jl
index 5d019507..3fc645f8 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -35,8 +35,6 @@ for T in (UInt8, UInt16, UInt32, UInt64)
     end
 end
 
-reinterpret(::Type{Normed{T,f}}, x::T) where {T <: Unsigned,f} = Normed{T,f}(x, 0)
-
 function rawone(::Type{Normed{T,f}}) where {T <: Unsigned, f}
     typemax(T) >> (bitwidth(T) - f)
 end
@@ -105,8 +103,6 @@ rem(x::Normed, ::Type{T}) where {T <: Normed} = reinterpret(T, _unsafe_trunc(raw
 rem(x::Real, ::Type{T}) where {T <: Normed} = reinterpret(T, _unsafe_trunc(rawtype(T), round(rawone(T)*x)))
 rem(x::Float16, ::Type{T}) where {T <: Normed} = rem(Float32(x), T)  # avoid overflow
 
-float(x::Normed) = convert(floattype(x), x)
-
 
 function (::Type{T})(x::Normed) where {T <: AbstractFloat}
     # The following optimization for constant division may cause rounding errors.
diff --git a/test/fixed.jl b/test/fixed.jl
index 290c04e0..adbc5dfb 100644
--- a/test/fixed.jl
+++ b/test/fixed.jl
@@ -50,6 +50,16 @@ function test_fixed(::Type{T}, f) where {T}
     end
 end
 
+@testset "reinterpret" begin
+    @test reinterpret(Q0f7, signed(0xa2)) === -0.734375Q0f7
+    @test reinterpret(Q5f10, signed(0x00a2)) === 0.158203125Q5f10
+
+    @test reinterpret(reinterpret(Q0f7, signed(0xa2))) === signed(0xa2)
+    @test reinterpret(reinterpret(Q5f10, signed(0x00a2))) === signed(0x00a2)
+
+    @test reinterpret(Int8, 0.5Q0f7) === signed(0x40)
+end
+
 @testset "inexactness" begin
     @test_throws InexactError Q0f7(-2)
     # TODO: change back to InexactError when it allows message strings
diff --git a/test/normed.jl b/test/normed.jl
index c663aaef..4c0430ea 100644
--- a/test/normed.jl
+++ b/test/normed.jl
@@ -14,6 +14,8 @@ using FixedPointNumbers: bitwidth
     @test reinterpret(reinterpret(N2f14, 0x00a2)) === 0x00a2
     @test reinterpret(reinterpret(N0f16, 0x00a2)) === 0x00a2
 
+    @test reinterpret(UInt8, 1N0f8) === 0xff
+
     @test 0.635N0f8   == N0f8(0.635)
     @test 0.635N6f10 == N6f10(0.635)
     @test 0.635N4f12 == N4f12(0.635)

From bbf96ff3fd26e25e0d7e27c4858191cfa7833efe Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 21:41:42 +0900
Subject: [PATCH 6/8] Add support for `minmax` and `bswap` for `Fixed`

Although `minmax` has been available, but this specializes it.
---
 src/FixedPointNumbers.jl | 7 +++++++
 src/normed.jl            | 8 --------
 test/fixed.jl            | 5 +++++
 3 files changed, 12 insertions(+), 8 deletions(-)

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index 0042f4e1..60911c42 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -116,6 +116,13 @@ floattype(::Type{X}) where {T <: LongInts, X <: FixedPoint{T}} = BigFloat
 
 float(x::FixedPoint) = convert(floattype(x), x)
 
+function minmax(x::X, y::X) where {X <: FixedPoint}
+    a, b = minmax(reinterpret(x), reinterpret(y))
+    X(a,0), X(b,0)
+end
+
+bswap(x::X) where {X <: FixedPoint} = sizeof(X) == 1 ? x : X(bswap(x.i), 0)
+
 for f in (:zero, :oneunit, :one, :eps, :rawone, :rawtype, :floattype)
     @eval begin
         $f(x::FixedPoint) = $f(typeof(x))
diff --git a/src/normed.jl b/src/normed.jl
index 3fc645f8..5acf0991 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -269,14 +269,6 @@ isfinite(x::Normed) = true
 isnan(x::Normed) = false
 isinf(x::Normed) = false
 
-bswap(x::Normed{UInt8,f}) where {f} = x
-bswap(x::Normed)  = typeof(x)(bswap(reinterpret(x)),0)
-
-function minmax(x::T, y::T) where {T <: Normed}
-    a, b = minmax(reinterpret(x), reinterpret(y))
-    T(a,0), T(b,0)
-end
-
 # Iteration
 # The main subtlety here is that iterating over N0f8(0):N0f8(1) will wrap around
 # unless we iterate using a wider type
diff --git a/test/fixed.jl b/test/fixed.jl
index adbc5dfb..96f7cc6a 100644
--- a/test/fixed.jl
+++ b/test/fixed.jl
@@ -190,6 +190,11 @@ end
     end
 end
 
+@testset "low-level arithmetic" begin
+    @test bswap(Q0f7(0.5)) === Q0f7(0.5)
+    @test bswap(Q0f15(0.5)) === reinterpret(Q0f15, signed(0x0040))
+end
+
 @testset "Promotion within Fixed" begin
     @test @inferred(promote(Q0f7(0.25), Q0f7(0.75))) ===
         (Q0f7(0.25), Q0f7(0.75))

From 7ede11b332f2e5859fe8cadddaf4034626d3219e Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 21:23:28 +0900
Subject: [PATCH 7/8] Remove useless specialized comparison methods

---
 src/normed.jl | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/src/normed.jl b/src/normed.jl
index 5acf0991..327c972a 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -239,10 +239,6 @@ abs(x::Normed) = x
 *(x::T, y::T) where {T <: Normed} = convert(T,convert(floattype(T), x)*convert(floattype(T), y))
 /(x::T, y::T) where {T <: Normed} = convert(T,convert(floattype(T), x)/convert(floattype(T), y))
 
-# Comparisons
- <(x::T, y::T) where {T <: Normed} = reinterpret(x) < reinterpret(y)
-<=(x::T, y::T) where {T <: Normed} = reinterpret(x) <= reinterpret(y)
-
 # Functions
 trunc(x::T) where {T <: Normed} = T(div(reinterpret(x), rawone(T))*rawone(T),0)
 floor(x::T) where {T <: Normed} = trunc(x)

From 763783ee026a3a7c048d9cd757de7ae23bb3e7be Mon Sep 17 00:00:00 2001
From: kimikage <kimikage.ceo@gmail.com>
Date: Mon, 23 Dec 2019 23:07:21 +0900
Subject: [PATCH 8/8] Clean arithmetic methods up

---
 src/FixedPointNumbers.jl | 13 +++++++------
 src/fixed.jl             |  7 +------
 src/normed.jl            |  7 +------
 3 files changed, 9 insertions(+), 18 deletions(-)

diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index 60911c42..c8ee8948 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -45,10 +45,6 @@ rawtype(::Type{X}) where {T, X <: FixedPoint{T}} = T
 # construction using the (approximate) intended value, i.e., N0f8
 *(x::Real, ::Type{X}) where {X<:FixedPoint} = X(x)
 
-# comparison
-==(x::T, y::T) where {T <: FixedPoint} = x.i == y.i
- <(x::T, y::T) where {T <: FixedPoint} = x.i  < y.i
-<=(x::T, y::T) where {T <: FixedPoint} = x.i <= y.i
 """
     isapprox(x::FixedPoint, y::FixedPoint; rtol=0, atol=max(eps(x), eps(y)))
 
@@ -128,12 +124,17 @@ for f in (:zero, :oneunit, :one, :eps, :rawone, :rawtype, :floattype)
         $f(x::FixedPoint) = $f(typeof(x))
     end
 end
-for f in (:div, :fld, :fld1)
+for f in (:(==), :<, :<=, :div, :fld, :fld1)
     @eval begin
         $f(x::X, y::X) where {X <: FixedPoint} = $f(x.i, y.i)
     end
 end
-for f in (:rem, :mod, :mod1, :min, :max)
+for f in (:-, :~, :abs)
+    @eval begin
+        $f(x::X) where {X <: FixedPoint} = X($f(x.i), 0)
+    end
+end
+for f in (:+, :-, :rem, :mod, :mod1, :min, :max)
     @eval begin
         $f(x::X, y::X) where {X <: FixedPoint} = X($f(x.i, y.i), 0)
     end
diff --git a/src/fixed.jl b/src/fixed.jl
index 60c97618..d4164b67 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -44,12 +44,7 @@ function rawone(::Type{Fixed{T,f}}) where {T, f}
     oneunit(T) << f
 end
 
-# basic operators
--(x::Fixed{T,f}) where {T,f} = Fixed{T,f}(-x.i,0)
-abs(x::Fixed{T,f}) where {T,f} = Fixed{T,f}(abs(x.i),0)
-
-+(x::Fixed{T,f}, y::Fixed{T,f}) where {T,f} = Fixed{T,f}(x.i+y.i,0)
--(x::Fixed{T,f}, y::Fixed{T,f}) where {T,f} = Fixed{T,f}(x.i-y.i,0)
+# unchecked arithmetic
 
 # with truncation:
 #*(x::Fixed{T,f}, y::Fixed{T,f}) = Fixed{T,f}(Base.widemul(x.i,y.i)>>f,0)
diff --git a/src/normed.jl b/src/normed.jl
index 327c972a..7e119bba 100644
--- a/src/normed.jl
+++ b/src/normed.jl
@@ -228,14 +228,9 @@ Base.Integer(x::Normed) = convert(Integer, x*1.0)
 Base.Rational{Ti}(x::Normed) where {Ti <: Integer} = convert(Ti, reinterpret(x))//convert(Ti, rawone(x))
 Base.Rational(x::Normed) = reinterpret(x)//rawone(x)
 
-# Traits
 abs(x::Normed) = x
 
-(-)(x::T) where {T <: Normed} = T(-reinterpret(x), 0)
-(~)(x::T) where {T <: Normed} = T(~reinterpret(x), 0)
-
-+(x::Normed{T,f}, y::Normed{T,f}) where {T,f} = Normed{T,f}(convert(T, x.i+y.i),0)
--(x::Normed{T,f}, y::Normed{T,f}) where {T,f} = Normed{T,f}(convert(T, x.i-y.i),0)
+# unchecked arithmetic
 *(x::T, y::T) where {T <: Normed} = convert(T,convert(floattype(T), x)*convert(floattype(T), y))
 /(x::T, y::T) where {T <: Normed} = convert(T,convert(floattype(T), x)/convert(floattype(T), y))