float.jl

# This file is a part of Julia. License is MIT: http://julialang.org/license

## floating point traits ##

"""
    Inf16

Positive infinity of type `Float16`.
"""
const Inf16 = bitcast(Float16, 0x7c00)
"""
    NaN16

A not-a-number value of type `Float16`.
"""
const NaN16 = bitcast(Float16, 0x7e00)
"""
    Inf32

Positive infinity of type `Float32`.
"""
const Inf32 = bitcast(Float32, 0x7f800000)
"""
    NaN32

A not-a-number value of type `Float32`.
"""
const NaN32 = bitcast(Float32, 0x7fc00000)
const Inf64 = bitcast(Float64, 0x7ff0000000000000)
const NaN64 = bitcast(Float64, 0x7ff8000000000000)

"""
    Inf

Positive infinity of type `Float64`.
"""
const Inf = Inf64
"""
    NaN

A not-a-number value of type `Float64`.
"""
const NaN = NaN64

## conversions to floating-point ##
convert(::Type{Float16}, x::Integer) = convert(Float16, convert(Float32, x))
for t in (Int8, Int16, Int32, Int64, Int128, UInt8, UInt16, UInt32, UInt64, UInt128)
    @eval promote_rule(::Type{Float16}, ::Type{$t}) = Float16
end
promote_rule(::Type{Float16}, ::Type{Bool}) = Float16

for t1 in (Float32, Float64)
    for st in (Int8, Int16, Int32, Int64)
        @eval begin
            convert(::Type{$t1}, x::($st)) = sitofp($t1, x)
            promote_rule(::Type{$t1}, ::Type{$st}) = $t1
        end
    end
    for ut in (Bool, UInt8, UInt16, UInt32, UInt64)
        @eval begin
            convert(::Type{$t1}, x::($ut)) = uitofp($t1, x)
            promote_rule(::Type{$t1}, ::Type{$ut}) = $t1
        end
    end
end
convert(::Type{Integer}, x::Float16) = convert(Integer, Float32(x))
convert{T<:Integer}(::Type{T}, x::Float16) = convert(T, Float32(x))


promote_rule(::Type{Float64}, ::Type{UInt128}) = Float64
promote_rule(::Type{Float64}, ::Type{Int128}) = Float64
promote_rule(::Type{Float32}, ::Type{UInt128}) = Float32
promote_rule(::Type{Float32}, ::Type{Int128}) = Float32

function convert(::Type{Float64}, x::UInt128)
    x == 0 && return 0.0
    n = 128-leading_zeros(x) # ndigits0z(x,2)
    if n <= 53
        y = ((x % UInt64) << (53-n)) & 0x000f_ffff_ffff_ffff
    else
        y = ((x >> (n-54)) % UInt64) & 0x001f_ffff_ffff_ffff # keep 1 extra bit
        y = (y+1)>>1 # round, ties up (extra leading bit in case of next exponent)
        y &= ~UInt64(trailing_zeros(x) == (n-54)) # fix last bit to round to even
    end
    d = ((n+1022) % UInt64) << 52
    reinterpret(Float64, d + y)
end

function convert(::Type{Float64}, x::Int128)
    x == 0 && return 0.0
    s = ((x >>> 64) % UInt64) & 0x8000_0000_0000_0000 # sign bit
    x = abs(x) % UInt128
    n = 128-leading_zeros(x) # ndigits0z(x,2)
    if n <= 53
        y = ((x % UInt64) << (53-n)) & 0x000f_ffff_ffff_ffff
    else
        y = ((x >> (n-54)) % UInt64) & 0x001f_ffff_ffff_ffff # keep 1 extra bit
        y = (y+1)>>1 # round, ties up (extra leading bit in case of next exponent)
        y &= ~UInt64(trailing_zeros(x) == (n-54)) # fix last bit to round to even
    end
    d = ((n+1022) % UInt64) << 52
    reinterpret(Float64, s | d + y)
end

function convert(::Type{Float32}, x::UInt128)
    x == 0 && return 0f0
    n = 128-leading_zeros(x) # ndigits0z(x,2)
    if n <= 24
        y = ((x % UInt32) << (24-n)) & 0x007f_ffff
    else
        y = ((x >> (n-25)) % UInt32) & 0x00ff_ffff # keep 1 extra bit
        y = (y+one(UInt32))>>1 # round, ties up (extra leading bit in case of next exponent)
        y &= ~UInt32(trailing_zeros(x) == (n-25)) # fix last bit to round to even
    end
    d = ((n+126) % UInt32) << 23
    reinterpret(Float32, d + y)
end

function convert(::Type{Float32}, x::Int128)
    x == 0 && return 0f0
    s = ((x >>> 96) % UInt32) & 0x8000_0000 # sign bit
    x = abs(x) % UInt128
    n = 128-leading_zeros(x) # ndigits0z(x,2)
    if n <= 24
        y = ((x % UInt32) << (24-n)) & 0x007f_ffff
    else
        y = ((x >> (n-25)) % UInt32) & 0x00ff_ffff # keep 1 extra bit
        y = (y+one(UInt32))>>1 # round, ties up (extra leading bit in case of next exponent)
        y &= ~UInt32(trailing_zeros(x) == (n-25)) # fix last bit to round to even
    end
    d = ((n+126) % UInt32) << 23
    reinterpret(Float32, s | d + y)
end

function convert(::Type{Float16}, val::Float32)
    f = reinterpret(UInt32, val)
    i = (f >> 23) & 0x1ff + 1
    sh = shifttable[i]
    f &= 0x007fffff
    h::UInt16 = basetable[i] + (f >> sh)
    # round
    # NOTE: we maybe should ignore NaNs here, but the payload is
    # getting truncated anyway so "rounding" it might not matter
    nextbit = (f >> (sh-1)) & 1
    if nextbit != 0
        # Round halfway to even or check lower bits
        if h&1 == 1 || (f & ((1<<(sh-1))-1)) != 0
            h += 1
        end
    end
    reinterpret(Float16, h)
end

function convert(::Type{Float32}, val::Float16)
    local ival::UInt32 = reinterpret(UInt16, val)
    local sign::UInt32 = (ival & 0x8000) >> 15
    local exp::UInt32  = (ival & 0x7c00) >> 10
    local sig::UInt32  = (ival & 0x3ff) >> 0
    local ret::UInt32

    if exp == 0
        if sig == 0
            sign = sign << 31
            ret = sign | exp | sig
        else
            n_bit = 1
            bit = 0x0200
            while (bit & sig) == 0
                n_bit = n_bit + 1
                bit = bit >> 1
            end
            sign = sign << 31
            exp = (-14 - n_bit + 127) << 23
            sig = ((sig & (~bit)) << n_bit) << (23 - 10)
            ret = sign | exp | sig
        end
    elseif exp == 0x1f
        if sig == 0  # Inf
            if sign == 0
                ret = 0x7f800000
            else
                ret = 0xff800000
            end
        else  # NaN
            ret = 0x7fc00000 | (sign<<31)
        end
    else
        sign = sign << 31
        exp  = (exp - 15 + 127) << 23
        sig  = sig << (23 - 10)
        ret = sign | exp | sig
    end
    return reinterpret(Float32, ret)
end

# Float32 -> Float16 algorithm from:
#   "Fast Half Float Conversion" by Jeroen van der Zijp
#   ftp://ftp.fox-toolkit.org/pub/fasthalffloatconversion.pdf

const basetable = Array{UInt16}(512)
const shifttable = Array{UInt8}(512)

for i = 0:255
    e = i - 127
    if e < -24  # Very small numbers map to zero
        basetable[i|0x000+1] = 0x0000
        basetable[i|0x100+1] = 0x8000
        shifttable[i|0x000+1] = 24
        shifttable[i|0x100+1] = 24
    elseif e < -14  # Small numbers map to denorms
        basetable[i|0x000+1] = (0x0400>>(-e-14))
        basetable[i|0x100+1] = (0x0400>>(-e-14)) | 0x8000
        shifttable[i|0x000+1] = -e-1
        shifttable[i|0x100+1] = -e-1
    elseif e <= 15  # Normal numbers just lose precision
        basetable[i|0x000+1] = ((e+15)<<10)
        basetable[i|0x100+1] = ((e+15)<<10) | 0x8000
        shifttable[i|0x000+1] = 13
        shifttable[i|0x100+1] = 13
    elseif e < 128  # Large numbers map to Infinity
        basetable[i|0x000+1] = 0x7C00
        basetable[i|0x100+1] = 0xFC00
        shifttable[i|0x000+1] = 24
        shifttable[i|0x100+1] = 24
    else  # Infinity and NaN's stay Infinity and NaN's
        basetable[i|0x000+1] = 0x7C00
        basetable[i|0x100+1] = 0xFC00
        shifttable[i|0x000+1] = 13
        shifttable[i|0x100+1] = 13
    end
end
#convert(::Type{Float16}, x::Float32) = fptrunc(Float16, x)
convert(::Type{Float32}, x::Float64) = fptrunc(Float32, x)
convert(::Type{Float16}, x::Float64) = convert(Float16, convert(Float32, x))

#convert(::Type{Float32}, x::Float16) = fpext(Float32, x)
convert(::Type{Float64}, x::Float32) = fpext(Float64, x)
convert(::Type{Float64}, x::Float16) = convert(Float64, convert(Float32, x))

convert(::Type{AbstractFloat}, x::Bool)    = convert(Float64, x)
convert(::Type{AbstractFloat}, x::Int8)    = convert(Float64, x)
convert(::Type{AbstractFloat}, x::Int16)   = convert(Float64, x)
convert(::Type{AbstractFloat}, x::Int32)   = convert(Float64, x)
convert(::Type{AbstractFloat}, x::Int64)   = convert(Float64, x) # LOSSY
convert(::Type{AbstractFloat}, x::Int128)  = convert(Float64, x) # LOSSY
convert(::Type{AbstractFloat}, x::UInt8)   = convert(Float64, x)
convert(::Type{AbstractFloat}, x::UInt16)  = convert(Float64, x)
convert(::Type{AbstractFloat}, x::UInt32)  = convert(Float64, x)
convert(::Type{AbstractFloat}, x::UInt64)  = convert(Float64, x) # LOSSY
convert(::Type{AbstractFloat}, x::UInt128) = convert(Float64, x) # LOSSY

"""
    float(x)

Convert a number or array to a floating point data type.
When passed a string, this function is equivalent to `parse(Float64, x)`.
"""
float(x) = convert(AbstractFloat, x)

"""
    float(T::Type)

Returns an appropriate type to represent a value of type `T` as a floating point value.
Equivalent to `typeof(float(zero(T)))`.

```jldoctest
julia> float(Complex{Int})
Complex{Float64}

julia> float(Int)
Float64
```
"""
float{T<:Number}(::Type{T}) = typeof(float(zero(T)))

for Ti in (Int8, Int16, Int32, Int64)
    @eval begin
        unsafe_trunc(::Type{$Ti}, x::Float32) = fptosi($Ti, x)
        unsafe_trunc(::Type{$Ti}, x::Float64) = fptosi($Ti, x)
    end
end
for Ti in (UInt8, UInt16, UInt32, UInt64)
    @eval begin
        unsafe_trunc(::Type{$Ti}, x::Float32) = fptoui($Ti, x)
        unsafe_trunc(::Type{$Ti}, x::Float64) = fptoui($Ti, x)
    end
end

function unsafe_trunc(::Type{UInt128}, x::Float64)
    xu = reinterpret(UInt64,x)
    k = Int(xu >> 52) & 0x07ff - 1075
    xu = (xu & 0x000f_ffff_ffff_ffff) | 0x0010_0000_0000_0000
    if k <= 0
        UInt128(xu >> -k)
    else
        UInt128(xu) << k
    end
end
function unsafe_trunc(::Type{Int128}, x::Float64)
    copysign(unsafe_trunc(UInt128,x) % Int128, x)
end

function unsafe_trunc(::Type{UInt128}, x::Float32)
    xu = reinterpret(UInt32,x)
    k = Int(xu >> 23) & 0x00ff - 150
    xu = (xu & 0x007f_ffff) | 0x0080_0000
    if k <= 0
        UInt128(xu >> -k)
    else
        UInt128(xu) << k
    end
end
function unsafe_trunc(::Type{Int128}, x::Float32)
    copysign(unsafe_trunc(UInt128,x) % Int128, x)
end


# matches convert methods
# also determines floor, ceil, round
trunc(::Type{Signed}, x::Float32) = trunc(Int,x)
trunc(::Type{Signed}, x::Float64) = trunc(Int,x)
trunc(::Type{Unsigned}, x::Float32) = trunc(UInt,x)
trunc(::Type{Unsigned}, x::Float64) = trunc(UInt,x)
trunc(::Type{Integer}, x::Float32) = trunc(Int,x)
trunc(::Type{Integer}, x::Float64) = trunc(Int,x)
trunc{T<:Integer}(::Type{T}, x::Float16) = trunc(T, Float32(x))

# fallbacks
floor{T<:Integer}(::Type{T}, x::AbstractFloat) = trunc(T,floor(x))
floor{T<:Integer}(::Type{T}, x::Float16) = floor(T, Float32(x))
ceil{ T<:Integer}(::Type{T}, x::AbstractFloat) = trunc(T,ceil(x))
ceil{ T<:Integer}(::Type{T}, x::Float16) = ceil(T, Float32(x))
round{T<:Integer}(::Type{T}, x::AbstractFloat) = trunc(T,round(x))
round{T<:Integer}(::Type{T}, x::Float16) = round(T, Float32(x))

trunc(x::Float64) = trunc_llvm(x)
trunc(x::Float32) = trunc_llvm(x)
trunc(x::Float16) = Float16(trunc(Float32(x)))

floor(x::Float64) = floor_llvm(x)
floor(x::Float32) = floor_llvm(x)
floor(x::Float16) = Float16(floor(Float32(x)))

ceil(x::Float64) = ceil_llvm(x)
ceil(x::Float32) = ceil_llvm(x)
ceil(x::Float16) = Float16( ceil(Float32(x)))

round(x::Float64) = rint_llvm(x)
round(x::Float32) = rint_llvm(x)
round(x::Float16) = Float16(round(Float32(x)))

## floating point promotions ##
promote_rule(::Type{Float32}, ::Type{Float16}) = Float32
promote_rule(::Type{Float64}, ::Type{Float16}) = Float64
promote_rule(::Type{Float64}, ::Type{Float32}) = Float64

widen(::Type{Float16}) = Float32
widen(::Type{Float32}) = Float64

_default_type(T::Union{Type{Real},Type{AbstractFloat}}) = Float64

## floating point arithmetic ##
-(x::Float64) = neg_float(x)
-(x::Float32) = neg_float(x)
-(x::Float16) = reinterpret(Float16, reinterpret(UInt16, x) ⊻ 0x8000)

for op in (:+, :-, :*, :/, :\, :^)
    @eval ($op)(a::Float16, b::Float16) = Float16(($op)(Float32(a), Float32(b)))
end
+(x::Float32, y::Float32) = add_float(x, y)
+(x::Float64, y::Float64) = add_float(x, y)
-(x::Float32, y::Float32) = sub_float(x, y)
-(x::Float64, y::Float64) = sub_float(x, y)
*(x::Float32, y::Float32) = mul_float(x, y)
*(x::Float64, y::Float64) = mul_float(x, y)
/(x::Float32, y::Float32) = div_float(x, y)
/(x::Float64, y::Float64) = div_float(x, y)

muladd(x::Float32, y::Float32, z::Float32) = muladd_float(x, y, z)
muladd(x::Float64, y::Float64, z::Float64) = muladd_float(x, y, z)
function muladd(a::Float16, b::Float16, c::Float16)
    Float16(muladd(Float32(a), Float32(b), Float32(c)))
end

# TODO: faster floating point div?
# TODO: faster floating point fld?
# TODO: faster floating point mod?

for func in (:div,:fld,:cld,:rem,:mod)
    @eval begin
        $func(a::Float16,b::Float16) = Float16($func(Float32(a),Float32(b)))
    end
end

rem(x::Float32, y::Float32) = rem_float(x, y)
rem(x::Float64, y::Float64) = rem_float(x, y)

cld{T<:AbstractFloat}(x::T, y::T) = -fld(-x,y)

function mod{T<:AbstractFloat}(x::T, y::T)
    r = rem(x,y)
    if r == 0
        copysign(r,y)
    elseif (r > 0) ⊻ (y > 0)
        r+y
    else
        r
    end
end

## floating point comparisons ##
function ==(x::Float16, y::Float16)
    ix = reinterpret(UInt16,x)
    iy = reinterpret(UInt16,y)
    if (ix|iy)&0x7fff > 0x7c00 #isnan(x) || isnan(y)
        return false
    end
    if (ix|iy)&0x7fff == 0x0000
        return true
    end
    return ix == iy
end
==(x::Float32, y::Float32) = eq_float(x, y)
==(x::Float64, y::Float64) = eq_float(x, y)
!=(x::Float32, y::Float32) = ne_float(x, y)
!=(x::Float64, y::Float64) = ne_float(x, y)
<( x::Float32, y::Float32) = lt_float(x, y)
<( x::Float64, y::Float64) = lt_float(x, y)
<=(x::Float32, y::Float32) = le_float(x, y)
<=(x::Float64, y::Float64) = le_float(x, y)

isequal(x::Float32, y::Float32) = fpiseq(x, y)
isequal(x::Float64, y::Float64) = fpiseq(x, y)
isless( x::Float32, y::Float32) = fpislt(x, y)
isless( x::Float64, y::Float64) = fpislt(x, y)
for op in (:<, :<=, :isless)
    @eval ($op)(a::Float16, b::Float16) = ($op)(Float32(a), Float32(b))
end

function cmp(x::AbstractFloat, y::AbstractFloat)
    (isnan(x) || isnan(y)) && throw(DomainError())
    ifelse(x<y, -1, ifelse(x>y, 1, 0))
end

function cmp(x::Real, y::AbstractFloat)
    isnan(y) && throw(DomainError())
    ifelse(x<y, -1, ifelse(x>y, 1, 0))
end

function cmp(x::AbstractFloat, y::Real)
    isnan(x) && throw(DomainError())
    ifelse(x<y, -1, ifelse(x>y, 1, 0))
end

# Exact Float (Tf) vs Integer (Ti) comparisons
# Assumes:
# - typemax(Ti) == 2^n-1
# - typemax(Ti) can't be exactly represented by Tf:
#   => Tf(typemax(Ti)) == 2^n or Inf
# - typemin(Ti) can be exactly represented by Tf
#
# 1. convert y::Ti to float fy::Tf
# 2. perform Tf comparison x vs fy
# 3. if x == fy, check if (1) resulted in rounding:
#  a. convert fy back to Ti and compare with original y
#  b. unsafe_convert undefined behaviour if fy == Tf(typemax(Ti))
#     (but consequently x == fy > y)
for Ti in (Int64,UInt64,Int128,UInt128)
    for Tf in (Float32,Float64)
        @eval begin
            function ==(x::$Tf, y::$Ti)
                fy = ($Tf)(y)
                (x == fy) & (fy != $(Tf(typemax(Ti)))) & (y == unsafe_trunc($Ti,fy))
            end
            ==(y::$Ti, x::$Tf) = x==y

            function <(x::$Ti, y::$Tf)
                fx = ($Tf)(x)
                (fx < y) | ((fx == y) & ((fx == $(Tf(typemax(Ti)))) | (x < unsafe_trunc($Ti,fx)) ))
            end
            function <=(x::$Ti, y::$Tf)
                fx = ($Tf)(x)
                (fx < y) | ((fx == y) & ((fx == $(Tf(typemax(Ti)))) | (x <= unsafe_trunc($Ti,fx)) ))
            end

            function <(x::$Tf, y::$Ti)
                fy = ($Tf)(y)
                (x < fy) | ((x == fy) & (fy < $(Tf(typemax(Ti)))) & (unsafe_trunc($Ti,fy) < y))
            end
            function <=(x::$Tf, y::$Ti)
                fy = ($Tf)(y)
                (x < fy) | ((x == fy) & (fy < $(Tf(typemax(Ti)))) & (unsafe_trunc($Ti,fy) <= y))
            end
        end
    end
end

==(x::Float32, y::Union{Int32,UInt32}) = Float64(x)==Float64(y)
==(x::Union{Int32,UInt32}, y::Float32) = Float64(x)==Float64(y)

<(x::Float32, y::Union{Int32,UInt32}) = Float64(x)<Float64(y)
<(x::Union{Int32,UInt32}, y::Float32) = Float64(x)<Float64(y)

<=(x::Float32, y::Union{Int32,UInt32}) = Float64(x)<=Float64(y)
<=(x::Union{Int32,UInt32}, y::Float32) = Float64(x)<=Float64(y)


abs(x::Float16) = reinterpret(Float16, reinterpret(UInt16, x) & 0x7fff)
abs(x::Float32) = abs_float(x)
abs(x::Float64) = abs_float(x)

"""
    isnan(f) -> Bool

Test whether a floating point number is not a number (NaN).
"""
isnan(x::AbstractFloat) = x != x
isnan(x::Float16)    = reinterpret(UInt16,x)&0x7fff  > 0x7c00
isnan(x::Real) = false

"""
    isfinite(f) -> Bool

Test whether a number is finite.

```jldoctest
julia> isfinite(5)
true

julia> isfinite(NaN32)
false
```
"""
isfinite(x::AbstractFloat) = x - x == 0
isfinite(x::Float16) = reinterpret(UInt16,x)&0x7c00 != 0x7c00
isfinite(x::Real) = decompose(x)[3] != 0
isfinite(x::Integer) = true

"""
    isinf(f) -> Bool

Test whether a number is infinite.
"""
isinf(x::Real) = !isnan(x) & !isfinite(x)

## hashing small, built-in numeric types ##

hx(a::UInt64, b::Float64, h::UInt) = hash_uint64((3a + reinterpret(UInt64,b)) - h)
const hx_NaN = hx(UInt64(0), NaN, UInt(0  ))

hash(x::UInt64,  h::UInt) = hx(x, Float64(x), h)
hash(x::Int64,   h::UInt) = hx(reinterpret(UInt64, abs(x)), Float64(x), h)
hash(x::Float64, h::UInt) = isnan(x) ? (hx_NaN ⊻ h) : hx(fptoui(UInt64, abs(x)), x, h)

hash(x::Union{Bool,Int8,UInt8,Int16,UInt16,Int32,UInt32}, h::UInt) = hash(Int64(x), h)
hash(x::Float32, h::UInt) = hash(Float64(x), h)

## precision, as defined by the effective number of bits in the mantissa ##
precision(::Type{Float16}) = 11
precision(::Type{Float32}) = 24
precision(::Type{Float64}) = 53
precision{T<:AbstractFloat}(::T) = precision(T)

"""
    uabs(x::Integer)

Returns the absolute value of `x`, possibly returning a different type should the
operation be susceptible to overflow. This typically arises when `x` is a two's complement
signed integer, so that `abs(typemin(x)) == typemin(x) < 0`, in which case the result of
`uabs(x)` will be an unsigned integer of the same size.
"""
uabs(x::Integer) = abs(x)
uabs(x::Signed) = unsigned(abs(x))


"""
    nextfloat(x::AbstractFloat, n::Integer)

The result of `n` iterative applications of `nextfloat` to `x` if `n >= 0`, or `-n`
applications of `prevfloat` if `n < 0`.
"""
function nextfloat(f::Union{Float16,Float32,Float64}, d::Integer)
    F = typeof(f)
    fumax = reinterpret(Unsigned, F(Inf))
    U = typeof(fumax)

    isnan(f) && return f
    fi = reinterpret(Signed, f)
    fneg = fi < 0
    fu = unsigned(fi & typemax(fi))

    dneg = d < 0
    da = uabs(d)
    if da > typemax(U)
        fneg = dneg
        fu = fumax
    else
        du = da % U
        if fneg ⊻ dneg
            if du > fu
                fu = min(fumax, du - fu)
                fneg = !fneg
            else
                fu = fu - du
            end
        else
            if fumax - fu < du
                fu = fumax
            else
                fu = fu + du
            end
        end
    end
    if fneg
        fu |= sign_mask(F)
    end
    reinterpret(F, fu)
end

"""
    nextfloat(x::AbstractFloat)

Returns the smallest floating point number `y` of the same type as `x` such `x < y`. If no
such `y` exists (e.g. if `x` is `Inf` or `NaN`), then returns `x`.
"""
nextfloat(x::AbstractFloat) = nextfloat(x,1)

"""
    prevfloat(x::AbstractFloat)

Returns the largest floating point number `y` of the same type as `x` such `y < x`. If no
such `y` exists (e.g. if `x` is `-Inf` or `NaN`), then returns `x`.
"""
prevfloat(x::AbstractFloat) = nextfloat(x,-1)

for Ti in (Int8, Int16, Int32, Int64, Int128, UInt8, UInt16, UInt32, UInt64, UInt128)
    for Tf in (Float32, Float64)
        if Ti <: Unsigned || sizeof(Ti) < sizeof(Tf)
            # Here `Tf(typemin(Ti))-1` is exact, so we can compare the lower-bound
            # directly. `Tf(typemax(Ti))+1` is either always exactly representable, or
            # rounded to `Inf` (e.g. when `Ti==UInt128 && Tf==Float32`).
            @eval begin
                function trunc(::Type{$Ti},x::$Tf)
                    if $(Tf(typemin(Ti))-one(Tf)) < x < $(Tf(typemax(Ti))+one(Tf))
                        return unsafe_trunc($Ti,x)
                    else
                        throw(InexactError())
                    end
                end
                function convert(::Type{$Ti}, x::$Tf)
                    if ($(Tf(typemin(Ti))) <= x <= $(Tf(typemax(Ti)))) && (trunc(x) == x)
                        return unsafe_trunc($Ti,x)
                    else
                        throw(InexactError())
                    end
                end
            end
        else
            # Here `eps(Tf(typemin(Ti))) > 1`, so the only value which can be truncated to
            # `Tf(typemin(Ti)` is itself. Similarly, `Tf(typemax(Ti))` is inexact and will
            # be rounded up. This assumes that `Tf(typemin(Ti)) > -Inf`, which is true for
            # these types, but not for `Float16` or larger integer types.
            @eval begin
                function trunc(::Type{$Ti},x::$Tf)
                    if $(Tf(typemin(Ti))) <= x < $(Tf(typemax(Ti)))
                        return unsafe_trunc($Ti,x)
                    else
                        throw(InexactError())
                    end
                end
                function convert(::Type{$Ti}, x::$Tf)
                    if ($(Tf(typemin(Ti))) <= x < $(Tf(typemax(Ti)))) && (trunc(x) == x)
                        return unsafe_trunc($Ti,x)
                    else
                        throw(InexactError())
                    end
                end
            end
        end
    end
end

@eval begin
    issubnormal(x::Float32) = (abs(x) < $(bitcast(Float32, 0x00800000))) & (x!=0)
    issubnormal(x::Float64) = (abs(x) < $(bitcast(Float64, 0x0010000000000000))) & (x!=0)

    typemin(::Type{Float16}) = $(bitcast(Float16, 0xfc00))
    typemax(::Type{Float16}) = $(Inf16)
    typemin(::Type{Float32}) = $(-Inf32)
    typemax(::Type{Float32}) = $(Inf32)
    typemin(::Type{Float64}) = $(-Inf64)
    typemax(::Type{Float64}) = $(Inf64)
    typemin{T<:Real}(x::T) = typemin(T)
    typemax{T<:Real}(x::T) = typemax(T)

    realmin(::Type{Float16}) = $(bitcast(Float16, 0x0400))
    realmin(::Type{Float32}) = $(bitcast(Float32, 0x00800000))
    realmin(::Type{Float64}) = $(bitcast(Float64, 0x0010000000000000))
    realmax(::Type{Float16}) = $(bitcast(Float16, 0x7bff))
    realmax(::Type{Float32}) = $(bitcast(Float32, 0x7f7fffff))
    realmax(::Type{Float64}) = $(bitcast(Float64, 0x7fefffffffffffff))
    realmin{T<:AbstractFloat}(x::T) = realmin(T)
    realmax{T<:AbstractFloat}(x::T) = realmax(T)
    realmin() = realmin(Float64)
    realmax() = realmax(Float64)

    eps(x::AbstractFloat) = isfinite(x) ? abs(x) >= realmin(x) ? ldexp(eps(typeof(x)), exponent(x)) : nextfloat(zero(x)) : oftype(x, NaN)
    eps(::Type{Float16}) = $(bitcast(Float16, 0x1400))
    eps(::Type{Float32}) = $(bitcast(Float32, 0x34000000))
    eps(::Type{Float64}) = $(bitcast(Float64, 0x3cb0000000000000))
    eps() = eps(Float64)
end

## byte order swaps for arbitrary-endianness serialization/deserialization ##
bswap(x::Float32) = bswap_int(x)
bswap(x::Float64) = bswap_int(x)

# bit patterns
reinterpret(::Type{Unsigned}, x::Float64) = reinterpret(UInt64, x)
reinterpret(::Type{Unsigned}, x::Float32) = reinterpret(UInt32, x)
reinterpret(::Type{Signed}, x::Float64) = reinterpret(Int64, x)
reinterpret(::Type{Signed}, x::Float32) = reinterpret(Int32, x)

sign_mask(::Type{Float64}) =        0x8000_0000_0000_0000
exponent_mask(::Type{Float64}) =    0x7ff0_0000_0000_0000
exponent_one(::Type{Float64}) =     0x3ff0_0000_0000_0000
exponent_half(::Type{Float64}) =    0x3fe0_0000_0000_0000
significand_mask(::Type{Float64}) = 0x000f_ffff_ffff_ffff

sign_mask(::Type{Float32}) =        0x8000_0000
exponent_mask(::Type{Float32}) =    0x7f80_0000
exponent_one(::Type{Float32}) =     0x3f80_0000
exponent_half(::Type{Float32}) =    0x3f00_0000
significand_mask(::Type{Float32}) = 0x007f_ffff

sign_mask(::Type{Float16}) =        0x8000
exponent_mask(::Type{Float16}) =    0x7c00
exponent_one(::Type{Float16}) =     0x3c00
exponent_half(::Type{Float16}) =    0x3800
significand_mask(::Type{Float16}) = 0x03ff

# integer size of float
fpinttype(::Type{Float64}) = UInt64
fpinttype(::Type{Float32}) = UInt32
fpinttype(::Type{Float16}) = UInt16

@pure significand_bits{T<:AbstractFloat}(::Type{T}) = trailing_ones(significand_mask(T))
@pure exponent_bits{T<:AbstractFloat}(::Type{T}) = sizeof(T)*8 - significand_bits(T) - 1
@pure exponent_bias{T<:AbstractFloat}(::Type{T}) = Int(exponent_one(T) >> significand_bits(T))
# maximum float exponent
@pure exponent_max{T<:AbstractFloat}(::Type{T}) = Int(exponent_mask(T) >> significand_bits(T)) - exponent_bias(T)
# maximum float exponent without bias
@pure exponent_raw_max{T<:AbstractFloat}(::Type{T}) = Int(exponent_mask(T) >> significand_bits(T))

## TwicePrecision utilities
# The numeric constants are half the number of bits in the mantissa
for (F, T, n) in ((Float16, UInt16, 5), (Float32, UInt32, 12), (Float64, UInt64, 26))
    @eval begin
        function truncbits(x::$F, nb)
            @_inline_meta
            truncmask(x, typemax($T) << nb)
        end
        function truncmask(x::$F, mask)
            @_inline_meta
            reinterpret($F, mask & reinterpret($T, x))
        end
        function splitprec(x::$F)
            @_inline_meta
            hi = truncmask(x, typemax($T) << $n)
            hi, x-hi
        end
    end
end

truncbits(x, nb) = x
truncmask(x, mask) = x

## Array operations on floating point numbers ##

float{T<:AbstractFloat}(A::AbstractArray{T}) = A

function float{T}(A::AbstractArray{T})
    if !isleaftype(T)
        error("`float` not defined on abstractly-typed arrays; please convert to a more specific type")
    end
    convert(AbstractArray{typeof(float(zero(T)))}, A)
end

float(r::StepRange) = float(r.start):float(r.step):float(last(r))
float(r::UnitRange) = float(r.start):float(last(r))
float(r::StepRangeLen) = StepRangeLen(float(r.ref), float(r.step), length(r), r.offset)
function float(r::LinSpace)
    LinSpace(float(r.start), float(r.stop), length(r))
end

# big, broadcast over arrays
# TODO: do the definitions below primarily pertaining to integers belong in float.jl?
function big end # no prior definitions of big in sysimg.jl, necessitating this
broadcast(::typeof(big), r::UnitRange) = big(r.start):big(last(r))
broadcast(::typeof(big), r::StepRange) = big(r.start):big(r.step):big(last(r))
broadcast(::typeof(big), r::StepRangeLen) = StepRangeLen(big(r.ref), big(r.step), length(r), r.offset)
function broadcast(::typeof(big), r::LinSpace)
    LinSpace(big(r.start), big(r.stop), length(r))
end