Revision 924b52adeb64d4f7a9663396dfead74fcb518424 authored by Keno Fischer on 26 March 2024, 05:51:40 UTC, committed by Keno Fischer on 26 March 2024, 08:50:14 UTC
In case the default argument of some other argument makes use of the destructured argument. Fixes #53832.
1 parent cd523fe
float.jl
# This file is a part of Julia. License is MIT: https://julialang.org/license
const IEEEFloat = Union{Float16, Float32, Float64}
## floating point traits ##
"""
Inf16
Positive infinity of type [`Float16`](@ref).
"""
const Inf16 = bitcast(Float16, 0x7c00)
"""
NaN16
A not-a-number value of type [`Float16`](@ref).
See also: [`NaN`](@ref).
"""
const NaN16 = bitcast(Float16, 0x7e00)
"""
Inf32
Positive infinity of type [`Float32`](@ref).
"""
const Inf32 = bitcast(Float32, 0x7f800000)
"""
NaN32
A not-a-number value of type [`Float32`](@ref).
See also: [`NaN`](@ref).
"""
const NaN32 = bitcast(Float32, 0x7fc00000)
const Inf64 = bitcast(Float64, 0x7ff0000000000000)
const NaN64 = bitcast(Float64, 0x7ff8000000000000)
const Inf = Inf64
"""
Inf, Inf64
Positive infinity of type [`Float64`](@ref).
See also: [`isfinite`](@ref), [`typemax`](@ref), [`NaN`](@ref), [`Inf32`](@ref).
# Examples
```jldoctest
julia> π/0
Inf
julia> +1.0 / -0.0
-Inf
julia> ℯ^-Inf
0.0
```
"""
Inf, Inf64
const NaN = NaN64
"""
NaN, NaN64
A not-a-number value of type [`Float64`](@ref).
See also: [`isnan`](@ref), [`missing`](@ref), [`NaN32`](@ref), [`Inf`](@ref).
# Examples
```jldoctest
julia> 0/0
NaN
julia> Inf - Inf
NaN
julia> NaN == NaN, isequal(NaN, NaN), isnan(NaN)
(false, true, true)
```
!!! note
Always use [`isnan`](@ref) or [`isequal`](@ref) for checking for `NaN`.
Using `x === NaN` may give unexpected results:
```julia-repl
julia> reinterpret(UInt32, NaN32)
0x7fc00000
julia> NaN32p1 = reinterpret(Float32, 0x7fc00001)
NaN32
julia> NaN32p1 === NaN32, isequal(NaN32p1, NaN32), isnan(NaN32p1)
(false, true, true)
```
"""
NaN, NaN64
# bit patterns
reinterpret(::Type{Unsigned}, x::Float64) = reinterpret(UInt64, x)
reinterpret(::Type{Unsigned}, x::Float32) = reinterpret(UInt32, x)
reinterpret(::Type{Unsigned}, x::Float16) = reinterpret(UInt16, x)
reinterpret(::Type{Signed}, x::Float64) = reinterpret(Int64, x)
reinterpret(::Type{Signed}, x::Float32) = reinterpret(Int32, x)
reinterpret(::Type{Signed}, x::Float16) = reinterpret(Int16, 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
mantissa(x::T) where {T} = reinterpret(Unsigned, x) & significand_mask(T)
for T in (Float16, Float32, Float64)
@eval significand_bits(::Type{$T}) = $(trailing_ones(significand_mask(T)))
@eval exponent_bits(::Type{$T}) = $(sizeof(T)*8 - significand_bits(T) - 1)
@eval exponent_bias(::Type{$T}) = $(Int(exponent_one(T) >> significand_bits(T)))
# maximum float exponent
@eval exponent_max(::Type{$T}) = $(Int(exponent_mask(T) >> significand_bits(T)) - exponent_bias(T) - 1)
# maximum float exponent without bias
@eval exponent_raw_max(::Type{$T}) = $(Int(exponent_mask(T) >> significand_bits(T)))
end
"""
exponent_max(T)
Maximum [`exponent`](@ref) value for a floating point number of type `T`.
# Examples
```jldoctest
julia> Base.exponent_max(Float64)
1023
```
Note, `exponent_max(T) + 1` is a possible value of the exponent field
with bias, which might be used as sentinel value for `Inf` or `NaN`.
"""
function exponent_max end
"""
exponent_raw_max(T)
Maximum value of the [`exponent`](@ref) field for a floating point number of type `T` without bias,
i.e. the maximum integer value representable by [`exponent_bits(T)`](@ref) bits.
"""
function exponent_raw_max end
"""
IEEE 754 definition of the minimum exponent.
"""
ieee754_exponent_min(::Type{T}) where {T<:IEEEFloat} = Int(1 - exponent_max(T))::Int
exponent_min(::Type{Float16}) = ieee754_exponent_min(Float16)
exponent_min(::Type{Float32}) = ieee754_exponent_min(Float32)
exponent_min(::Type{Float64}) = ieee754_exponent_min(Float64)
function ieee754_representation(
::Type{F}, sign_bit::Bool, exponent_field::Integer, significand_field::Integer
) where {F<:IEEEFloat}
T = uinttype(F)
ret::T = sign_bit
ret <<= exponent_bits(F)
ret |= exponent_field
ret <<= significand_bits(F)
ret |= significand_field
end
# ±floatmax(T)
function ieee754_representation(
::Type{F}, sign_bit::Bool, ::Val{:omega}
) where {F<:IEEEFloat}
ieee754_representation(F, sign_bit, exponent_raw_max(F) - 1, significand_mask(F))
end
# NaN or an infinity
function ieee754_representation(
::Type{F}, sign_bit::Bool, significand_field::Integer, ::Val{:nan}
) where {F<:IEEEFloat}
ieee754_representation(F, sign_bit, exponent_raw_max(F), significand_field)
end
# NaN with default payload
function ieee754_representation(
::Type{F}, sign_bit::Bool, ::Val{:nan}
) where {F<:IEEEFloat}
ieee754_representation(F, sign_bit, one(uinttype(F)) << (significand_bits(F) - 1), Val(:nan))
end
# Infinity
function ieee754_representation(
::Type{F}, sign_bit::Bool, ::Val{:inf}
) where {F<:IEEEFloat}
ieee754_representation(F, sign_bit, false, Val(:nan))
end
# Subnormal or zero
function ieee754_representation(
::Type{F}, sign_bit::Bool, significand_field::Integer, ::Val{:subnormal}
) where {F<:IEEEFloat}
ieee754_representation(F, sign_bit, false, significand_field)
end
# Zero
function ieee754_representation(
::Type{F}, sign_bit::Bool, ::Val{:zero}
) where {F<:IEEEFloat}
ieee754_representation(F, sign_bit, false, Val(:subnormal))
end
"""
uabs(x::Integer)
Return 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::BitSigned) = unsigned(abs(x))
## conversions to floating-point ##
# TODO: deprecate in 2.0
Float16(x::Integer) = convert(Float16, convert(Float32, x)::Float32)
for t1 in (Float16, Float32, Float64)
for st in (Int8, Int16, Int32, Int64)
@eval begin
(::Type{$t1})(x::($st)) = sitofp($t1, x)
promote_rule(::Type{$t1}, ::Type{$st}) = $t1
end
end
for ut in (Bool, UInt8, UInt16, UInt32, UInt64)
@eval begin
(::Type{$t1})(x::($ut)) = uitofp($t1, x)
promote_rule(::Type{$t1}, ::Type{$ut}) = $t1
end
end
end
Bool(x::Real) = x==0 ? false : x==1 ? true : throw(InexactError(:Bool, Bool, 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
promote_rule(::Type{Float16}, ::Type{UInt128}) = Float16
promote_rule(::Type{Float16}, ::Type{Int128}) = Float16
function Float64(x::UInt128)
if x < UInt128(1) << 104 # Can fit it in two 52 bits mantissas
low_exp = 0x1p52
high_exp = 0x1p104
low_bits = (x % UInt64) & Base.significand_mask(Float64)
low_value = reinterpret(Float64, reinterpret(UInt64, low_exp) | low_bits) - low_exp
high_bits = ((x >> 52) % UInt64)
high_value = reinterpret(Float64, reinterpret(UInt64, high_exp) | high_bits) - high_exp
low_value + high_value
else # Large enough that low bits only affect rounding, pack low bits
low_exp = 0x1p76
high_exp = 0x1p128
low_bits = ((x >> 12) % UInt64) >> 12 | (x % UInt64) & 0xFFFFFF
low_value = reinterpret(Float64, reinterpret(UInt64, low_exp) | low_bits) - low_exp
high_bits = ((x >> 76) % UInt64)
high_value = reinterpret(Float64, reinterpret(UInt64, high_exp) | high_bits) - high_exp
low_value + high_value
end
end
function Float64(x::Int128)
sign_bit = ((x >> 127) % UInt64) << 63
ux = uabs(x)
if ux < UInt128(1) << 104 # Can fit it in two 52 bits mantissas
low_exp = 0x1p52
high_exp = 0x1p104
low_bits = (ux % UInt64) & Base.significand_mask(Float64)
low_value = reinterpret(Float64, reinterpret(UInt64, low_exp) | low_bits) - low_exp
high_bits = ((ux >> 52) % UInt64)
high_value = reinterpret(Float64, reinterpret(UInt64, high_exp) | high_bits) - high_exp
reinterpret(Float64, sign_bit | reinterpret(UInt64, low_value + high_value))
else # Large enough that low bits only affect rounding, pack low bits
low_exp = 0x1p76
high_exp = 0x1p128
low_bits = ((ux >> 12) % UInt64) >> 12 | (ux % UInt64) & 0xFFFFFF
low_value = reinterpret(Float64, reinterpret(UInt64, low_exp) | low_bits) - low_exp
high_bits = ((ux >> 76) % UInt64)
high_value = reinterpret(Float64, reinterpret(UInt64, high_exp) | high_bits) - high_exp
reinterpret(Float64, sign_bit | reinterpret(UInt64, low_value + high_value))
end
end
function Float32(x::UInt128)
x == 0 && return 0f0
n = top_set_bit(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 Float32(x::Int128)
x == 0 && return 0f0
s = ((x >>> 96) % UInt32) & 0x8000_0000 # sign bit
x = abs(x) % UInt128
n = top_set_bit(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
# TODO: optimize
Float16(x::UInt128) = convert(Float16, Float64(x))
Float16(x::Int128) = convert(Float16, Float64(x))
Float16(x::Float32) = fptrunc(Float16, x)
Float16(x::Float64) = fptrunc(Float16, x)
Float32(x::Float64) = fptrunc(Float32, x)
Float32(x::Float16) = fpext(Float32, x)
Float64(x::Float32) = fpext(Float64, x)
Float64(x::Float16) = fpext(Float64, x)
AbstractFloat(x::Bool) = Float64(x)
AbstractFloat(x::Int8) = Float64(x)
AbstractFloat(x::Int16) = Float64(x)
AbstractFloat(x::Int32) = Float64(x)
AbstractFloat(x::Int64) = Float64(x) # LOSSY
AbstractFloat(x::Int128) = Float64(x) # LOSSY
AbstractFloat(x::UInt8) = Float64(x)
AbstractFloat(x::UInt16) = Float64(x)
AbstractFloat(x::UInt32) = Float64(x)
AbstractFloat(x::UInt64) = Float64(x) # LOSSY
AbstractFloat(x::UInt128) = Float64(x) # LOSSY
Bool(x::Float16) = x==0 ? false : x==1 ? true : throw(InexactError(:Bool, Bool, x))
"""
float(x)
Convert a number or array to a floating point data type.
See also: [`complex`](@ref), [`oftype`](@ref), [`convert`](@ref).
# Examples
```jldoctest
julia> float(1:1000)
1.0:1.0:1000.0
julia> float(typemax(Int32))
2.147483647e9
```
"""
float(x) = AbstractFloat(x)
"""
float(T::Type)
Return an appropriate type to represent a value of type `T` as a floating point value.
Equivalent to `typeof(float(zero(T)))`.
# Examples
```jldoctest
julia> float(Complex{Int})
ComplexF64 (alias for Complex{Float64})
julia> float(Int)
Float64
```
"""
float(::Type{T}) where {T<:Number} = typeof(float(zero(T)))
float(::Type{T}) where {T<:AbstractFloat} = T
float(::Type{Union{}}, slurp...) = Union{}(0.0)
"""
unsafe_trunc(T, x)
Return the nearest integral value of type `T` whose absolute value is
less than or equal to the absolute value of `x`. If the value is not representable by `T`,
an arbitrary value will be returned.
See also [`trunc`](@ref).
# Examples
```jldoctest
julia> unsafe_trunc(Int, -2.2)
-2
julia> unsafe_trunc(Int, NaN)
-9223372036854775808
```
"""
function unsafe_trunc end
for Ti in (Int8, Int16, Int32, Int64)
@eval begin
unsafe_trunc(::Type{$Ti}, x::IEEEFloat) = fptosi($Ti, x)
end
end
for Ti in (UInt8, UInt16, UInt32, UInt64)
@eval begin
unsafe_trunc(::Type{$Ti}, x::IEEEFloat) = 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
unsafe_trunc(::Type{UInt128}, x::Float16) = unsafe_trunc(UInt128, Float32(x))
unsafe_trunc(::Type{Int128}, x::Float16) = unsafe_trunc(Int128, Float32(x))
# matches convert methods
# also determines trunc, floor, ceil
round(::Type{Signed}, x::IEEEFloat, r::RoundingMode) = round(Int, x, r)
round(::Type{Unsigned}, x::IEEEFloat, r::RoundingMode) = round(UInt, x, r)
round(::Type{Integer}, x::IEEEFloat, r::RoundingMode) = round(Int, x, r)
round(x::IEEEFloat, ::RoundingMode{:ToZero}) = trunc_llvm(x)
round(x::IEEEFloat, ::RoundingMode{:Down}) = floor_llvm(x)
round(x::IEEEFloat, ::RoundingMode{:Up}) = ceil_llvm(x)
round(x::IEEEFloat, ::RoundingMode{:Nearest}) = rint_llvm(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
## floating point arithmetic ##
-(x::IEEEFloat) = neg_float(x)
+(x::T, y::T) where {T<:IEEEFloat} = add_float(x, y)
-(x::T, y::T) where {T<:IEEEFloat} = sub_float(x, y)
*(x::T, y::T) where {T<:IEEEFloat} = mul_float(x, y)
/(x::T, y::T) where {T<:IEEEFloat} = div_float(x, y)
muladd(x::T, y::T, z::T) where {T<:IEEEFloat} = muladd_float(x, y, z)
# TODO: faster floating point div?
# TODO: faster floating point fld?
# TODO: faster floating point mod?
function unbiased_exponent(x::T) where {T<:IEEEFloat}
return (reinterpret(Unsigned, x) & exponent_mask(T)) >> significand_bits(T)
end
function explicit_mantissa_noinfnan(x::T) where {T<:IEEEFloat}
m = mantissa(x)
issubnormal(x) || (m |= significand_mask(T) + uinttype(T)(1))
return m
end
function _to_float(number::U, ep) where {U<:Unsigned}
F = floattype(U)
S = signed(U)
epint = unsafe_trunc(S,ep)
lz::signed(U) = unsafe_trunc(S, Core.Intrinsics.ctlz_int(number) - U(exponent_bits(F)))
number <<= lz
epint -= lz
bits = U(0)
if epint >= 0
bits = number & significand_mask(F)
bits |= ((epint + S(1)) << significand_bits(F)) & exponent_mask(F)
else
bits = (number >> -epint) & significand_mask(F)
end
return reinterpret(F, bits)
end
@assume_effects :terminates_locally :nothrow function rem_internal(x::T, y::T) where {T<:IEEEFloat}
xuint = reinterpret(Unsigned, x)
yuint = reinterpret(Unsigned, y)
if xuint <= yuint
if xuint < yuint
return x
end
return zero(T)
end
e_x = unbiased_exponent(x)
e_y = unbiased_exponent(y)
# Most common case where |y| is "very normal" and |x/y| < 2^EXPONENT_WIDTH
if e_y > (significand_bits(T)) && (e_x - e_y) <= (exponent_bits(T))
m_x = explicit_mantissa_noinfnan(x)
m_y = explicit_mantissa_noinfnan(y)
d = urem_int((m_x << (e_x - e_y)), m_y)
iszero(d) && return zero(T)
return _to_float(d, e_y - uinttype(T)(1))
end
# Both are subnormals
if e_x == 0 && e_y == 0
return reinterpret(T, urem_int(xuint, yuint) & significand_mask(T))
end
m_x = explicit_mantissa_noinfnan(x)
e_x -= uinttype(T)(1)
m_y = explicit_mantissa_noinfnan(y)
lz_m_y = uinttype(T)(exponent_bits(T))
if e_y > 0
e_y -= uinttype(T)(1)
else
m_y = mantissa(y)
lz_m_y = Core.Intrinsics.ctlz_int(m_y)
end
tz_m_y = Core.Intrinsics.cttz_int(m_y)
sides_zeroes_cnt = lz_m_y + tz_m_y
# n>0
exp_diff = e_x - e_y
# Shift hy right until the end or n = 0
right_shift = min(exp_diff, tz_m_y)
m_y >>= right_shift
exp_diff -= right_shift
e_y += right_shift
# Shift hx left until the end or n = 0
left_shift = min(exp_diff, uinttype(T)(exponent_bits(T)))
m_x <<= left_shift
exp_diff -= left_shift
m_x = urem_int(m_x, m_y)
iszero(m_x) && return zero(T)
iszero(exp_diff) && return _to_float(m_x, e_y)
while exp_diff > sides_zeroes_cnt
exp_diff -= sides_zeroes_cnt
m_x <<= sides_zeroes_cnt
m_x = urem_int(m_x, m_y)
end
m_x <<= exp_diff
m_x = urem_int(m_x, m_y)
return _to_float(m_x, e_y)
end
function rem(x::T, y::T) where {T<:IEEEFloat}
if isfinite(x) && !iszero(x) && isfinite(y) && !iszero(y)
return copysign(rem_internal(abs(x), abs(y)), x)
elseif isinf(x) || isnan(y) || iszero(y) # y can still be Inf
return T(NaN)
else
return x
end
end
function mod(x::T, y::T) where {T<:AbstractFloat}
r = rem(x,y)
if r == 0
copysign(r,y)
elseif (r > 0) ⊻ (y > 0)
r+y
else
r
end
end
## floating point comparisons ##
==(x::T, y::T) where {T<:IEEEFloat} = eq_float(x, y)
!=(x::T, y::T) where {T<:IEEEFloat} = ne_float(x, y)
<( x::T, y::T) where {T<:IEEEFloat} = lt_float(x, y)
<=(x::T, y::T) where {T<:IEEEFloat} = le_float(x, y)
isequal(x::T, y::T) where {T<:IEEEFloat} = fpiseq(x, y)
# interpret as sign-magnitude integer
@inline function _fpint(x)
IntT = inttype(typeof(x))
ix = reinterpret(IntT, x)
return ifelse(ix < zero(IntT), ix ⊻ typemax(IntT), ix)
end
@inline function isless(a::T, b::T) where T<:IEEEFloat
(isnan(a) || isnan(b)) && return !isnan(a)
return _fpint(a) < _fpint(b)
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
for op in (:(==), :<, :<=)
@eval begin
($op)(x::Float16, y::Union{Int128,UInt128,Int64,UInt64}) = ($op)(Float64(x), Float64(y))
($op)(x::Union{Int128,UInt128,Int64,UInt64}, y::Float16) = ($op)(Float64(x), Float64(y))
($op)(x::Union{Float16,Float32}, y::Union{Int32,UInt32}) = ($op)(Float64(x), Float64(y))
($op)(x::Union{Int32,UInt32}, y::Union{Float16,Float32}) = ($op)(Float64(x), Float64(y))
($op)(x::Float16, y::Union{Int16,UInt16}) = ($op)(Float32(x), Float32(y))
($op)(x::Union{Int16,UInt16}, y::Float16) = ($op)(Float32(x), Float32(y))
end
end
abs(x::IEEEFloat) = abs_float(x)
"""
isnan(f) -> Bool
Test whether a number value is a NaN, an indeterminate value which is neither an infinity
nor a finite number ("not a number").
See also: [`iszero`](@ref), [`isone`](@ref), [`isinf`](@ref), [`ismissing`](@ref).
"""
isnan(x::AbstractFloat) = (x != x)::Bool
isnan(x::Number) = false
isfinite(x::AbstractFloat) = !isnan(x - x)
isfinite(x::Real) = decompose(x)[3] != 0
isfinite(x::Integer) = true
"""
isinf(f) -> Bool
Test whether a number is infinite.
See also: [`Inf`](@ref), [`iszero`](@ref), [`isfinite`](@ref), [`isnan`](@ref).
"""
isinf(x::Real) = !isnan(x) & !isfinite(x)
isinf(x::IEEEFloat) = abs(x) === oftype(x, Inf)
const hx_NaN = hash_uint64(reinterpret(UInt64, NaN))
function hash(x::Float64, h::UInt)
# see comments on trunc and hash(Real, UInt)
if typemin(Int64) <= x < typemax(Int64)
xi = fptosi(Int64, x)
if isequal(xi, x)
return hash(xi, h)
end
elseif typemin(UInt64) <= x < typemax(UInt64)
xu = fptoui(UInt64, x)
if isequal(xu, x)
return hash(xu, h)
end
elseif isnan(x)
return hx_NaN ⊻ h # NaN does not have a stable bit pattern
end
return hash_uint64(bitcast(UInt64, x)) - 3h
end
hash(x::Float32, h::UInt) = hash(Float64(x), h)
function hash(x::Float16, h::UInt)
# see comments on trunc and hash(Real, UInt)
if isfinite(x) # all finite Float16 fit in Int64
xi = fptosi(Int64, x)
if isequal(xi, x)
return hash(xi, h)
end
elseif isnan(x)
return hx_NaN ⊻ h # NaN does not have a stable bit pattern
end
return hash_uint64(bitcast(UInt64, Float64(x))) - 3h
end
## generic hashing for rational values ##
function hash(x::Real, h::UInt)
# decompose x as num*2^pow/den
num, pow, den = decompose(x)
# handle special values
num == 0 && den == 0 && return hash(NaN, h)
num == 0 && return hash(ifelse(den > 0, 0.0, -0.0), h)
den == 0 && return hash(ifelse(num > 0, Inf, -Inf), h)
# normalize decomposition
if den < 0
num = -num
den = -den
end
num_z = trailing_zeros(num)
num >>= num_z
den_z = trailing_zeros(den)
den >>= den_z
pow += num_z - den_z
# If the real can be represented as an Int64, UInt64, or Float64, hash as those types.
# To be an Integer the denominator must be 1 and the power must be non-negative.
if den == 1
# left = ceil(log2(num*2^pow))
left = top_set_bit(abs(num)) + pow
# 2^-1074 is the minimum Float64 so if the power is smaller, not a Float64
if -1074 <= pow
if 0 <= pow # if pow is non-negative, it is an integer
left <= 63 && return hash(Int64(num) << Int(pow), h)
left <= 64 && !signbit(num) && return hash(UInt64(num) << Int(pow), h)
end # typemin(Int64) handled by Float64 case
# 2^1024 is the maximum Float64 so if the power is greater, not a Float64
# Float64s only have 53 mantisa bits (including implicit bit)
left <= 1024 && left - pow <= 53 && return hash(ldexp(Float64(num), pow), h)
end
else
h = hash_integer(den, h)
end
# handle generic rational values
h = hash_integer(pow, h)
h = hash_integer(num, h)
return h
end
#=
`decompose(x)`: non-canonical decomposition of rational values as `num*2^pow/den`.
The decompose function is the point where rational-valued numeric types that support
hashing hook into the hashing protocol. `decompose(x)` should return three integer
values `num, pow, den`, such that the value of `x` is mathematically equal to
num*2^pow/den
The decomposition need not be canonical in the sense that it just needs to be *some*
way to express `x` in this form, not any particular way – with the restriction that
`num` and `den` may not share any odd common factors. They may, however, have powers
of two in common – the generic hashing code will normalize those as necessary.
Special values:
- `x` is zero: `num` should be zero and `den` should have the same sign as `x`
- `x` is infinite: `den` should be zero and `num` should have the same sign as `x`
- `x` is not a number: `num` and `den` should both be zero
=#
decompose(x::Integer) = x, 0, 1
function decompose(x::Float16)::NTuple{3,Int}
isnan(x) && return 0, 0, 0
isinf(x) && return ifelse(x < 0, -1, 1), 0, 0
n = reinterpret(UInt16, x)
s = (n & 0x03ff) % Int16
e = ((n & 0x7c00) >> 10) % Int
s |= Int16(e != 0) << 10
d = ifelse(signbit(x), -1, 1)
s, e - 25 + (e == 0), d
end
function decompose(x::Float32)::NTuple{3,Int}
isnan(x) && return 0, 0, 0
isinf(x) && return ifelse(x < 0, -1, 1), 0, 0
n = reinterpret(UInt32, x)
s = (n & 0x007fffff) % Int32
e = ((n & 0x7f800000) >> 23) % Int
s |= Int32(e != 0) << 23
d = ifelse(signbit(x), -1, 1)
s, e - 150 + (e == 0), d
end
function decompose(x::Float64)::Tuple{Int64, Int, Int}
isnan(x) && return 0, 0, 0
isinf(x) && return ifelse(x < 0, -1, 1), 0, 0
n = reinterpret(UInt64, x)
s = (n & 0x000fffffffffffff) % Int64
e = ((n & 0x7ff0000000000000) >> 52) % Int
s |= Int64(e != 0) << 52
d = ifelse(signbit(x), -1, 1)
s, e - 1075 + (e == 0), d
end
"""
precision(num::AbstractFloat; base::Integer=2)
precision(T::Type; base::Integer=2)
Get the precision of a floating point number, as defined by the effective number of bits in
the significand, or the precision of a floating-point type `T` (its current default, if
`T` is a variable-precision type like [`BigFloat`](@ref)).
If `base` is specified, then it returns the maximum corresponding
number of significand digits in that base.
!!! compat "Julia 1.8"
The `base` keyword requires at least Julia 1.8.
"""
function precision end
_precision_with_base_2(::Type{Float16}) = 11
_precision_with_base_2(::Type{Float32}) = 24
_precision_with_base_2(::Type{Float64}) = 53
function _precision(x, base::Integer)
base > 1 || throw(DomainError(base, "`base` cannot be less than 2."))
p = _precision_with_base_2(x)
return base == 2 ? Int(p) : floor(Int, p / log2(base))
end
precision(::Type{T}; base::Integer=2) where {T<:AbstractFloat} = _precision(T, base)
precision(::T; base::Integer=2) where {T<:AbstractFloat} = precision(T; base)
"""
nextfloat(x::AbstractFloat, n::Integer)
The result of `n` iterative applications of `nextfloat` to `x` if `n >= 0`, or `-n`
applications of [`prevfloat`](@ref) if `n < 0`.
"""
function nextfloat(f::IEEEFloat, 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)
Return 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 return `x`.
See also: [`prevfloat`](@ref), [`eps`](@ref), [`issubnormal`](@ref).
"""
nextfloat(x::AbstractFloat) = nextfloat(x,1)
"""
prevfloat(x::AbstractFloat, n::Integer)
The result of `n` iterative applications of `prevfloat` to `x` if `n >= 0`, or `-n`
applications of [`nextfloat`](@ref) if `n < 0`.
"""
prevfloat(x::AbstractFloat, d::Integer) = nextfloat(x, -d)
"""
prevfloat(x::AbstractFloat)
Return 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 return `x`.
"""
prevfloat(x::AbstractFloat) = nextfloat(x,-1)
for Ti in (Int8, Int16, Int32, Int64, Int128, UInt8, UInt16, UInt32, UInt64, UInt128)
for Tf in (Float16, 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 round(::Type{$Ti},x::$Tf,::RoundingMode{:ToZero})
if $(Tf(typemin(Ti))-one(Tf)) < x < $(Tf(typemax(Ti))+one(Tf))
return unsafe_trunc($Ti,x)
else
throw(InexactError(:round, $Ti, x, RoundToZero))
end
end
function (::Type{$Ti})(x::$Tf)
# When typemax(Ti) is not representable by Tf but typemax(Ti) + 1 is,
# then < Tf(typemax(Ti) + 1) is stricter than <= Tf(typemax(Ti)). Using
# the former causes us to throw on UInt64(Float64(typemax(UInt64))+1)
if ($(Tf(typemin(Ti))) <= x < $(Tf(typemax(Ti))+one(Tf))) && isinteger(x)
return unsafe_trunc($Ti,x)
else
throw(InexactError($(Expr(:quote,Ti.name.name)), $Ti, x))
end
end
end
else
# Here `eps(Tf(typemin(Ti))) > 1`, so the only value which can be truncated to
# `Tf(typemin(Ti)` is itself. Similarly, `Tf(typemax(Ti))` is inexact and will
# be rounded up. This assumes that `Tf(typemin(Ti)) > -Inf`, which is true for
# these types, but not for `Float16` or larger integer types.
@eval begin
function round(::Type{$Ti},x::$Tf,::RoundingMode{:ToZero})
if $(Tf(typemin(Ti))) <= x < $(Tf(typemax(Ti)))
return unsafe_trunc($Ti,x)
else
throw(InexactError(:round, $Ti, x, RoundToZero))
end
end
function (::Type{$Ti})(x::$Tf)
if ($(Tf(typemin(Ti))) <= x < $(Tf(typemax(Ti)))) && isinteger(x)
return unsafe_trunc($Ti,x)
else
throw(InexactError($(Expr(:quote,Ti.name.name)), $Ti, x))
end
end
end
end
end
end
"""
issubnormal(f) -> Bool
Test whether a floating point number is subnormal.
An IEEE floating point number is [subnormal](https://en.wikipedia.org/wiki/Subnormal_number)
when its exponent bits are zero and its significand is not zero.
# Examples
```jldoctest
julia> floatmin(Float32)
1.1754944f-38
julia> issubnormal(1.0f-37)
false
julia> issubnormal(1.0f-38)
true
```
"""
function issubnormal(x::T) where {T<:IEEEFloat}
y = reinterpret(Unsigned, x)
(y & exponent_mask(T) == 0) & (y & significand_mask(T) != 0)
end
ispow2(x::AbstractFloat) = !iszero(x) && frexp(x)[1] == 0.5
iseven(x::AbstractFloat) = isinteger(x) && (abs(x) > maxintfloat(x) || iseven(Integer(x)))
isodd(x::AbstractFloat) = isinteger(x) && abs(x) ≤ maxintfloat(x) && isodd(Integer(x))
@eval begin
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(x::T) where {T<:Real} = typemin(T)
typemax(x::T) where {T<:Real} = typemax(T)
floatmin(::Type{Float16}) = $(bitcast(Float16, 0x0400))
floatmin(::Type{Float32}) = $(bitcast(Float32, 0x00800000))
floatmin(::Type{Float64}) = $(bitcast(Float64, 0x0010000000000000))
floatmax(::Type{Float16}) = $(bitcast(Float16, 0x7bff))
floatmax(::Type{Float32}) = $(bitcast(Float32, 0x7f7fffff))
floatmax(::Type{Float64}) = $(bitcast(Float64, 0x7fefffffffffffff))
eps(::Type{Float16}) = $(bitcast(Float16, 0x1400))
eps(::Type{Float32}) = $(bitcast(Float32, 0x34000000))
eps(::Type{Float64}) = $(bitcast(Float64, 0x3cb0000000000000))
eps() = eps(Float64)
end
eps(x::AbstractFloat) = isfinite(x) ? abs(x) >= floatmin(x) ? ldexp(eps(typeof(x)), exponent(x)) : nextfloat(zero(x)) : oftype(x, NaN)
function eps(x::T) where T<:IEEEFloat
# For isfinite(x), toggling the LSB will produce either prevfloat(x) or
# nextfloat(x) but will never change the sign or exponent.
# For !isfinite(x), this will map Inf to NaN and NaN to NaN or Inf.
y = reinterpret(T, reinterpret(Unsigned, x) ⊻ true)
# The absolute difference between these values is eps(x). This is true even
# for Inf/NaN values.
return abs(x - y)
end
"""
floatmin(T = Float64)
Return the smallest positive normal number representable by the floating-point
type `T`.
# Examples
```jldoctest
julia> floatmin(Float16)
Float16(6.104e-5)
julia> floatmin(Float32)
1.1754944f-38
julia> floatmin()
2.2250738585072014e-308
```
"""
floatmin(x::T) where {T<:AbstractFloat} = floatmin(T)
"""
floatmax(T = Float64)
Return the largest finite number representable by the floating-point type `T`.
See also: [`typemax`](@ref), [`floatmin`](@ref), [`eps`](@ref).
# Examples
```jldoctest
julia> floatmax(Float16)
Float16(6.55e4)
julia> floatmax(Float32)
3.4028235f38
julia> floatmax()
1.7976931348623157e308
julia> typemax(Float64)
Inf
```
"""
floatmax(x::T) where {T<:AbstractFloat} = floatmax(T)
floatmin() = floatmin(Float64)
floatmax() = floatmax(Float64)
"""
eps(::Type{T}) where T<:AbstractFloat
eps()
Return the *machine epsilon* of the floating point type `T` (`T = Float64` by
default). This is defined as the gap between 1 and the next largest value representable by
`typeof(one(T))`, and is equivalent to `eps(one(T))`. (Since `eps(T)` is a
bound on the *relative error* of `T`, it is a "dimensionless" quantity like [`one`](@ref).)
# Examples
```jldoctest
julia> eps()
2.220446049250313e-16
julia> eps(Float32)
1.1920929f-7
julia> 1.0 + eps()
1.0000000000000002
julia> 1.0 + eps()/2
1.0
```
"""
eps(::Type{<:AbstractFloat})
"""
eps(x::AbstractFloat)
Return the *unit in last place* (ulp) of `x`. This is the distance between consecutive
representable floating point values at `x`. In most cases, if the distance on either side
of `x` is different, then the larger of the two is taken, that is
eps(x) == max(x-prevfloat(x), nextfloat(x)-x)
The exceptions to this rule are the smallest and largest finite values
(e.g. `nextfloat(-Inf)` and `prevfloat(Inf)` for [`Float64`](@ref)), which round to the
smaller of the values.
The rationale for this behavior is that `eps` bounds the floating point rounding
error. Under the default `RoundNearest` rounding mode, if ``y`` is a real number and ``x``
is the nearest floating point number to ``y``, then
```math
|y-x| \\leq \\operatorname{eps}(x)/2.
```
See also: [`nextfloat`](@ref), [`issubnormal`](@ref), [`floatmax`](@ref).
# Examples
```jldoctest
julia> eps(1.0)
2.220446049250313e-16
julia> eps(prevfloat(2.0))
2.220446049250313e-16
julia> eps(2.0)
4.440892098500626e-16
julia> x = prevfloat(Inf) # largest finite Float64
1.7976931348623157e308
julia> x + eps(x)/2 # rounds up
Inf
julia> x + prevfloat(eps(x)/2) # rounds down
1.7976931348623157e308
```
"""
eps(::AbstractFloat)
## byte order swaps for arbitrary-endianness serialization/deserialization ##
bswap(x::IEEEFloat) = bswap_int(x)
# integer size of float
uinttype(::Type{Float64}) = UInt64
uinttype(::Type{Float32}) = UInt32
uinttype(::Type{Float16}) = UInt16
inttype(::Type{Float64}) = Int64
inttype(::Type{Float32}) = Int32
inttype(::Type{Float16}) = Int16
# float size of integer
floattype(::Type{UInt64}) = Float64
floattype(::Type{UInt32}) = Float32
floattype(::Type{UInt16}) = Float16
floattype(::Type{Int64}) = Float64
floattype(::Type{Int32}) = Float32
floattype(::Type{Int16}) = Float16
## Array operations on floating point numbers ##
float(A::AbstractArray{<:AbstractFloat}) = A
function float(A::AbstractArray{T}) where T
if !isconcretetype(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{T}) where {T} =
StepRangeLen{typeof(float(T(r.ref)))}(float(r.ref), float(r.step), length(r), r.offset)
function float(r::LinRange)
LinRange(float(r.start), float(r.stop), length(r))
end
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