869 lines
28 KiB
Julia
869 lines
28 KiB
Julia
# This file is a part of Julia. License is MIT: https://julialang.org/license
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## floating point traits ##
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"""
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Inf16
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Positive infinity of type [`Float16`](@ref).
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"""
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const Inf16 = bitcast(Float16, 0x7c00)
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"""
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NaN16
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A not-a-number value of type [`Float16`](@ref).
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"""
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const NaN16 = bitcast(Float16, 0x7e00)
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"""
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Inf32
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Positive infinity of type [`Float32`](@ref).
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"""
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const Inf32 = bitcast(Float32, 0x7f800000)
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"""
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NaN32
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A not-a-number value of type [`Float32`](@ref).
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"""
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const NaN32 = bitcast(Float32, 0x7fc00000)
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const Inf64 = bitcast(Float64, 0x7ff0000000000000)
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const NaN64 = bitcast(Float64, 0x7ff8000000000000)
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"""
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Inf
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Positive infinity of type [`Float64`](@ref).
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"""
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const Inf = Inf64
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"""
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NaN
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A not-a-number value of type [`Float64`](@ref).
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"""
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const NaN = NaN64
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## conversions to floating-point ##
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convert(::Type{Float16}, x::Integer) = convert(Float16, convert(Float32, x))
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for t in (Int8, Int16, Int32, Int64, Int128, UInt8, UInt16, UInt32, UInt64, UInt128)
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@eval promote_rule(::Type{Float16}, ::Type{$t}) = Float16
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end
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promote_rule(::Type{Float16}, ::Type{Bool}) = Float16
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for t1 in (Float32, Float64)
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for st in (Int8, Int16, Int32, Int64)
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@eval begin
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convert(::Type{$t1}, x::($st)) = sitofp($t1, x)
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promote_rule(::Type{$t1}, ::Type{$st}) = $t1
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end
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end
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for ut in (Bool, UInt8, UInt16, UInt32, UInt64)
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@eval begin
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convert(::Type{$t1}, x::($ut)) = uitofp($t1, x)
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promote_rule(::Type{$t1}, ::Type{$ut}) = $t1
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end
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end
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end
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convert(::Type{Integer}, x::Float16) = convert(Integer, Float32(x))
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convert(::Type{T}, x::Float16) where {T<:Integer} = convert(T, Float32(x))
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promote_rule(::Type{Float64}, ::Type{UInt128}) = Float64
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promote_rule(::Type{Float64}, ::Type{Int128}) = Float64
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promote_rule(::Type{Float32}, ::Type{UInt128}) = Float32
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promote_rule(::Type{Float32}, ::Type{Int128}) = Float32
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function convert(::Type{Float64}, x::UInt128)
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x == 0 && return 0.0
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n = 128-leading_zeros(x) # ndigits0z(x,2)
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if n <= 53
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y = ((x % UInt64) << (53-n)) & 0x000f_ffff_ffff_ffff
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else
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y = ((x >> (n-54)) % UInt64) & 0x001f_ffff_ffff_ffff # keep 1 extra bit
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y = (y+1)>>1 # round, ties up (extra leading bit in case of next exponent)
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y &= ~UInt64(trailing_zeros(x) == (n-54)) # fix last bit to round to even
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end
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d = ((n+1022) % UInt64) << 52
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reinterpret(Float64, d + y)
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end
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function convert(::Type{Float64}, x::Int128)
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x == 0 && return 0.0
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s = ((x >>> 64) % UInt64) & 0x8000_0000_0000_0000 # sign bit
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x = abs(x) % UInt128
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n = 128-leading_zeros(x) # ndigits0z(x,2)
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if n <= 53
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y = ((x % UInt64) << (53-n)) & 0x000f_ffff_ffff_ffff
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else
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y = ((x >> (n-54)) % UInt64) & 0x001f_ffff_ffff_ffff # keep 1 extra bit
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y = (y+1)>>1 # round, ties up (extra leading bit in case of next exponent)
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y &= ~UInt64(trailing_zeros(x) == (n-54)) # fix last bit to round to even
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end
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d = ((n+1022) % UInt64) << 52
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reinterpret(Float64, s | d + y)
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end
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function convert(::Type{Float32}, x::UInt128)
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x == 0 && return 0f0
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n = 128-leading_zeros(x) # ndigits0z(x,2)
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if n <= 24
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y = ((x % UInt32) << (24-n)) & 0x007f_ffff
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else
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y = ((x >> (n-25)) % UInt32) & 0x00ff_ffff # keep 1 extra bit
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y = (y+one(UInt32))>>1 # round, ties up (extra leading bit in case of next exponent)
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y &= ~UInt32(trailing_zeros(x) == (n-25)) # fix last bit to round to even
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end
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d = ((n+126) % UInt32) << 23
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reinterpret(Float32, d + y)
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end
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function convert(::Type{Float32}, x::Int128)
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x == 0 && return 0f0
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s = ((x >>> 96) % UInt32) & 0x8000_0000 # sign bit
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x = abs(x) % UInt128
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n = 128-leading_zeros(x) # ndigits0z(x,2)
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if n <= 24
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y = ((x % UInt32) << (24-n)) & 0x007f_ffff
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else
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y = ((x >> (n-25)) % UInt32) & 0x00ff_ffff # keep 1 extra bit
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y = (y+one(UInt32))>>1 # round, ties up (extra leading bit in case of next exponent)
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y &= ~UInt32(trailing_zeros(x) == (n-25)) # fix last bit to round to even
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end
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d = ((n+126) % UInt32) << 23
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reinterpret(Float32, s | d + y)
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end
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function convert(::Type{Float16}, val::Float32)
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f = reinterpret(UInt32, val)
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i = (f >> 23) & 0x1ff + 1
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sh = shifttable[i]
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f &= 0x007fffff
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h::UInt16 = basetable[i] + (f >> sh)
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# round
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# NOTE: we maybe should ignore NaNs here, but the payload is
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# getting truncated anyway so "rounding" it might not matter
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nextbit = (f >> (sh-1)) & 1
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if nextbit != 0
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# Round halfway to even or check lower bits
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if h&1 == 1 || (f & ((1<<(sh-1))-1)) != 0
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h += 1
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end
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end
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reinterpret(Float16, h)
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end
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function convert(::Type{Float32}, val::Float16)
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local ival::UInt32 = reinterpret(UInt16, val)
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local sign::UInt32 = (ival & 0x8000) >> 15
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local exp::UInt32 = (ival & 0x7c00) >> 10
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local sig::UInt32 = (ival & 0x3ff) >> 0
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local ret::UInt32
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if exp == 0
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if sig == 0
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sign = sign << 31
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ret = sign | exp | sig
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else
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n_bit = 1
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bit = 0x0200
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while (bit & sig) == 0
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n_bit = n_bit + 1
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bit = bit >> 1
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end
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sign = sign << 31
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exp = (-14 - n_bit + 127) << 23
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sig = ((sig & (~bit)) << n_bit) << (23 - 10)
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ret = sign | exp | sig
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end
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elseif exp == 0x1f
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if sig == 0 # Inf
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if sign == 0
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ret = 0x7f800000
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else
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ret = 0xff800000
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end
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else # NaN
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ret = 0x7fc00000 | (sign<<31)
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end
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else
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sign = sign << 31
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exp = (exp - 15 + 127) << 23
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sig = sig << (23 - 10)
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ret = sign | exp | sig
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end
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return reinterpret(Float32, ret)
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end
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# Float32 -> Float16 algorithm from:
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# "Fast Half Float Conversion" by Jeroen van der Zijp
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# ftp://ftp.fox-toolkit.org/pub/fasthalffloatconversion.pdf
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const basetable = Vector{UInt16}(512)
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const shifttable = Vector{UInt8}(512)
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for i = 0:255
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e = i - 127
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if e < -24 # Very small numbers map to zero
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basetable[i|0x000+1] = 0x0000
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basetable[i|0x100+1] = 0x8000
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shifttable[i|0x000+1] = 24
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shifttable[i|0x100+1] = 24
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elseif e < -14 # Small numbers map to denorms
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basetable[i|0x000+1] = (0x0400>>(-e-14))
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basetable[i|0x100+1] = (0x0400>>(-e-14)) | 0x8000
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shifttable[i|0x000+1] = -e-1
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shifttable[i|0x100+1] = -e-1
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elseif e <= 15 # Normal numbers just lose precision
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basetable[i|0x000+1] = ((e+15)<<10)
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basetable[i|0x100+1] = ((e+15)<<10) | 0x8000
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shifttable[i|0x000+1] = 13
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shifttable[i|0x100+1] = 13
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elseif e < 128 # Large numbers map to Infinity
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basetable[i|0x000+1] = 0x7C00
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basetable[i|0x100+1] = 0xFC00
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shifttable[i|0x000+1] = 24
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shifttable[i|0x100+1] = 24
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else # Infinity and NaN's stay Infinity and NaN's
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basetable[i|0x000+1] = 0x7C00
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basetable[i|0x100+1] = 0xFC00
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shifttable[i|0x000+1] = 13
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shifttable[i|0x100+1] = 13
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end
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end
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#convert(::Type{Float16}, x::Float32) = fptrunc(Float16, x)
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convert(::Type{Float32}, x::Float64) = fptrunc(Float32, x)
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convert(::Type{Float16}, x::Float64) = convert(Float16, convert(Float32, x))
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#convert(::Type{Float32}, x::Float16) = fpext(Float32, x)
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convert(::Type{Float64}, x::Float32) = fpext(Float64, x)
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convert(::Type{Float64}, x::Float16) = convert(Float64, convert(Float32, x))
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convert(::Type{AbstractFloat}, x::Bool) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::Int8) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::Int16) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::Int32) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::Int64) = convert(Float64, x) # LOSSY
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convert(::Type{AbstractFloat}, x::Int128) = convert(Float64, x) # LOSSY
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convert(::Type{AbstractFloat}, x::UInt8) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::UInt16) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::UInt32) = convert(Float64, x)
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convert(::Type{AbstractFloat}, x::UInt64) = convert(Float64, x) # LOSSY
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convert(::Type{AbstractFloat}, x::UInt128) = convert(Float64, x) # LOSSY
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"""
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float(x)
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Convert a number or array to a floating point data type.
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When passed a string, this function is equivalent to `parse(Float64, x)`.
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"""
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float(x) = convert(AbstractFloat, x)
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"""
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float(T::Type)
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Returns an appropriate type to represent a value of type `T` as a floating point value.
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Equivalent to `typeof(float(zero(T)))`.
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```jldoctest
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julia> float(Complex{Int})
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Complex{Float64}
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julia> float(Int)
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Float64
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```
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"""
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float(::Type{T}) where {T<:Number} = typeof(float(zero(T)))
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for Ti in (Int8, Int16, Int32, Int64)
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@eval begin
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unsafe_trunc(::Type{$Ti}, x::Float16) = unsafe_trunc($Ti, Float32(x))
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unsafe_trunc(::Type{$Ti}, x::Float32) = fptosi($Ti, x)
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unsafe_trunc(::Type{$Ti}, x::Float64) = fptosi($Ti, x)
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end
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end
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for Ti in (UInt8, UInt16, UInt32, UInt64)
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@eval begin
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unsafe_trunc(::Type{$Ti}, x::Float16) = unsafe_trunc($Ti, Float32(x))
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unsafe_trunc(::Type{$Ti}, x::Float32) = fptoui($Ti, x)
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unsafe_trunc(::Type{$Ti}, x::Float64) = fptoui($Ti, x)
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end
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end
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function unsafe_trunc(::Type{UInt128}, x::Float64)
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xu = reinterpret(UInt64,x)
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k = Int(xu >> 52) & 0x07ff - 1075
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xu = (xu & 0x000f_ffff_ffff_ffff) | 0x0010_0000_0000_0000
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if k <= 0
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UInt128(xu >> -k)
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else
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UInt128(xu) << k
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end
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end
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function unsafe_trunc(::Type{Int128}, x::Float64)
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copysign(unsafe_trunc(UInt128,x) % Int128, x)
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end
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function unsafe_trunc(::Type{UInt128}, x::Float32)
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xu = reinterpret(UInt32,x)
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k = Int(xu >> 23) & 0x00ff - 150
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xu = (xu & 0x007f_ffff) | 0x0080_0000
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if k <= 0
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UInt128(xu >> -k)
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else
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UInt128(xu) << k
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end
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end
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function unsafe_trunc(::Type{Int128}, x::Float32)
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copysign(unsafe_trunc(UInt128,x) % Int128, x)
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end
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unsafe_trunc(::Type{UInt128}, x::Float16) = unsafe_trunc(UInt128, Float32(x))
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unsafe_trunc(::Type{Int128}, x::Float16) = unsafe_trunc(Int128, Float32(x))
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# matches convert methods
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# also determines floor, ceil, round
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trunc(::Type{Signed}, x::Float32) = trunc(Int,x)
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trunc(::Type{Signed}, x::Float64) = trunc(Int,x)
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trunc(::Type{Unsigned}, x::Float32) = trunc(UInt,x)
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trunc(::Type{Unsigned}, x::Float64) = trunc(UInt,x)
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trunc(::Type{Integer}, x::Float32) = trunc(Int,x)
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trunc(::Type{Integer}, x::Float64) = trunc(Int,x)
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trunc(::Type{T}, x::Float16) where {T<:Integer} = trunc(T, Float32(x))
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# fallbacks
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floor(::Type{T}, x::AbstractFloat) where {T<:Integer} = trunc(T,floor(x))
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floor(::Type{T}, x::Float16) where {T<:Integer} = floor(T, Float32(x))
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ceil(::Type{T}, x::AbstractFloat) where {T<:Integer} = trunc(T,ceil(x))
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ceil(::Type{T}, x::Float16) where {T<:Integer} = ceil(T, Float32(x))
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round(::Type{T}, x::AbstractFloat) where {T<:Integer} = trunc(T,round(x))
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round(::Type{T}, x::Float16) where {T<:Integer} = round(T, Float32(x))
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trunc(x::Float64) = trunc_llvm(x)
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trunc(x::Float32) = trunc_llvm(x)
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trunc(x::Float16) = Float16(trunc(Float32(x)))
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floor(x::Float64) = floor_llvm(x)
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floor(x::Float32) = floor_llvm(x)
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floor(x::Float16) = Float16(floor(Float32(x)))
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ceil(x::Float64) = ceil_llvm(x)
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ceil(x::Float32) = ceil_llvm(x)
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ceil(x::Float16) = Float16( ceil(Float32(x)))
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round(x::Float64) = rint_llvm(x)
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round(x::Float32) = rint_llvm(x)
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round(x::Float16) = Float16(round(Float32(x)))
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## floating point promotions ##
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promote_rule(::Type{Float32}, ::Type{Float16}) = Float32
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promote_rule(::Type{Float64}, ::Type{Float16}) = Float64
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promote_rule(::Type{Float64}, ::Type{Float32}) = Float64
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widen(::Type{Float16}) = Float32
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widen(::Type{Float32}) = Float64
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_default_type(T::Union{Type{Real},Type{AbstractFloat}}) = Float64
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## floating point arithmetic ##
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-(x::Float64) = neg_float(x)
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-(x::Float32) = neg_float(x)
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-(x::Float16) = reinterpret(Float16, reinterpret(UInt16, x) ⊻ 0x8000)
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for op in (:+, :-, :*, :/, :\, :^)
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@eval ($op)(a::Float16, b::Float16) = Float16(($op)(Float32(a), Float32(b)))
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end
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+(x::Float32, y::Float32) = add_float(x, y)
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+(x::Float64, y::Float64) = add_float(x, y)
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-(x::Float32, y::Float32) = sub_float(x, y)
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-(x::Float64, y::Float64) = sub_float(x, y)
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*(x::Float32, y::Float32) = mul_float(x, y)
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*(x::Float64, y::Float64) = mul_float(x, y)
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/(x::Float32, y::Float32) = div_float(x, y)
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/(x::Float64, y::Float64) = div_float(x, y)
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muladd(x::Float32, y::Float32, z::Float32) = muladd_float(x, y, z)
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muladd(x::Float64, y::Float64, z::Float64) = muladd_float(x, y, z)
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function muladd(a::Float16, b::Float16, c::Float16)
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Float16(muladd(Float32(a), Float32(b), Float32(c)))
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end
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# TODO: faster floating point div?
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# TODO: faster floating point fld?
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# TODO: faster floating point mod?
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for func in (:div,:fld,:cld,:rem,:mod)
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@eval begin
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$func(a::Float16,b::Float16) = Float16($func(Float32(a),Float32(b)))
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end
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end
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rem(x::Float32, y::Float32) = rem_float(x, y)
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rem(x::Float64, y::Float64) = rem_float(x, y)
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cld(x::T, y::T) where {T<:AbstractFloat} = -fld(-x,y)
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function mod(x::T, y::T) where T<:AbstractFloat
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r = rem(x,y)
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if r == 0
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copysign(r,y)
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elseif (r > 0) ⊻ (y > 0)
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r+y
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else
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r
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end
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end
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## floating point comparisons ##
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function ==(x::Float16, y::Float16)
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ix = reinterpret(UInt16,x)
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iy = reinterpret(UInt16,y)
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if (ix|iy)&0x7fff > 0x7c00 #isnan(x) || isnan(y)
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return false
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end
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if (ix|iy)&0x7fff == 0x0000
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return true
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end
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return ix == iy
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end
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==(x::Float32, y::Float32) = eq_float(x, y)
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==(x::Float64, y::Float64) = eq_float(x, y)
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!=(x::Float32, y::Float32) = ne_float(x, y)
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!=(x::Float64, y::Float64) = ne_float(x, y)
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<( x::Float32, y::Float32) = lt_float(x, y)
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<( 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) where {T<:AbstractFloat} = 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(x::T) where {T<:Real} = typemin(T)
|
|
typemax(x::T) where {T<:Real} = 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(x::T) where {T<:AbstractFloat} = realmin(T)
|
|
realmax(x::T) where {T<:AbstractFloat} = 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
|
|
|
|
"""
|
|
eps(::Type{T}) where T<:AbstractFloat
|
|
eps()
|
|
|
|
Returns 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
|
|
`T`, and is equivalent to `eps(one(T))`.
|
|
|
|
```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)
|
|
|
|
Returns 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.
|
|
```
|
|
|
|
```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::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
|
|
|
|
## 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(A::AbstractArray{<:AbstractFloat}) = A
|
|
|
|
function float(A::AbstractArray{T}) where 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
|