# This file is a part of Julia. License is MIT: https://julialang.org/license
# Twice-precision arithmetic.
# Necessary for creating nicely-behaved ranges like r = 0.1:0.1:0.3
# that return r[3] == 0.3. Otherwise, we have roundoff error due to
# 0.1 + 2*0.1 = 0.30000000000000004
"""
hi, lo = splitprec(F::Type{<:AbstractFloat}, i::Integer)
Represent an integer `i` as a pair of floating-point numbers `hi` and
`lo` (of type `F`) such that:
- `widen(hi) + widen(lo) ≈ i`. It is exact if 1.5 * (number of precision bits in `F`) is greater than the number of bits in `i`.
- all bits in `hi` are more significant than any of the bits in `lo`
- `hi` can be exactly multiplied by the `hi` component of another call to `splitprec`.
In particular, while `convert(Float64, i)` can be lossy since Float64
has only 53 bits of precision, `splitprec(Float64, i)` is exact for
any Int64/UInt64.
"""
function splitprec(::Type{F}, i::Integer) where {F<:AbstractFloat}
hi = truncbits(F(i), cld(precision(F), 2))
ihi = oftype(i, hi)
hi, F(i - ihi)
end
function truncmask(x::F, mask) where {F<:IEEEFloat}
reinterpret(F, mask & reinterpret(uinttype(F), x))
end
truncmask(x, mask) = x
function truncbits(x::F, nb) where {F<:IEEEFloat}
truncmask(x, typemax(uinttype(F)) << nb)
end
truncbits(x, nb) = x
## Dekker arithmetic
"""
hi, lo = canonicalize2(big, little)
Generate a representation where all the nonzero bits in `hi` are more
significant than any of the nonzero bits in `lo`. `big` must be larger
in absolute value than `little`.
"""
function canonicalize2(big, little)
h = big+little
h, (big - h) + little
end
"""
zhi, zlo = add12(x, y)
A high-precision representation of `x + y` for floating-point
numbers. Mathematically, `zhi + zlo = x + y`, where `zhi` contains the
most significant bits and `zlo` the least significant.
Because of the way floating-point numbers are printed, `lo` may not
look the way you might expect from the standpoint of decimal
representation, even though it is exact from the standpoint of binary
representation.
Example:
```julia
julia> 1.0 + 1.0001e-15
1.000000000000001
julia> big(1.0) + big(1.0001e-15)
1.000000000000001000100000000000020165767380775934141445417482375879192346701529
julia> hi, lo = Base.add12(1.0, 1.0001e-15)
(1.000000000000001, -1.1012302462515652e-16)
julia> big(hi) + big(lo)
1.000000000000001000100000000000020165767380775934141445417482375879192346701529
```
`lo` differs from 1.0e-19 because `hi` is not exactly equal to
the first 16 decimal digits of the answer.
"""
function add12(x::T, y::T) where {T}
x, y = ifelse(abs(y) > abs(x), (y, x), (x, y))
canonicalize2(x, y)
end
add12(x, y) = add12(promote(x, y)...)
"""
zhi, zlo = mul12(x, y)
A high-precision representation of `x * y` for floating-point
numbers. Mathematically, `zhi + zlo = x * y`, where `zhi` contains the
most significant bits and `zlo` the least significant.
Example:
```julia
julia> x = Float32(π)
3.1415927f0
julia> x * x
9.869605f0
julia> Float64(x) * Float64(x)
9.869604950382893
julia> hi, lo = Base.mul12(x, x)
(9.869605f0, -1.140092f-7)
julia> Float64(hi) + Float64(lo)
9.869604950382893
```
"""
function mul12(x::T, y::T) where {T<:AbstractFloat}
h = x * y
ifelse(iszero(h) | !isfinite(h), (h, h), canonicalize2(h, fma(x, y, -h)))
end
mul12(x::T, y::T) where {T} = (p = x * y; (p, zero(p)))
mul12(x, y) = mul12(promote(x, y)...)
"""
zhi, zlo = div12(x, y)
A high-precision representation of `x / y` for floating-point
numbers. Mathematically, `zhi + zlo ≈ x / y`, where `zhi` contains the
most significant bits and `zlo` the least significant.
Example:
```julia
julia> x, y = Float32(π), 3.1f0
(3.1415927f0, 3.1f0)
julia> x / y
1.013417f0
julia> Float64(x) / Float64(y)
1.0134170444063078
julia> hi, lo = Base.div12(x, y)
(1.013417f0, 3.8867366f-8)
julia> Float64(hi) + Float64(lo)
1.0134170444063066
"""
function div12(x::T, y::T) where {T<:AbstractFloat}
# We lose precision if any intermediate calculation results in a subnormal.
# To prevent this from happening, standardize the values.
xs, xe = frexp(x)
ys, ye = frexp(y)
r = xs / ys
rh, rl = canonicalize2(r, -fma(r, ys, -xs)/ys)
ifelse(iszero(r) | !isfinite(r), (r, r), (ldexp(rh, xe-ye), ldexp(rl, xe-ye)))
end
div12(x::T, y::T) where {T} = (p = x / y; (p, zero(p)))
div12(x, y) = div12(promote(x, y)...)
## TwicePrecision
"""
TwicePrecision{T}(hi::T, lo::T)
TwicePrecision{T}((num, denom))
A number with twice the precision of `T`, e.g., quad-precision if `T =
Float64`. `hi` represents the high bits (most significant bits) and
`lo` the low bits (least significant bits). Rational values
`num//denom` can be approximated conveniently using the syntax
`TwicePrecision{T}((num, denom))`.
When used with `T<:Union{Float16,Float32,Float64}` to construct an "exact"
`StepRangeLen`, `ref` should be the range element with smallest
magnitude and `offset` set to the corresponding index. For
efficiency, multiplication of `step` by the index is not performed at
twice precision: `step.hi` should have enough trailing zeros in its
`bits` representation that `(0:len-1)*step.hi` is exact (has no
roundoff error). If `step` has an exact rational representation
`num//denom`, then you can construct `step` using
step = TwicePrecision{T}((num, denom), nb)
where `nb` is the number of trailing zero bits of `step.hi`. For
ranges, you can set `nb = ceil(Int, log2(len-1))`.
"""
struct TwicePrecision{T}
hi::T # most significant bits
lo::T # least significant bits
end
TwicePrecision{T}(x::T) where {T} = TwicePrecision{T}(x, zero(T))
function TwicePrecision{T}(x) where {T}
xT = convert(T, x)
Δx = x - xT
TwicePrecision{T}(xT, T(Δx))
end
TwicePrecision{T}(i::Integer) where {T<:AbstractFloat} =
TwicePrecision{T}(canonicalize2(splitprec(T, i)...)...)
TwicePrecision(x) = TwicePrecision{typeof(x)}(x)
# Numerator/Denominator constructors
function TwicePrecision{T}(nd::Tuple{Integer,Integer}) where {T<:Union{Float16,Float32}}
n, d = nd
TwicePrecision{T}(n/d)
end
function TwicePrecision{T}(nd::Tuple{Any,Any}) where {T}
n, d = nd
TwicePrecision{T}(n) / d
end
function TwicePrecision{T}(nd::Tuple{I,I}, nb::Integer) where {T,I}
twiceprecision(TwicePrecision{T}(nd), nb)
end
# Truncating constructors. Useful for generating values that can be
# exactly multiplied by small integers.
function twiceprecision(val::T, nb::Integer) where {T<:IEEEFloat}
hi = truncbits(val, nb)
TwicePrecision{T}(hi, val - hi)
end
function twiceprecision(val::TwicePrecision{T}, nb::Integer) where {T<:IEEEFloat}
hi = truncbits(val.hi, nb)
TwicePrecision{T}(hi, (val.hi - hi) + val.lo)
end
nbitslen(r::StepRangeLen) = nbitslen(eltype(r), length(r), r.offset)
nbitslen(::Type{T}, len, offset) where {T<:IEEEFloat} =
min(cld(precision(T), 2), nbitslen(len, offset))
# The +1 here is for safety, because the precision of the significand
# is 1 bit higher than the number that are explicitly stored.
nbitslen(len, offset) = len < 2 ? 0 : ceil(Int, log2(max(offset-1, len-offset))) + 1
eltype(::Type{TwicePrecision{T}}) where {T} = T
promote_rule(::Type{TwicePrecision{R}}, ::Type{TwicePrecision{S}}) where {R,S} =
TwicePrecision{promote_type(R,S)}
promote_rule(::Type{TwicePrecision{R}}, ::Type{S}) where {R,S<:Number} =
TwicePrecision{promote_type(R,S)}
(::Type{T})(x::TwicePrecision) where {T<:Number} = T(x.hi + x.lo)::T
TwicePrecision{T}(x::Number) where {T} = TwicePrecision{T}(T(x), zero(T))
convert(::Type{TwicePrecision{T}}, x::TwicePrecision{T}) where {T} = x
convert(::Type{TwicePrecision{T}}, x::TwicePrecision) where {T} =
TwicePrecision{T}(convert(T, x.hi), convert(T, x.lo))
convert(::Type{T}, x::TwicePrecision) where {T<:Number} = T(x)
convert(::Type{TwicePrecision{T}}, x::Number) where {T} = TwicePrecision{T}(x)
float(x::TwicePrecision{<:AbstractFloat}) = x
float(x::TwicePrecision) = TwicePrecision(float(x.hi), float(x.lo))
big(x::TwicePrecision) = big(x.hi) + big(x.lo)
-(x::TwicePrecision) = TwicePrecision(-x.hi, -x.lo)
zero(::Type{TwicePrecision{T}}) where {T} = TwicePrecision{T}(0, 0)
# Arithmetic
function +(x::TwicePrecision, y::Number)
s_hi, s_lo = add12(x.hi, y)
TwicePrecision(canonicalize2(s_hi, s_lo+x.lo)...)
end
+(x::Number, y::TwicePrecision) = y+x
function +(x::TwicePrecision{T}, y::TwicePrecision{T}) where T
r = x.hi + y.hi
s = abs(x.hi) > abs(y.hi) ? (((x.hi - r) + y.hi) + y.lo) + x.lo : (((y.hi - r) + x.hi) + x.lo) + y.lo
TwicePrecision(canonicalize2(r, s)...)
end
+(x::TwicePrecision, y::TwicePrecision) = +(promote(x, y)...)
-(x::TwicePrecision, y::TwicePrecision) = x + (-y)
-(x::TwicePrecision, y::Number) = x + (-y)
-(x::Number, y::TwicePrecision) = x + (-y)
function *(x::TwicePrecision, v::Number)
v == 0 && return TwicePrecision(x.hi*v, x.lo*v)
x * TwicePrecision(oftype(x.hi*v, v))
end
function *(x::TwicePrecision{<:IEEEFloat}, v::Integer)
v == 0 && return TwicePrecision(x.hi*v, x.lo*v)
nb = ceil(Int, log2(abs(v)))
u = truncbits(x.hi, nb)
TwicePrecision(canonicalize2(u*v, ((x.hi-u) + x.lo)*v)...)
end
*(v::Number, x::TwicePrecision) = x*v
function *(x::TwicePrecision{T}, y::TwicePrecision{T}) where {T}
zh, zl = mul12(x.hi, y.hi)
ret = TwicePrecision{T}(canonicalize2(zh, (x.hi * y.lo + x.lo * y.hi) + zl)...)
ifelse(iszero(zh) | !isfinite(zh), TwicePrecision{T}(zh, zh), ret)
end
*(x::TwicePrecision, y::TwicePrecision) = *(promote(x, y)...)
function /(x::TwicePrecision, v::Number)
x / TwicePrecision(oftype(x.hi/v, v))
end
function /(x::TwicePrecision, y::TwicePrecision)
hi = x.hi / y.hi
uh, ul = mul12(hi, y.hi)
lo = ((((x.hi - uh) - ul) + x.lo) - hi*y.lo)/y.hi
ret = TwicePrecision(canonicalize2(hi, lo)...)
ifelse(iszero(hi) | !isfinite(hi), TwicePrecision(hi, hi), ret)
end
## StepRangeLen
# Use TwicePrecision only for Float64; use Float64 for T<:Union{Float16,Float32}
# See also _linspace1
# Ratio-of-integers constructors
function steprangelen_hp(::Type{Float64}, ref::Tuple{Integer,Integer},
step::Tuple{Integer,Integer}, nb::Integer,
len::Integer, offset::Integer)
StepRangeLen(TwicePrecision{Float64}(ref),
TwicePrecision{Float64}(step, nb), Int(len), offset)
end
function steprangelen_hp(::Type{T}, ref::Tuple{Integer,Integer},
step::Tuple{Integer,Integer}, nb::Integer,
len::Integer, offset::Integer) where {T<:IEEEFloat}
StepRangeLen{T}(ref[1]/ref[2], step[1]/step[2], Int(len), offset)
end
# AbstractFloat constructors (can supply a single number or a 2-tuple
const F_or_FF = Union{AbstractFloat, Tuple{AbstractFloat,AbstractFloat}}
asF64(x::AbstractFloat) = Float64(x)
asF64(x::Tuple{AbstractFloat,AbstractFloat}) = Float64(x[1]) + Float64(x[2])
function steprangelen_hp(::Type{Float64}, ref::F_or_FF,
step::F_or_FF, nb::Integer,
len::Integer, offset::Integer)
StepRangeLen(TwicePrecision{Float64}(ref...),
twiceprecision(TwicePrecision{Float64}(step...), nb), Int(len), offset)
end
function steprangelen_hp(::Type{T}, ref::F_or_FF,
step::F_or_FF, nb::Integer,
len::Integer, offset::Integer) where {T<:IEEEFloat}
StepRangeLen{T}(asF64(ref),
asF64(step), Int(len), offset)
end
StepRangeLen(ref::TwicePrecision{T}, step::TwicePrecision{T},
len::Integer, offset::Integer=1) where {T} =
StepRangeLen{T,TwicePrecision{T},TwicePrecision{T}}(ref, step, len, offset)
# Construct range for rational start=start_n/den, step=step_n/den
function floatrange(::Type{T}, start_n::Integer, step_n::Integer, len::Integer, den::Integer) where T
if len < 2
return steprangelen_hp(T, (start_n, den), (step_n, den), 0, Int(len), 1)
end
# index of smallest-magnitude value
imin = clamp(round(Int, -start_n/step_n+1), 1, Int(len))
# Compute smallest-magnitude element to 2x precision
ref_n = start_n+(imin-1)*step_n # this shouldn't overflow, so don't check
nb = nbitslen(T, len, imin)
steprangelen_hp(T, (ref_n, den), (step_n, den), nb, Int(len), imin)
end
function floatrange(a::AbstractFloat, st::AbstractFloat, len::Real, divisor::AbstractFloat)
T = promote_type(typeof(a), typeof(st), typeof(divisor))
m = maxintfloat(T, Int)
if abs(a) <= m && abs(st) <= m && abs(divisor) <= m
ia, ist, idivisor = round(Int, a), round(Int, st), round(Int, divisor)
if ia == a && ist == st && idivisor == divisor
# We can return the high-precision range
return floatrange(T, ia, ist, Int(len), idivisor)
end
end
# Fallback (misses the opportunity to set offset different from 1,
# but otherwise this is still high-precision)
steprangelen_hp(T, (a,divisor), (st,divisor), nbitslen(T, len, 1), Int(len), 1)
end
function (:)(start::T, step::T, stop::T) where T<:Union{Float16,Float32,Float64}
step == 0 && throw(ArgumentError("range step cannot be zero"))
# see if the inputs have exact rational approximations (and if so,
# perform all computations in terms of the rationals)
step_n, step_d = rat(step)
if step_d != 0 && T(step_n/step_d) == step
start_n, start_d = rat(start)
stop_n, stop_d = rat(stop)
if start_d != 0 && stop_d != 0 &&
T(start_n/start_d) == start && T(stop_n/stop_d) == stop
den = lcm(start_d, step_d) # use same denominator for start and step
m = maxintfloat(T, Int)
if den != 0 && abs(start*den) <= m && abs(step*den) <= m && # will round succeed?
rem(den, start_d) == 0 && rem(den, step_d) == 0 # check lcm overflow
start_n = round(Int, start*den)
step_n = round(Int, step*den)
len = max(0, div(den*stop_n - stop_d*start_n + step_n*stop_d, step_n*stop_d))
# Integer ops could overflow, so check that this makes sense
if isbetween(start, start + (len-1)*step, stop + step/2) &&
!isbetween(start, start + len*step, stop)
# Return a 2x precision range
return floatrange(T, start_n, step_n, len, den)
end
end
end
end
# Fallback, taking start and step literally
lf = (stop-start)/step
if lf < 0
len = 0
elseif lf == 0
len = 1
else
len = round(Int, lf) + 1
stop′ = start + (len-1)*step
# if we've overshot the end, subtract one:
len -= (start < stop < stop′) + (start > stop > stop′)
end
steprangelen_hp(T, start, step, 0, len, 1)
end
function _range(a::T, st::T, ::Nothing, len::Integer) where T<:Union{Float16,Float32,Float64}
start_n, start_d = rat(a)
step_n, step_d = rat(st)
if start_d != 0 && step_d != 0 &&
T(start_n/start_d) == a && T(step_n/step_d) == st
den = lcm(start_d, step_d)
m = maxintfloat(T, Int)
if abs(den*a) <= m && abs(den*st) <= m &&
rem(den, start_d) == 0 && rem(den, step_d) == 0
start_n = round(Int, den*a)
step_n = round(Int, den*st)
return floatrange(T, start_n, step_n, len, den)
end
end
steprangelen_hp(T, a, st, 0, len, 1)
end
# This assumes that r.step has already been split so that (0:len-1)*r.step.hi is exact
function unsafe_getindex(r::StepRangeLen{T,<:TwicePrecision,<:TwicePrecision}, i::Integer) where T
# Very similar to _getindex_hiprec, but optimized to avoid a 2nd call to add12
@_inline_meta
u = i - r.offset
shift_hi, shift_lo = u*r.step.hi, u*r.step.lo
x_hi, x_lo = add12(r.ref.hi, shift_hi)
T(x_hi + (x_lo + (shift_lo + r.ref.lo)))
end
function _getindex_hiprec(r::StepRangeLen{<:Any,<:TwicePrecision,<:TwicePrecision}, i::Integer)
u = i - r.offset
shift_hi, shift_lo = u*r.step.hi, u*r.step.lo
x_hi, x_lo = add12(r.ref.hi, shift_hi)
x_hi, x_lo = add12(x_hi, x_lo + (shift_lo + r.ref.lo))
TwicePrecision(x_hi, x_lo)
end
function getindex(r::StepRangeLen{T,<:TwicePrecision,<:TwicePrecision}, s::OrdinalRange{<:Integer}) where T
@boundscheck checkbounds(r, s)
soffset = 1 + round(Int, (r.offset - first(s))/step(s))
soffset = clamp(soffset, 1, length(s))
ioffset = first(s) + (soffset-1)*step(s)
if step(s) == 1 || length(s) < 2
newstep = r.step
else
newstep = twiceprecision(r.step*step(s), nbitslen(T, length(s), soffset))
end
if ioffset == r.offset
StepRangeLen(r.ref, newstep, length(s), max(1,soffset))
else
StepRangeLen(r.ref + (ioffset-r.offset)*r.step, newstep, length(s), max(1,soffset))
end
end
*(x::Real, r::StepRangeLen{<:Real,<:TwicePrecision}) =
StepRangeLen(x*r.ref, twiceprecision(x*r.step, nbitslen(r)), length(r), r.offset)
*(r::StepRangeLen{<:Real,<:TwicePrecision}, x::Real) = x*r
/(r::StepRangeLen{<:Real,<:TwicePrecision}, x::Real) =
StepRangeLen(r.ref/x, twiceprecision(r.step/x, nbitslen(r)), length(r), r.offset)
StepRangeLen{T,R,S}(r::StepRangeLen{T,R,S}) where {T<:AbstractFloat,R<:TwicePrecision,S<:TwicePrecision} = r
StepRangeLen{T,R,S}(r::StepRangeLen) where {T<:AbstractFloat,R<:TwicePrecision,S<:TwicePrecision} =
_convertSRL(StepRangeLen{T,R,S}, r)
(::Type{StepRangeLen{Float64}})(r::StepRangeLen) =
_convertSRL(StepRangeLen{Float64,TwicePrecision{Float64},TwicePrecision{Float64}}, r)
StepRangeLen{T}(r::StepRangeLen) where {T<:IEEEFloat} =
_convertSRL(StepRangeLen{T,Float64,Float64}, r)
(::Type{StepRangeLen{Float64}})(r::AbstractRange) =
_convertSRL(StepRangeLen{Float64,TwicePrecision{Float64},TwicePrecision{Float64}}, r)
StepRangeLen{T}(r::AbstractRange) where {T<:IEEEFloat} =
_convertSRL(StepRangeLen{T,Float64,Float64}, r)
function _convertSRL(::Type{StepRangeLen{T,R,S}}, r::StepRangeLen{<:Integer}) where {T,R,S}
StepRangeLen{T,R,S}(R(r.ref), S(r.step), length(r), r.offset)
end
function _convertSRL(::Type{StepRangeLen{T,R,S}}, r::AbstractRange{<:Integer}) where {T,R,S}
StepRangeLen{T,R,S}(R(first(r)), S(step(r)), length(r))
end
function _convertSRL(::Type{StepRangeLen{T,R,S}}, r::AbstractRange{U}) where {T,R,S,U}
# if start and step have a rational approximation in the old type,
# then we transfer that rational approximation to the new type
f, s = first(r), step(r)
start_n, start_d = rat(f)
step_n, step_d = rat(s)
if start_d != 0 && step_d != 0 &&
U(start_n/start_d) == f && U(step_n/step_d) == s
den = lcm(start_d, step_d)
m = maxintfloat(T, Int)
if den != 0 && abs(f*den) <= m && abs(s*den) <= m &&
rem(den, start_d) == 0 && rem(den, step_d) == 0
start_n = round(Int, f*den)
step_n = round(Int, s*den)
return floatrange(T, start_n, step_n, length(r), den)
end
end
__convertSRL(StepRangeLen{T,R,S}, r)
end
function __convertSRL(::Type{StepRangeLen{T,R,S}}, r::StepRangeLen{U}) where {T,R,S,U}
StepRangeLen{T,R,S}(R(r.ref), S(r.step), length(r), r.offset)
end
function __convertSRL(::Type{StepRangeLen{T,R,S}}, r::AbstractRange{U}) where {T,R,S,U}
StepRangeLen{T,R,S}(R(first(r)), S(step(r)), length(r))
end
function sum(r::StepRangeLen)
l = length(r)
# Compute the contribution of step over all indices.
# Indexes on opposite side of r.offset contribute with opposite sign,
# r.step * (sum(1:np) - sum(1:nn))
np, nn = l - r.offset, r.offset - 1 # positive, negative
# To prevent overflow in sum(1:n), multiply its factors by the step
sp, sn = sumpair(np), sumpair(nn)
tp = _tp_prod(r.step, sp[1], sp[2])
tn = _tp_prod(r.step, sn[1], sn[2])
s_hi, s_lo = add12(tp.hi, -tn.hi)
s_lo += tp.lo - tn.lo
# Add in contributions of ref
ref = r.ref * l
sm_hi, sm_lo = add12(s_hi, ref.hi)
add12(sm_hi, sm_lo + ref.lo)[1]
end
# sum(1:n) as a product of two integers
sumpair(n::Integer) = iseven(n) ? (n+1, n>>1) : (n, (n+1)>>1)
function +(r1::StepRangeLen{T,R}, r2::StepRangeLen{T,R}) where T where R<:TwicePrecision
len = length(r1)
(len == length(r2) ||
throw(DimensionMismatch("argument dimensions must match")))
if r1.offset == r2.offset
imid = r1.offset
ref = r1.ref + r2.ref
else
imid = round(Int, (r1.offset+r2.offset)/2)
ref1mid = _getindex_hiprec(r1, imid)
ref2mid = _getindex_hiprec(r2, imid)
ref = ref1mid + ref2mid
end
step = twiceprecision(r1.step + r2.step, nbitslen(T, len, imid))
StepRangeLen{T,typeof(ref),typeof(step)}(ref, step, len, imid)
end
## LinRange
# For Float16, Float32, and Float64, this returns a StepRangeLen
function _range(start::T, ::Nothing, stop::T, len::Integer) where {T<:IEEEFloat}
len < 2 && return _linspace1(T, start, stop, len)
if start == stop
return steprangelen_hp(T, start, zero(T), 0, len, 1)
end
# Attempt to find exact rational approximations
start_n, start_d = rat(start)
stop_n, stop_d = rat(stop)
if start_d != 0 && stop_d != 0
den = lcm(start_d, stop_d)
m = maxintfloat(T, Int)
if den != 0 && abs(den*start) <= m && abs(den*stop) <= m
start_n = round(Int, den*start)
stop_n = round(Int, den*stop)
if T(start_n/den) == start && T(stop_n/den) == stop
return _linspace(T, start_n, stop_n, len, den)
end
end
end
_linspace(start, stop, len)
end
function _linspace(start::T, stop::T, len::Integer) where {T<:IEEEFloat}
(isfinite(start) && isfinite(stop)) || throw(ArgumentError("start and stop must be finite, got $start and $stop"))
# Find the index that returns the smallest-magnitude element
Δ, Δfac = stop-start, 1
if !isfinite(Δ) # handle overflow for large endpoints
Δ, Δfac = stop/len - start/len, Int(len)
end
tmin = -(start/Δ)/Δfac # t such that (1-t)*start + t*stop == 0
imin = round(Int, tmin*(len-1)+1) # index approximately corresponding to t
if 1 < imin < len
# The smallest-magnitude element is in the interior
t = (imin-1)/(len-1)
ref = T((1-t)*start + t*stop)
step = imin-1 < len-imin ? (ref-start)/(imin-1) : (stop-ref)/(len-imin)
elseif imin <= 1
imin = 1
ref = start
step = (Δ/(len-1))*Δfac
else
imin = Int(len)
ref = stop
step = (Δ/(len-1))*Δfac
end
if len == 2 && !isfinite(step)
# For very large endpoints where step overflows, exploit the
# split-representation to handle the overflow
return steprangelen_hp(T, start, (-start, stop), 0, 2, 1)
end
# 2x calculations to get high precision endpoint matching while also
# preventing overflow in ref_hi+(i-offset)*step_hi
m, k = prevfloat(realmax(T)), max(imin-1, len-imin)
step_hi_pre = clamp(step, max(-(m+ref)/k, (-m+ref)/k), min((m-ref)/k, (m+ref)/k))
nb = nbitslen(T, len, imin)
step_hi = truncbits(step_hi_pre, nb)
x1_hi, x1_lo = add12((1-imin)*step_hi, ref)
x2_hi, x2_lo = add12((len-imin)*step_hi, ref)
a, b = (start - x1_hi) - x1_lo, (stop - x2_hi) - x2_lo
step_lo = (b - a)/(len - 1)
ref_lo = a - (1 - imin)*step_lo
steprangelen_hp(T, (ref, ref_lo), (step_hi, step_lo), 0, Int(len), imin)
end
# range for rational numbers, start = start_n/den, stop = stop_n/den
# Note this returns a StepRangeLen
_linspace(::Type{T}, start::Integer, stop::Integer, len::Integer) where {T<:IEEEFloat} = _linspace(T, start, stop, len, 1)
function _linspace(::Type{T}, start_n::Integer, stop_n::Integer, len::Integer, den::Integer) where T<:IEEEFloat
len < 2 && return _linspace1(T, start_n/den, stop_n/den, len)
start_n == stop_n && return steprangelen_hp(T, (start_n, den), (zero(start_n), den), 0, len, 1)
tmin = -start_n/(Float64(stop_n) - Float64(start_n))
imin = round(Int, tmin*(len-1)+1)
imin = clamp(imin, 1, Int(len))
ref_num = Int128(len-imin) * start_n + Int128(imin-1) * stop_n
ref_denom = Int128(len-1) * den
ref = (ref_num, ref_denom)
step_full = (Int128(stop_n) - Int128(start_n), ref_denom)
steprangelen_hp(T, ref, step_full, nbitslen(T, len, imin), Int(len), imin)
end
# For len < 2
function _linspace1(::Type{T}, start, stop, len::Integer) where T<:IEEEFloat
len >= 0 || throw(ArgumentError("range($start, stop=$stop, length=$len): negative length"))
if len <= 1
len == 1 && (start == stop || throw(ArgumentError("range($start, stop=$stop, length=$len): endpoints differ")))
# Ensure that first(r)==start and last(r)==stop even for len==0
# The output type must be consistent with steprangelen_hp
if T<:Union{Float32,Float16}
return StepRangeLen{T}(Float64(start), Float64(start) - Float64(stop), len, 1)
else
return StepRangeLen(TwicePrecision(start, zero(T)), TwicePrecision(start, -stop), len, 1)
end
end
throw(ArgumentError("should only be called for len < 2, got $len"))
end
### Numeric utilities
# Approximate x with a rational representation. Guaranteed to return,
# but not guaranteed to return a precise answer.
# https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations
function rat(x)
y = x
a = d = 1
b = c = 0
m = maxintfloat(narrow(typeof(x)), Int)
while abs(y) <= m
f = trunc(Int,y)
y -= f
a, c = f*a + c, a
b, d = f*b + d, b
max(abs(a), abs(b)) <= convert(Int,m) || return c, d
oftype(x,a)/oftype(x,b) == x && break
y = inv(y)
end
return a, b
end
narrow(::Type{T}) where {T<:AbstractFloat} = Float64
narrow(::Type{Float64}) = Float32
narrow(::Type{Float32}) = Float16
narrow(::Type{Float16}) = Float16
function _tp_prod(t::TwicePrecision, x, y...)
@_inline_meta
_tp_prod(t * x, y...)
end
_tp_prod(t::TwicePrecision) = t
<(x::TwicePrecision{T}, y::TwicePrecision{T}) where {T} =
x.hi < y.hi || ((x.hi == y.hi) & (x.lo < y.lo))
isbetween(a, x, b) = a <= x <= b || b <= x <= a