julia/base/twiceprecision.jl

# 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:
```jldoctest
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:
```jldoctest
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, l) = Math.two_mul(x, y)
    ifelse(!isfinite(h), (h, h), (h, l))
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:
```jldoctest
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`.

!!! warning
    `TwicePrecision` is an internal type used to increase the
    precision of floating-point ranges, and not intended for external use.
    If you encounter them in real code, the most likely explanation is
    that you are directly accessing the fields of a range. Use
    the function interface instead, `step(r)` rather than `r.step`

# Extended help

`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))

TwicePrecision{T}(x::TwicePrecision{T}) where {T} = x

function TwicePrecision{T}(x) where {T}
    xT = 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}(TwicePrecision{T}(n) / d)
end

function TwicePrecision{T}(nd::Tuple{I,I}, nb::Integer) where {T,I}
    twiceprecision(TwicePrecision{T}(nd), nb)
end

# Fix #39798
# See steprangelen_hp(::Type{Float64}, ref::Tuple{Integer,Integer},
#                         step::Tuple{Integer,Integer}, nb::Integer,
#                         len::Integer, offset::Integer)
function TwicePrecision{T}(nd::Tuple{Integer,Integer}, nb::Integer) where T
    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 : top_set_bit(max(offset-1, len-offset) - 1) + 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) + T(x.lo))::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))::TwicePrecision{T}

convert(::Type{T}, x::TwicePrecision) where {T<:Number} = T(x)::T
convert(::Type{TwicePrecision{T}}, x::Number) where {T} = TwicePrecision{T}(x)::TwicePrecision{T}

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(x::TwicePrecision) = zero(typeof(x))
function zero(::Type{TwicePrecision{T}}) where {T}
    z = zero(T)
    TwicePrecision{T}(z, z)
end

# 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 = top_set_bit(abs(v)-1)
    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), 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], 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), 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), 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
    len = len + 0 # promote with Int
    if len < 2 || step_n == 0
        return steprangelen_hp(T, (start_n, den), (step_n, den), 0, len, oneunit(len))
    end
    # index of smallest-magnitude value
    L = typeof(len)
    imin = clamp(round(typeof(len), -start_n/step_n+1), oneunit(L), 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, len, imin)
end

function (:)(start::T, step::T, stop::T) where T<:IEEEFloat
    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_unchecked(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, Int(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
    # n.b. we use Int as the default length type for IEEEFloats
    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

step(r::StepRangeLen{T,TwicePrecision{T},TwicePrecision{T}}) where {T<:AbstractFloat} = T(r.step)
step(r::StepRangeLen{T,TwicePrecision{T},TwicePrecision{T}}) where {T} = T(r.step)

range_start_step_length(a::Real, st::IEEEFloat, len::Integer) =
    range_start_step_length(promote(a, st)..., len)

range_start_step_length(a::IEEEFloat, st::Real, len::Integer) =
    range_start_step_length(promote(a, st)..., len)

range_start_step_length(a::IEEEFloat, st::IEEEFloat, len::Integer) =
    range_start_step_length(promote(a, st)..., len)

function range_start_step_length(a::T, st::T, len::Integer) where T<:IEEEFloat
    len = len + 0 # promote with Int
    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_unchecked(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

range_step_stop_length(step::Real, stop::IEEEFloat, len::Integer) =
    range_step_stop_length(promote(step, stop)..., len)

range_step_stop_length(step::IEEEFloat, stop::Real, len::Integer) =
    range_step_stop_length(promote(step, stop)..., len)

function range_step_stop_length(step::IEEEFloat, stop::IEEEFloat, len::Integer)
    r = range_start_step_length(stop, negate(step), len)
    reverse(r)
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
    u = oftype(r.offset, 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)
    i isa Bool && throw(ArgumentError("invalid index: $i of type Bool"))
    u = oftype(r.offset, 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{S}) where {T, S<:Integer}
    @boundscheck checkbounds(r, s)
    len = length(s)
    L = typeof(len)
    sstep = step_hp(s)
    rstep = step_hp(r)
    if S === Bool
        #rstep *= one(sstep)
        if len == 0
            return StepRangeLen{T}(first(r), rstep, zero(L), oneunit(L))
        elseif len == 1
            if first(s)
                return StepRangeLen{T}(first(r), rstep, oneunit(L), oneunit(L))
            else
                return StepRangeLen{T}(first(r), rstep, zero(L), oneunit(L))
            end
        else # len == 2
            return StepRangeLen{T}(last(r), step(r), oneunit(L), oneunit(L))
        end
    else
        soffset = round(L, (r.offset - first(s))/sstep + 1)
        soffset = clamp(soffset, oneunit(L), len)
        ioffset = L(first(s) + (soffset - oneunit(L)) * sstep)
        if sstep == 1 || len < 2
            newstep = rstep #* one(sstep)
        else
            newstep = rstep * sstep
            newstep = twiceprecision(newstep, nbitslen(T, len, soffset))
        end
        soffset = max(oneunit(L), soffset)
        if ioffset == r.offset
            return StepRangeLen{T}(r.ref, newstep, len, soffset)
        else
            return StepRangeLen{T}(r.ref + (ioffset-r.offset)*rstep, newstep, len, soffset)
        end
    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,L}(r::StepRangeLen{T,R,S,L}) where {T<:AbstractFloat,R<:TwicePrecision,S<:TwicePrecision,L} = r

StepRangeLen{T,R,S,L}(r::StepRangeLen) where {T<:AbstractFloat,R<:TwicePrecision,S<:TwicePrecision,L} =
    _convertSRL(StepRangeLen{T,R,S,L}, r)

StepRangeLen{Float64}(r::StepRangeLen) =
    _convertSRL(StepRangeLen{Float64,TwicePrecision{Float64},TwicePrecision{Float64},Int}, r)
StepRangeLen{T}(r::StepRangeLen) where {T<:IEEEFloat} =
    _convertSRL(StepRangeLen{T,Float64,Float64,Int}, r)

StepRangeLen{Float64}(r::AbstractRange) =
    _convertSRL(StepRangeLen{Float64,TwicePrecision{Float64},TwicePrecision{Float64},Int}, r)
StepRangeLen{T}(r::AbstractRange) where {T<:IEEEFloat} =
    _convertSRL(StepRangeLen{T,Float64,Float64,Int}, r)

function _convertSRL(::Type{StepRangeLen{T,R,S,L}}, r::StepRangeLen{<:Integer}) where {T,R,S,L}
    StepRangeLen{T,R,S,L}(R(r.ref), S(r.step), L(length(r)), L(r.offset))
end

function _convertSRL(::Type{StepRangeLen{T,R,S,L}}, r::AbstractRange{<:Integer}) where {T,R,S,L}
    StepRangeLen{T,R,S,L}(R(first(r)), S(step(r)), L(length(r)))
end

function _convertSRL(::Type{StepRangeLen{T,R,S,L}}, r::AbstractRange{U}) where {T,R,S,L,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_unchecked(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, L(length(r)), den)
        end
    end
    return __convertSRL(StepRangeLen{T,R,S,L}, r)
end

function __convertSRL(::Type{StepRangeLen{T,R,S,L}}, r::StepRangeLen{U}) where {T,R,S,L,U}
    StepRangeLen{T,R,S,L}(R(r.ref), S(r.step), L(length(r)), L(r.offset))
end
function __convertSRL(::Type{StepRangeLen{T,R,S,L}}, r::AbstractRange{U}) where {T,R,S,L,U}
    StepRangeLen{T,R,S,L}(R(first(r)), S(step(r)), L(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)
    W = widen(typeof(l))
    Δn = W(sp[1]) * W(sp[2]) - W(sn[1]) * W(sn[2])
    s = r.step * Δn
    # Add in contributions of ref
    ref = r.ref * l
    convert(eltype(r), s + ref)
end
function sum(r::StepRangeLen{<:Any,<:TwicePrecision,<:TwicePrecision})
    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(typeof(len), (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),typeof(len)}(ref, step, len, imid)
end

## LinRange

# For Float16, Float32, and Float64, this returns a StepRangeLen
function range_start_stop_length(start::T, stop::T, len::Integer) where {T<:IEEEFloat}
    len = len + 0 # promote with Int
    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_unchecked(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}
    len = len + 0 # promote with Int
    (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, len
    end
    tmin = -(start/Δ)/Δfac            # t such that (1-t)*start + t*stop == 0
    L = typeof(len)
    lenn1 = len - oneunit(L)
    imin = round(L, tmin*lenn1 + 1) # index approximately corresponding to t
    if 1 < imin < len
        # The smallest-magnitude element is in the interior
        t = (imin - 1)/lenn1
        ref = T((1-t)*start + t*stop)
        step = imin-1 < len-imin ? (ref-start)/(imin-1) : (stop-ref)/(len-imin)
    elseif imin <= 1
        imin = oneunit(L)
        ref = start
        step = (Δ/(lenn1))*Δfac
    else
        imin = len
        ref = stop
        step = (Δ/(lenn1))*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, len, oneunit(L))
    end
    # 2x calculations to get high precision endpoint matching while also
    # preventing overflow in ref_hi+(i-offset)*step_hi
    m, k = prevfloat(floatmax(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, 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, one(start))
function _linspace(::Type{T}, start_n::Integer, stop_n::Integer, len::Integer, den::Integer) where T<:IEEEFloat
    len = len + 0 # promote with Int
    len < 2 && return _linspace1(T, start_n/den, stop_n/den, len)
    L = typeof(len)
    start_n == stop_n && return steprangelen_hp(T, (start_n, den), (zero(start_n), den), 0, len, oneunit(L))
    tmin = -start_n/(Float64(stop_n) - Float64(start_n))
    imin = round(typeof(len), tmin*(len-1)+1)
    imin = clamp(imin, oneunit(L), len)
    W = widen(L)
    start_n = W(start_n)
    stop_n = W(stop_n)
    ref_num = W(len-imin) * start_n + W(imin-1) * stop_n
    ref_denom = W(len-1) * den
    ref = (ref_num, ref_denom)
    step_full = (stop_n - start_n, ref_denom)
    steprangelen_hp(T, ref, step_full, nbitslen(T, len, imin), 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 # T == Float64
            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 as a pair of Int values.
# 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

# This version of lcm does not check for overflows
lcm_unchecked(a::T, b::T) where T<:Integer = a * div(b, gcd(a, b))

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
    _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

# These functions exist for use in LogRange:

_exp_allowing_twice64(x::Number) = exp(x)
_exp_allowing_twice64(x::TwicePrecision{Float64}) = Math.exp_impl(x.hi, x.lo, Val(:ℯ))

# No error on negative x, and for NaN/Inf this returns junk:
function _log_twice64_unchecked(x::Float64)
    xu = reinterpret(UInt64, x)
    if xu < (UInt64(1)<<52) # x is subnormal
        xu = reinterpret(UInt64, x * 0x1p52) # normalize x
        xu &= ~sign_mask(Float64)
        xu -= UInt64(52) << 52 # mess with the exponent
    end
    TwicePrecision(Math._log_ext(xu)...)
end