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

## Veltkamp splitting
# Multiplication is based on fma, but splitting routines are still
# useful esp. for handling precise Integer->AbstractFloat conversions.

"""
    xh = halfprec(x)

Return a value close to `x` that has at most half of the precision
available to `typeof(x)`, removing `x`'s trailing bits. This is a
component of Veltkamp splitting, see "Handbook of Floating-point
Arithmetic", §4.4.1.

Note that this may overflow for large `x`. A safe strategy is
```julia
xs, xe = frexp(x)
xh = ldexp(halfprec(xs), xe)
```
"""
function halfprec(x::T) where {T<:AbstractFloat}
    c = T((1 << cld(precision(T), 2)) + 1)
    p = x*c
    (x-p)+p
end

"""
    hi, lo = splitprec(x)

Return numbers `hi` and `lo` corresponding to the most and least
significant bits of `x`, respectively, such that `x = hi + lo`. Moreover, if

    hi1, lo1 = splitprec(x1)
    hi2, lo2 = splitprec(x2)

then pairwise products (e.g., `hi1 * hi2`, `lo1 * hi2`, etc.) can be
performed exactly (without loss of precision compared to widening,
unless one of the numbers is subnormal).

This is known as Veltkamp splitting, see "Handbook of Floating-point
Arithmetic", §4.4.1.
"""
function splitprec(x::T) where {T<:AbstractFloat}
    xs, xe = frexp(x)
    hie = halfprec(xs)
    loe = xs - hie
    ldexp(hie, xe), ldexp(loe, xe)
end

"""
    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`.
- `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.

Note that `lo` may have too many bits to be precise under
multiplication by other "split" floating point numbers---you may need
to call `splitprec(lo)` or use the Dekker arithmetic routines when
performing operations with `lo`.
"""
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(fpinttype(F), x))
end
truncmask(x, mask) = x

function truncbits(x::F, nb) where {F<:IEEEFloat}
    truncmask(x, typemax(fpinttype(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_noncircular(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_noncircular(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_noncircular(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} =
    TwicePrecision{promote_type(R,S)}

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} = convert(T, x.hi + x.lo)
convert(::Type{TwicePrecision{T}}, x::Number) where {T} = TwicePrecision{T}(convert(T, x), zero(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(::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_noncircular(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{typeof(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_noncircular(x, y)...)

function /(x::TwicePrecision, v::Number)
    x / TwicePrecision{typeof(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}
# 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 colon(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, 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)

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

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

convert(::Type{StepRangeLen{T}}, r::StepRangeLen) where {T<:IEEEFloat} =
    _convertSRL(StepRangeLen{T,TwicePrecision{T},TwicePrecision{T}}, r)

convert(::Type{StepRangeLen{T}}, r::Range) where {T<:IEEEFloat} =
    _convertSRL(StepRangeLen{T,TwicePrecision{T},TwicePrecision{T}}, 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::Range{<: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::Range{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::Range{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 indexes.
    # 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 = _prod(r.step, sp[1], sp[2])
    tn = _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

## LinSpace

# For Float16, Float32, and Float64, linspace returns a StepRangeLen
function linspace(start::T, 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

# linspace for rational numbers, start = start_n/den, stop = stop_n/den
# Note this returns a StepRangeLen
function linspace(::Type{T}, start_n::Integer, stop_n::Integer, len::Integer, den::Integer) where T
    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)
    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
    len >= 0 || throw(ArgumentError("linspace($start, $stop, $len): negative length"))
    if len <= 1
        len == 1 && (start == stop || throw(ArgumentError("linspace($start, $stop, $len): endpoints differ")))
        # Ensure that first(r)==start and last(r)==stop even for len==0
        return StepRangeLen(TwicePrecision(start, zero(T)), TwicePrecision(start, -stop), len, 1)
    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)))
    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{Float64}) = Float32
narrow(::Type{Float32}) = Float16
narrow(::Type{Float16}) = Float16

function _prod(t::TwicePrecision, x, y...)
    @_inline_meta
    _prod(t * x, y...)
end
_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