Float32 results are computed using Float64s

There is one export, the type Single32. Use it in place of Float32 for results that are more reliable, reporting numerical results with greater accuracy through the low-order bits of significands' precision.

Single32 values look like Float32 values and act like Float32 values. Their computational results are exceed the expectations for accuracy with Float32s. With relatively stable algorithms, Single32s are much better at preserving information present in significands' low-order bits.

Mathematical operations with Single32s are computed using Float64 internally. To get the benefit that they afford, it is necessary that you do not reach inside these values to see or to use any part that is not shown in regular use. This works only if the values you provide are Float32s and the values you obtain are Float32s.

The translation from an array of Float32 to an array of Single32 is done with broadcasting. After completing the computational work, the translation back to Float32 is just as easy:

using SingleFloats

               # original_data must be Float32

data_at_work   = Single32.(original_data)
value_obtains  = process(data_at_work)
result_of_work = Float32.(value_obtains)

               # result_of_work must be Float32

The intent is to provide robust coverage of Float32 ops. Let me know of requests. PRs welcome.

using SingleFloats

xs_fwd(::Type{T}) where T = T.(collect(1.0:20.0))
ys_fwd(::Type{T}) where T = cot.(xs_fwd(T))
sumfwd(::Type{T}) where T = sum(ys_fwd(T))

xs_rev(::Type{T}) where T = reverse(xs_fwd(T))
ys_rev(::Type{T}) where T = cot.(xs_rev(T))
sumrev(::Type{T}) where T = sum(ys_rev(T))

epsmax(a, b) = eps(max(a, b))

function muddybits(::Type{T}) where T
   fwd = sumfwd(T)
   rev = sumrev(T)
   muddy = round(Int32, abs(fwd - rev) / epsmax(fwd, rev))

   lsbits = 31 - leading_zeros(muddy)
   return max(0, lsbits)
end


#  How many low-order bits of these type's significands have become
#  opaque, replacing confirmatory valuation with inessential noise?

(Single32 = muddybits(Single32),
 Float32  = muddybits(Float32),
 Float64  = muddybits(Float64))

(Single32 = 0, Float32 = 6, Float64 = 7)