Actually this is really smart. You can represent exact values (lsp == 0) or open intervals one ulp wide... This is a good chunk of the value of fixed sized unums. I'll be keeping a close eye on the package. Thanks Jeffrey.
On Wednesday, November 25, 2015, Jeffrey Sarnoff <[email protected]> wrote: > > These are distinct operators that substitute directly for (+),(-),(*),(/) > in situations where one wants to obtain more of mathematically true result > than is usually available: > > two = 2.0; sqrt2 = sqrt(2); > residualValueRoundedAway = Float64(sqrt(big(2)) - sqrt2) # > -9.667293313452913e-17 > > mostSignficantPart, leastSignificantPart = eftSqrt(two) > mostSignificantPart == 1.4142135623730951 > leastSignificantPart == -9.667293313452912e-17 # we recover the residual > value, itself at Float64 precision > > so we obtain the arithmetic result at twice the 'working' precision (in > two parts, the mspart == the usual result). > > exp1log2 = exp(1.0)*log(2.0); > # 1.88416938536372 > residualValueRoundedAway = Float64(exp(big(1))*log(big(2)) - exp1log2) # > 8.146538547111741e-17 > > mostSignficantPart, leastSignificantPart = eftProd2( exp(1.0), log(2.0) ) > # (1.88416938536372, -8.177744937186283e-17) > > ------ > These transformations have the additional benefit that the two parts are > well separated, they do not overlap in the working precision. > So, in all cases, mostSignificantPart + leastSignificantPart == > mostSignificantPart. > They are as well separated as possible, without losing information. > > These functions are well-suited to assisting the implementation of > extended precision Floating Point math. > Another application (that, until otherwise informed, I'll say is from me) > is to accelerate inline rounding: > (RoundFast.jl <https://github.com/J-Sarnoff/RoundFast.jl>, there to see > how). > > Assuming one had a Float64 unum-ish capability, a double-double float > would extend the precision. > (Ultimately, all these parts should meld) > > > On Wednesday, November 25, 2015 at 9:19:08 AM UTC-5, Tom Breloff wrote: >> >> Thanks Jeffrey. Can you expand on the specifics of the package? What >> would you say are the primary use cases? How does this differ from interval >> arithmetic or Unums? >> >> On Wednesday, November 25, 2015, Jeffrey Sarnoff <[email protected]> >> wrote: >> >>> ErrorFreeArith.jl <https://github.com/J-Sarnoff/ErrorFreeArith.jl> offers >>> error-free transformations not (yet?) included in the ErrorFreeTransforms >>> package by dsiem. >>> >>> These operations convey the usual arithmetic result accompanied by a >>> residual value that is usually lost to rounding. >>> This gives the correct value at twice the working precision (correctly >>> rounded for +,-,*,/; still 1/(1/x) = x or x ± ulp(x)). >>> >>> >>> >>>
