(42) Double precision summation and DOT

[Albert Chan]

For decimal version, we can get precision P with less code

3 1/X 3 × 1 −                                  // -10^(-P), or 0 if binary version (*)

this is longer, but will work for binary version too

3 1/X Enter Enter Enter 1 − + +        // -10^(-P), or (-2)^(-P) for binary version

Example, on python 2.6:

>>> x = 1/3.
>>> print (x-1+x+x).hex()
-0x1.0000000000000p-53

(*) with binary, x = 1/3 has error of ±1/3 ULP
2^P ULP = 1/2 → 3x = 1 ± 1/2^(P+1), which always rounds back to 1