FloatHigher.jl

accurate floating point math at extended precisions

        Jeffrey A. Sarnoff                           2015-Dec-30 03:21:21 UTC America/New_York
                                                     2015-Dec-31 06:00:00 UTC (multityped ops)
                                                     2016-Jan-03 21:21:21 UTC (DigitsNN renamed)

+
Calculations are substantially faster than BigFloat.
Accuracy more consistently tracks displayed precision than BigFloat.

-
Pulls in all of Nemo to use some of Arb.
Does not implement some math functions and other operators available with Float64.

★
Fredrik Johannson has written 2015's best intermediate precision floating point math software. The right next step would be to study William Hart's Julia interface and make Arb's Real and Complex number support directly available in Julia. Matching missing operators and smoothly adding in his other math functions is straightforward. Julia would loose BigFloat issues working at precisions up through 10,000 decimal digits, and gain speed and provide more transparent accuracy when working with extended precision real/complex floats.

###∵
This is as far as I am comfortable taking Julia's search for a first class Float128 implementation.
                          .. with the hope this will have helped ..

=====

Digits30, Digits70, Digits140, or Digits300 are selected before 'using' this module
( or, in separate modules: DIGITS30.jl, DIGITS70.jl, DIGITS140.jl, DIGITS300.jl ).
When none are explicitly selected, Digits30 is used.

using FloatHigher
julia> a=Digits30(2);sqrt2=sqrt(a)
1.41421356237309504880168872421
julia> showball(sqrt2)
[1.4142135623730950488016887242096980786 +/- 4.87e-38]
quit()

UseDigits70=true
using FloatHigher
julia> a=Digits70(2);sqrt2=sqrt(a)
1.414213562373095048801688724209698078569671875376948073176679737990732
julia> showball(sqrt2)
 showball(sqrt2)
[1.4142135623730950488016887242096980785696718753769480731766797379907324784621 +/- 4.99e-77]
quit()

UseDigits140=true
using FloatHigher
julia> a=Digits140(0.5);println(exp(a));a-log(exp(a))
1.648721270700128146848650787814163571653776100710148011575079311640661
 0211942140244768312883565706777193388588425688402538063164289228771285
0

UseDigits300=true
using FloatHigher
julia> asin(Digits300(0.5))*6
3.141592653589793238462643383279502884197169399375105820974944592307816
 4062861980294536250318213496466758841295330717990647287577300387921745
 5543798124334760862857823598496915024953863910429488250583465990554483
 29813888808104138661292381584644317626953125
julia> showball(ans)
[3.141592653589793238462643383279502884197169399375105820974944592307816
  4062862089986280348253421170679821480865132823066470938446095505822317
  2535940812848111745028410270193852110555964462294895493038196442881097
  5665933446128475648233786783165271201909145648566923460348610454326648
  213393607260249141273724587 +/- 2.53e-307]

  
 

Digits30, Digits70, Digits140, Digits300 can be used together.
Intertype promotion defers to the smaller type (otherwise the result could become quite inaccurate invisibly):

UseDigits70=true
UseDigits300=true
using FloatHigher

a=sqrt(Digits70(256))
16
b=sqrt(Digits300(1024))
32
c=a+b; c, typeof(c)
48, Digits70

This relies entirely on Fredrik Johansson's Arb software, which is included with William Hart's Nemo package.
The Arb documentation is at http://fredrikj.net/arb/. The documentation for Nemo is at http://nemocas.org.
You will see Nemo's welcome message. Nemo is required unless Arb is separately available to Julia.