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ArbFloats

Much faster than BigFloat at precisions up to 3,500 bits (1050 digits)

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ArbFloats.jl

Copyright © 2016 by Jeffrey Sarnoff. Released under the MIT License.

Build Status

ArbFloats calculate faster than BigFloats at medium precisions

These results were obtained using BenchmarkTools.jl on one desktop system.

  • Relative Speed
    • big ≝ mean(execution time using BigFloats at the given precision)
    • arb ≝ mean(execution time using ArbFloats at the given precision)
    • Relative Speed = Speedup + 1 = round( abs(arb-big) / arb ) + 1

ArbFloat operations performed during one BigFloat operation

precisions
256 bits 1024 bits 2048 bits 3000 bits
function
add 1 3 2 2
multiply 2 4 3 4
divide 3 8 35 60
sine 10 12 10 12
arctangent 16 18 64 45
exponential 18 68 20 24
logarithm 25 68 140 200
Riemann zeta 40 100 24 28
Overview

This package is a faster alternative to BigFloats when working with significands
that do not exceed ~3,500 bits (~1050 digits).

The base C library implements floating point intervals and operations thereupon
which are guaranteed to produce results that enclose the theoretical math value.
While not the package focus, full access to interval-based functions is present.

ArbFloats provides more performant extended precision floating point math
and will show results as accurately as possible by using a precision that
does not misrepresent the information content of the underlying interval.

Version 0.1.00 released 2016-Sep-15.
Version 0.1.14 released 2016-Dec-23.

Install

Pkg.update()
Pkg.add("Nemo") # for Win, if Nemo was not already present, follow this with Pkg.build("Nemo")
Pkg.add("ArbFloats")

If you have not installed Nemo before, you will see compilation notes and maybe warnings.
Ignore them. This is a good time to walk the dog, go for coffee, or play shuffleboard.
When the prompt comes back, quit() and restart Julia and julia> using ArbFloats
should precompile quickly and work well. This is what I do, to get things set up:

Pkg.update()
# get current Nemo, if you have an old version of Nemo, do
# Pkg.rm("Nemo"); Pkg.rm("Nemo");
Pkg.add("Nemo")
quit()
# get current ArbFloats, if you have an older realization do
# Pkg.rm("ArbFloats");Pkg.rm("ArbFloats");
Pkg.add("ArbFloats")
Pkg.update()
using ArbFloats
quit()
using ArbFloats
quit()

It is helpful to add Nemo first, quit, then add ArbFloats and quit.

Initializing ArbFloats

ArbFloats can be initialized from Integers, Floats, Rationals, and Strings

using ArbFloats

precision(ArbFloat) # show the current default precision
# 116
setprecision(ArbFloat, 200) # change the current default precision
# 200
typealias ArbFloat{200} Arb200 # A Good Idea, and shaves cycles in use

a = ArbFloat(12);          # use the default precision, at run time
b = @ArbFloat(12);         # use the default precision, at compile time
c = ArbFloat{200}(golden); # use specified precision, at run time
d = @ArbFloat(200,golden); # use specified precision, at compile time
e = Arb200(12);            # use named precision, assuming prior typealias

# setprecision(ArbFloat, 53+0); # akin to setprecision(BigFloat, 53)
# to see elementary function evaluations rounded to (at least) N significand bits, 
#   using setprecision(ArbFloat, N+10) is recommended and at least N+7 is suggested
#   setprecisionAugmented(ArbFloat, N) does the N+10 automatically
#   setprecisionAugmented(ArbFloat, N, d) uses N+d for the precision

#==
          remember to do this        and           to avoid this
==#
    goodValue = @ArbFloat(1.2345);        wrongValue = ArbFloat(1.2345);
#       1.234500000000000000                   1.2344999999999999307
    ArbFloat(12345)/ArbFloat(1000);       ArbFloat(12.345)/ArbFloat(10)
#       1.234500000000000000                   1.234500000000000064

@ArbFloat(1.2345) == ArbFloat("1.2345")

Use

using ArbFloats

setprecision(ArbFloat, 80)

exp1 = exp(ArbFloat(1))
# 2.71828182845904523536029..

stringsmall(exp1),stringcompact(exp1),string(exp1),stringall(exp1)
("2.7182818","2.71828182845905","2.71828182845904523536029","2.71828182845904523536029")
showall_pm(exp1)
# 2.718281828459045235360286±3.3216471534462276e-24
bounds(exp1)
# ( 2.71828182845904523536028,  2.718281828459045235360293 )

setprecision(ArbFloat, 116); # the initial default precision

fuzzed_e = tan(atanh(tanh(atan(exp(one(ArbFloat))))))
# 2.718281828459045235360287
showall(fuzzed_e)
# 2.7182818284590452353602874713527

bounds(fuzzed_e)
# ( 2.718281828459045235360287,
#   2.718281828459045235360287 )
# they are not really the same ...    
lo, hi = bounds(fuzzed_e);
showall(lo,hi)
# ( 2.7182818284590452353602874713526543,
    2.7182818284590452353602874713526701 )

# use values of the same precision with interval operators

precision(exp1), precision(fuzzed_e)
# 80, 116
overlap(exp1, fuzzed_e), contains(fuzzed_e, exp1), iscontainedby(exp1, fuzzed_e)
# ( true. false, false )
exp1 = exp(ArbFloat(1.0))
precision(exp1), precision(fuzzed_e)
# (116, 116)
overlap(exp1, fuzzed_e), contains(fuzzed_e, exp1), iscontainedby(exp1, fuzzed_e)
# ( true. true, true )


smartstring(exp1)
# "2.71828182845904523536028747135266+"
smartstring(fuzzed_e)
# "2.7182818284590452353602874713527-"

Float32 and ArbFloat32

typealias ArbFloat32 ArbFloat{24} # Float32 has 24 significand bits
setprecision(ArbFloat, 24) # it is good to keep precisions in concert

fpOneThird = 1.0f0 / 3.0f0
# 0.3333334f0

oneThird = ArbFloat32(1) / ArbFloat32(3)
# 0.3333333..s
show_pm(oneThird)
# 0.33333331±2.98023223877e-8


# gamma(1/3) is 2.6789_3853_4707_7476_3365_5692_9409_7467_7644~
gamma( fpOneThird )
# 2.6789_384f0

gamma_oneThird = gamma( oneThird )
# 2.6789_4..
bounds(gamma_oneThird)
# (2.6789_362, 2.6789_401)
showsmall(gamma_oneThird)
# 2.67894

Display


# e.g. stringsmall & showsmall, stringsmall_pm & showsmall_pm
# {string,show}{small, compact, all, small_pm, compact_pm, all_pm}
stringsmall(oneThird), stringsmall_pm(oneThird)
("0.3333333",  "0.33333331±2.98e-8")

# show works with vectors and tuples and varargs of ArbFloat
showsmall([oneThird, oneThird]);showsmall((oneThird,oneThird));showsmall(oneThird,oneThird)
# [ 0.3333333,      ( 0.3333333,      ( 0.3333333,
#   0.3333333 ]       0.3333333 )       0.3333333 )


ArbFloat("Inf"), ArbFloat("-Inf"), ArbFloat("NaN")
# +Inf, -Inf, NaN
one(ArbFloat)/ArbFloat(Inf), ArbFloat("Inf")+ArbFloat("-Inf")
# 0, NaN

showmart(exp1)
# 2.71828182845904523536028747135266+
showsmart(fuzzed_e)
# 2.7182818284590452353602874713527-

pi66bits=ArbFloat{66}(pi)
# 3.141592653589793238
showpretty(ArbFloat{66}(pi))
# 3.141_592_653_589_793_238

pi67bits=ArbFloat{67}(pi)
# 3.1415926535897932385
showpretty(ArbFloat{67}(pi),5)
# 3.14159_26535_89793_2385

Non-Strict Total Ordering

thinner = midpoint_radius( 1000.0, 1.0);
thicker = midpoint_radius( "1000.0", "2.0");

thicker≻ thinner, thinner  ⪯  thicker, succ(thicker, thinner),
# (true, true, true)
thicker  ⪯  thinner, thinner ≻  thicker, preceq(thicker, thinner)
# (false, false, false)
succ(thicker, thinner), succ(thinner, thicker)
# false, true

Compatible Packages

using ArbFloats # goes anywhere
DifferentialEquations, DualNumbers, ForwardDiff, HyperDualNumbers, MappedArrays,
Plots, Polynomials, Quaternions, others

using ArbFloats # goes last!
TaylorSeries

partially compatible
Roots (accepts ArbFloats, results are Float64)

If you have a package that accepts AbstractFloats or Reals and does not “just work”
with ArbFloats, please note it as an issue. If you have a package that works well
with ArbFloats, do let us know.

More Information

Please the notes directory for more information about ArbFloats.

Hewing to the sensible

Arb is happiest, and performs most admirably using intervals where the radius is
a very small portion of the working precision. Ideally, the radius is kept within
8*eps(midpoint). With Arb, you are likely ok up to twice that. And should your
approach generate overly wide intervals, then it is worth trying an algorithm
that is designed to visit each extended precision variable less often. (prefer projection techniques to recursively applicative transforms), perhaps run
at higher working precision, is worth trying. A toy version is likely to behave
in the same manner as your the more refined software. It is worth the look.

The intervals underlying this package are kept by Arb as an extended precision
midpoint and a radius (halfwidth) as a float of low precision & high range.
The radius is stored as a 30 bit significand and a ~60 bit exponent. The radius
is like a Float32 (24bit significand) value with a much larger exponent.

Warp and Weft

One way of think of these midpoint+radius intervals is as cereal and milk.
The cereal sources nourishment and the milk makes it easy to digest.
The midpoint associates as a valuation, and the radius engages as a capacity-
limiting store of value. The more extensive the radius, the more spread out,
dilute is any value stored. Value concentrates as the midpoint magnitude
increases relative to the radius.

Another is to use the pairing of midpoint with its immediate locale (diameter)
as a semantic descriptor and quantify the semantics. The veridical presentment
of floating point quantities is one of the primary motivators for this package.
And there is software which moves from two floats, midpoint+radius, through
the active preternatural simplicty of most informing whilst least misleading,
into the floating point value that best reflects the crispness of its novelty.

Rough Spots

This package does whatever it may through the Arb C library. On rare occasion,
this may give a result which makes Arb sense yet appears counter-intuitive here.
One example is Arb's ability to work with and to return projective infinity (±Inf).
This package now does now provide a means of working with Arb's complex intervals,
nor is their access to any of Arb's matrix routines (det, inv, lu, maybe charpoly).

ArbFloats do not lend themselves easily to higher matrix algebra (svd, eigenvals).
If someone implements one of the known good algorithms for getting the eigenvalues
or the svd of a matrix with interval-valued entries, this package is at the ready.

We use some of Nemo's libraries. Nemo is very large, and this work needs less than 1/8th of it.

About Arb and using Nemo's libraries

This work is constructed atop a state-of-the-art C library for working with
midpoint ± radius intervals, Arb. Arb is designed and written by Fredrik
Johansson, who graciously allows Julia to use it under the MIT License.

The C libraries that this package accesses are some of the shared libraries that
Nemo.jl requires and builds; and, with permission, I call them directly.

It is a useful fiction to think of ArbFloats as Arb values with a zero radius
– and sometimes they are. When an ArbFloat has a nonzero radius, the user sees
only those digits that remain after rounding the ArbFloat to subsume the radius.

Appropriateness

This package is appropriate to use for extending the precision of floating point
computations from 64 bits [~17 digits] up to 3,250 bits [~1000 digits].
While Testing on many different hosts is needed to characterize a most performant
precision range, I have found working with 800 bits (~240 digits) a welcome change.

Conceptual Background

Transparency: a desirable quality that may obtain in the presentation of
numerical quantity. Where transparency exists, it may well not persist.
A diminution of transparency increases opacity, and vice versa. Presentation
of a floating point value either evinces transparency or furthers opacity.
With transparent values, ‘looking at a value’ is ‘looking through to see the
knowable value’. With opaque values, ‘looking at a value’ is ‘looking away from’
that. And it is that nonresponsive, nonparticipative engagement of cognitive
attention that is the opaqueness underlying opacity.

Presented with a transparent floating point value, the perceiver is become
best informed. There is no other rendition of that floating point realization
which is intrinsically more informing and none which relays the value of that
floating point realization more accurately – none with fewer digits, none with
more digits, none of greater magnitude, none of lesser magnitude.

An ArbFloat is an extended precision float architected to evince transparency.
It informs without leading or misleading. An ArbFloat, when viewed, appears as
an extended precision floating point value. When any of the exported arithmetic,
elementary or special functions is applied to an ArbFloat, the value transforms
as an extended precision floating point interval.

Exports (including re-exports)

used with ArbFloat nature
precision, setprecision as with BigFloat
Arb values are intervals nature
midpoint, radius, lowerbound, upperbound, bounds, Arb’s constituent parts
isexact, notexact, float-y or interval-y
overlap, donotoverlap, of interval suborder
contains, iscontainedby, doesnotcontain, isnotcontainedby, of interval partial order
ArbFloat attributes nature
isnan, isinf, isfinite, issubnormal, isinteger, notinteger, floatingpoint predicates
iszero, notzero, nonzero, isone, notone, number predicates
ispositive, notpositive, isnegative, notnegative, numerical predicates

copy, deepcopy, zero, one, eps, epsilon, isequal, notequal, isless,
(==), (!=), (<), (<=), (>=), (>), # Arb, strict: a < b iff upperbound(a) < lowerbound(b)
(≃), (≄), (≺), (⪯), (≻), (⪰), # non-strict total ordering (best for convergence tests)
simeq, nsime, prec, preceq, succ, succeq, # names matching binops above
approxeq, ≊, min, max, minmax,

signbit, sign, flipsign, copysign, abs, (+),(-),(*),(/),(),(%),(^), inv,
sqrt, invsqrt, hypot, factorial, doublefactorial, risingfactorial, trunc,
round, ceil, floor, trunc, fld, cld, fmod, modf, integerpart, fractionalpart,

pow, root, exp, expm1, log, log1p, log2, log10, logbase, sin, cos, sincos,
sincospi, tan, csc, sec, cot, asin, acos, atan, atan2, sinh, cosh, sinhcosh,
tanh, csch, sech, coth, asinh, acosh, atanh,

gamma, lgamma, digamma, sinc, zeta, polylog, agm

string, stringsmall, stringlarge, stringall, smartstring, smartvalue, smartmodf, decimalpart, # both use smartvalue(fractionalpart)

Credits, References, Thanks

This work relies on Fredrik Johansson's Arb software, using parts of that extensive C library.
He has been greatly helpful. The Arb library documentation is here.

Much of the early development was well informed from study of Nemo.jl, a number theory and
numerical algebra package that incorporates some of Arb's capabilities along with many others.
William Hart and Tommy Hofmann have been gracious with their work and generous with their time.

=====

Many have helped me. Some with their prior acts of good will.
Others by explaining subtleties, sharing exemplary Julian ways,
suggesting improvements, providing fixes, or doing testing.
The list of names outgrew this space, see this for more.

Please alert me to any issues, miscues or inartful expressions.

If you find something to be an issue for you, submit it as an issue.
If you write something that improves this for others, submit it as a pull request.

First Commit

06/29/2016

Last Touched

3 months ago

Commits

1091 commits

Used By: