# ArbFloats.jl

## ArbNumerics.jl is the successor to this package

#### Copyright © 2016-2019 by Jeffrey Sarnoff. Released under the MIT License.

### 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> add ArbFloats
pkg> build ArbFloats # if you did not see it build
pkg> precompile
```

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
const Arb200 = 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 type const
# 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

```
const 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, TaylorSeries, others

*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.