Jeffrey Sarnoff © 2016˗May˗26 in New York CityPreferred for extending the precision of floating point computations from 64 bits [17 digits] up to 512 bits [150 digits].
Recommended for use where elementary or special functions are evaluated to obtain results with up to 250 digits [800 bits].
Transparency: a desireable 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.
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 when it is installed; and I am calling them directly. Nemo is a computational environment where the most important software for number theory and related work. Julia is used to create a cohesive whole that shares a manner of use. Fredrik Johansson, William Hart, and Tommy Hoffman have been especially helpful, taking the time to explain details of Arb as I was working on ArbFloats.
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 _dont care_: the digits which remain after rounding the ArbFloat` so that the radius is subsumed (as if 0.0).
Pkg.add("Nemo")
Pkg.build("Nemo") # this takes a while
quit()Pkg.clone("https://github.com/jsarnoff-juliacon/ReadableNumbers.jl") # requires a recent Julia v0.5.0-dev
Pkg.clone("https://github.com/jsarnoff-juliacon/ArbFloats.jl") # requires a recent Julia v0.5.0-dev
quit()Pkg.test("ArbFloats") # should work without errorusing ArbFloats
five = ArbFloat(5)
5
e = exp(ArbFloat(1))
2.7182_8182_8459_0452_3536_0287_4713_5266_2 ± 4.8148250e-35
fuzzed_e = tan(atanh(tanh(atan(e))))
2.7182_8182_8459_0452_3536_0287_4713_52662 ± 7.8836806e-33
bounds(e)
( 2.7182_8182_8459_0452_3536_0287_4713_52663,
2.7182_8182_8459_0452_3536_0287_4713_52664 )
smartstring(e)
2.7182_8182_8459_0452_3536_0287_4713_5266₊
bounds(fuzzed_e)
( 2.7182_8182_8459_0452_3536_0287_4713_52654,
2.7182_8182_8459_0452_3536_0287_4713_52670 )
smartstring(fuzzed_e)
2.7182_8182_8459_0452_3536_0287_4713_527₋
# Float32 and ArbFloat32
# typealias ArbFloat32 ArbFloat{24} # predefined
setprecision(ArbFloat, 24)
fpOneThird = 1.0f0 / 3.0f0
0.3333_334f0
oneThird = ArbFloat(1) / ArbFloat(3)
0.3333_333 ± 2.9803_322e-8
# gamma(1/3) is 2.6789_3853_4707_7476_3365_5692_9409_7467_7644~
gamma_oneThird = gamma( oneThird )
2.6789_380 ± 1.8211887e-6
bounds(gamma_oneThird)
(2.6789_360, 2.6789_400)
gamma( fpOneThird )
2.6789_384f0| used with Arb and 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 values: Arb seen as precisely accurate floats
elevates transparant information over number mumble
each digit shown is an accurate refinement of value
The least significant digit observable, through show(af) or with string(af),
is smallest transparent _(intrisically non-misleading)_ refinement of value.
| 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, (==),(!=),(<),(<=),(>=),(>), approxeq, ≊,
min, max, minmax,
signbit, sign, flipsign, copysign, abs,
(+),(-),(*),(/),(),(%),(^), inv, sqrt, invsqrt, hypot,
factorial, doublefactorial, risingfactorial, trunc, round, ceil, floor,
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