Please use the current repository
Jeffrey Sarnoff © 2016˗May˗26 in New York City#####This is for Julia v0.5.
####Appropriateness
Preferred 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].
####Conceptual Background
Transparency: a desireable quality that may obtain in the presentation of numerical quantity. Where transparency exists, it may not persist. A diminution of transparency increases opacity, and vice versa. Presentation of a floating point value either evinces transparency or furthers opacity. With the transparent, Looking at is looking through. With the opaque looking at is neither looking through nor looking into nor gazing upon -- it is the remaining sort of "to be looking", looking away. 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, looks just like an extended precision floating point value; yet when mapped through mathematical functions, transforms as an extended precision floating point interval. Their confluence clarifys what distinguishes obtained value from calculated value.
###About Arb
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.
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 don`t care:
the digits which remain after rounding the ArbFloat so that the radius is subsumed (as if 0.0).
####Install
Pkg.clone("https://github.com/Jeffrey-Sarnoff/ArbFloats.jl") # requires a recent Julia v0.5.0-dev####Use
using 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