Around

A number x with an uncertainty dx, propagated to first order through arithmetic and functions -- the same uncertainty-arithmetic layer as Interval and CenteredInterval. Provided by @enumeratio/analytic, though the head's natural home is @enumeratio/statistics.

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Around(x, dx)x with uncertainty dx, propagated as Around(f(x), |f′(x)|·dx) through a function f.

Typed signatures

Type	Package	Applies
`(number, real?) -> number`	`analytic`	always

Domain: Interval arithmetic

Details

Add sums several independent uncertainties in quadrature (√Σdxᵢ²); a scalar Multiply scales the uncertainty linearly; several Around factors multiplied together combine their RELATIVE uncertainties in quadrature (the same rule, since d(∏xᵢ) = Σⱼ(∏_{i≠j}xᵢ)dxⱼ in quadrature, divided back out by the product); Power with a concrete exponent and Exp (Wolfram's Exp, canonicalized to Power(E, ·)) use their own closed-form derivatives.
Every other function -- the elementary heads, Log in any base, and the special functions -- takes f′ from compute-engine's symbolic D where it resolves, and from a central difference where it doesn't.
A head is propagated through as ONE function of its uncertain argument, before its own definition expands it: Multinomial(Around(2, 0.01), 2) is (a+2)(a+1)/2 at a = 2 ± 0.01, so 6 ± 0.035. Expanding first would count the same uncertainty in a numerator and a denominator as if they were independent.

Plot

Examples

2 more kept as data, checked against the oracles

Basic #

Independent uncertainties add in quadrature

In

\mathrm{Around}(1, 0.1)+\mathrm{Around}(2, 0.2)

Out

\mathrm{Around}(3, 0.223\,606\,797\,749\,979)

Scaling scales the uncertainty

In

3\mathrm{Around}(2, 0.1)

Out

\mathrm{Around}(6, 0.3)

Independent uncertainties add in quadrature

In

\mathrm{Around}(2, 0.01)+\mathrm{Around}(3, 0.02)

Out

\mathrm{Around}(5, 0.022\,360\,679\,774\,997\,897)

In

3\mathrm{Around}(2, 0.01)

Out

\mathrm{Around}(6, 0.03)

Independent uncertainties add in quadrature: $\sqrt{0.1^2 + 0.2^2}$

In

\mathrm{Around}(5, 0.1)+\mathrm{Around}(3, 0.2)

Out

\mathrm{Around}(8, 0.223\,606\,797\,749\,979)

Relative uncertainties combine: $6\sqrt{(0.1/2)^2 + (0.2/3)^2} = 0.5$

In

\mathrm{Around}(2, 0.1)\mathrm{Around}(3, 0.2)

Out

\mathrm{Around}(6, 0.5)

Scope #

Through a function: $|f'(x)|\,\delta x$

In

\mathrm{Around}(2, 0.1)^2

Out

\mathrm{Around}(4, 0.4)

In

\sqrt{\mathrm{Around}(4, 0.4)}

Out

\mathrm{Around}(2, 0.1)

Through a function, to first order: $f'(2) = 3.5$ , so $\mathrm{Multinomial}(2 \pm 0.01, 2) = 6 \pm 0.035$

In

\mathrm{Multinomial}(\mathrm{Around}(2, 0.01), 2)

Out

\mathrm{Around}(6, 0.035)

A power scales the uncertainty by the derivative, $2 \cdot 3 \cdot 0.1$

In

\mathrm{Around}(3, 0.1)^2

Out

\mathrm{Around}(9, 0.600\,000\,000\,000\,000\,1)

$\frac{0.2}{2\sqrt4}$

In

\sqrt{\mathrm{Around}(4, 0.2)}

Out

\mathrm{Around}(2, 0.05)

An exact factor scales the uncertainty

In

10\mathrm{Around}(1.5, 0.02)

Out

\mathrm{Around}(15, 0.2)

Through an elementary function: $\delta \cdot e^2$

In

e^{\mathrm{Around}(2, 0.01)}

Out

\mathrm{Around}(7.389\,056\,098\,930\,65, 0.073\,890\,560\,989\,306\,5)

Through a special function, with its derivative as slope

In

\operatorname{erf}(\mathrm{Around}(2, 0.01))

Out

\mathrm{Around}(0.995\,322\,265\,018\,952\,7, 0.000\,206\,669\,853\,540\,920\,54)

First-order propagation: the uncertainty scales by $|\cos 1|$

In

N(\sin(\mathrm{Around}(1, 0.1)))

Out

\mathrm{Around}(0.841\,470\,984\,807\,896\,5, 0.054\,030\,230\,586\,813\,98)

In

\ln(\mathrm{Around}(2, 0.01))

Out

\mathrm{Around}(0.693\,147\,180\,559\,945\,3, 0.005)

In

\mathrm{Around}(21^{21}, 19^{19})

Out

\mathrm{Around}(5\,842\,587\,018\,385\,982\,521\,381\,124\,421, 1\,978\,419\,655\,660\,313\,589\,123\,979)

Properties #

Squaring an uncertain value: the error doubles with the value, $2\cdot10\cdot1 = 20$

In

\mathrm{Around}(10, 1)^2

Out

\mathrm{Around}(100, 20)

About

Examples	20
Checked against	Wolfram Language 17 agree3 differ
Crosswalk	Wolfram Language

Implementation

mappedsympypackages/oracle/src/mappings.ts

SymPy has no value with an uncertainty, nor first-order propagation through arithmetic (that is the separate uncertainties package, not SymPy), so there is nothing for an Around row to run on; sympy.stats is probabilistic, not error propagation.

See also: Interval, CenteredInterval