Computational noise can corrupt NaN-safe computations

Different amounts of computational noise in the value and partials of a dual number has the potential to corrupt even NaN-safe computations.

Minimum working example:

using ForwardDiff, LinearAlgebra

# start with zero valued vector of dual numbers
v = zeros(ForwardDiff.Dual{Nothing, Float64, 1}, 3);

# assume a perturbation of one component exists due to some computational noise
value = 1.0e-200 # so that value^2 == 0.0 (due to machine precision)
partial = 1.0e-100 # so that 2*value*partial != 0.0 (due to machine precision)
v[1] = ForwardDiff.Dual{Nothing}(value, partial);

# try out two functions
norm(v)
# Dual{Nothing}(1.0e-200,NaN)
sqrt(sum(v.^2))
# Dual{Nothing}(0.0,Inf)

Both implementations will result in NaNs propagated throughout the function, even in NaN-safe mode. I first encountered this after propagating derivatives through a Newton solve.

Originally posted by @taylormcd in https://github.com/JuliaDiff/ForwardDiff.jl/issues/243#issuecomment-539291591

Reproduce the minimum working example using ForwardDiff.Dual, norm(v), and sqrt(sum(v.^2)) as shown in the issue. Compare the NaN-safe behavior of both computations under the stated value and partial inputs, then determine a project-consistent expected result; done means the demonstrated noise case no longer propagates NaN or Inf unexpectedly.