`
``
use Math::Symbolic qw/parse_from_string/;
use Math::Symbolic::Custom::ErrorPropagation;
# Force is mass times acceleration.
my $force = parse_from_string(m*a);
# The measurements of the acceleration and the mass are prone to
# statistical errors. (Hence have variances themselves.)
# Thus, the variance in the force is:
my $variance = $force->apply_error_propagation(a, m);
print $variance;
# prints:
# (
# ((sigma_a ^ 2) * ((partial_derivative(m * a, a)) ^ 2)) +
# ((sigma_m ^ 2) * ((partial_derivative(m * a, m)) ^ 2))
# ) ^ 0.5
`

This module extends the functionality of Math::Symbolic by offering
facilities to calculate the propagated variance of a function of
variables with variances themselves.
The module adds a method to all Math::Symbolic objects.

This method does not modify the Math::Symbolic tree itself, but instead
calculates and returns its variance based on its variable dependencies which
are expected to be passed as arguments to this method in form of a list
of variable names.
The variance is returned as a Math::Symbolic tree itself. It is calculated
using the Gaussian error propagation formula for uncorrelated variances:

`
`

`
variance( f(x_1, x_2, ..., x_n ) ) =
sqrt(
sum_over_i=1_to_n(
variance(x_i)^2 * (df/dx_i)^2
)
)
`

In the above formula, the derivatives are partial derivatives and the
component variances `variance(x_i)` are represented as sigma_x_i in the
resulting formula. (The x_i is replaced by the variable name, though.)

Please refer to the SYNOPSIS for an example.