Define @Uncertain macro to calculate uncertainty for any function

Limitation: function must take only one real argument.

The derivative is calculated using Calculus package.  We need numerical
derivatives because we want to support any possible function, not only
those implemented in Julia, so I don’t think automatic differentiation
is an option in this case.

Fixes issue #4.

REQUIRE

@@ -1 +1,2 @@
 julia 0.4
+Calculus

src/Measurements.jl

@@ -19,11 +19,14 @@
 
 module Measurements
 
+# This is used to calculate numerical derivatives in "@uncertain" macro.
+using Calculus
+
 # Functions to handle new type
 import Base: show, convert, promote_rule, float
 
 # Functions provided by this package and exposed to users
-export Measurement, ±, stdscore, weightedmean
+export Measurement, ±, stdscore, weightedmean, @uncertain
 
 # Define the new type
 immutable Measurement{T<:AbstractFloat} <: AbstractFloat
@@ -62,6 +65,21 @@ function show(io::IO, measure::Measurement)
     print(io, measure.val, " ± ", measure.err)
 end
 
+# @uncertain macro
+"""
+    @uncertain f(value ± stddev)
+
+A macro to calculate \$f(value) ± uncertainty\$, with \$uncertainty\$ derived
+from \$stddev\$ according to rules of linear error propagation theory.  Function
+\$f\$ must accept only one real argument, the type of the argument provided must
+be `Measurement`.
+"""
+macro uncertain(expr::Expr)
+    f = esc(expr.args[1]) # Function name
+    a = esc(expr.args[2]) # Argument, of Measurement type
+    return :( Measurement($f($a.val), abs(Calculus.derivative($f, $a.val)*$a.err)) )
+end
+
 # Standard Score
 """
     stdscore(measure::Measurement, expected_value::Real) -> standard_score

test/runtests.jl

@@ -9,6 +9,8 @@ test_approx_eq{T1<:Real,T2<:Real}(a::Complex{Measurement{T1}}, b::Measurement{T2
     test_approx_eq(a, complex(b))
 test_approx_eq{T1<:Real,T2<:Real}(a::Measurement{T1}, b::Complex{Measurement{T2}}) =
     test_approx_eq(complex(a), b)
+test_approx_eq_eps(a::Measurement, b::Measurement, tol::Real) =
+    (@test_approx_eq_eps(a.val, b.val, tol) ; @test_approx_eq_eps(a.err, b.err, tol))
 
 w = -0.5 ± 0.03
 x = 3 ± 0.1
@@ -320,3 +322,9 @@ test_approx_eq(mean((w, x, y)), (w + x + y)/3)
 
 # sort
 @test sort([y, w, x]) == [w, x, y]
+
+##### Test @uncertain macro
+test_approx_eq_eps(@uncertain(tan(x)), tan(x), 2e-11)
+test_approx_eq_eps(@uncertain((a -> a + a + a)(x)), 3x, 3e-12)
+test_approx_eq(@uncertain(zeta(x)),
+               Measurement(1.2020569031595951, 0.019812624290876782))