In the following examples, various functions are written in different algebraic forms and the results for the different forms is shown to be exactly same (e.g. \cos(x) vs. \sqrt{1-\sin^2(x)}, \tan(x) vs. \sin(x)/\cos(x)). These examples also verify that the combination of a function and its inverse simply returns the argument (e.g \arcsin(\sin(x))=x), as well as functions like \sinh(x)/((\exp(x)-\exp(-x))/2) (which is really a complicated way of writing 1!) returns a value of 1 with no error. However, if the values of two independent variates x_1 and x_2 and their corresponding errors are same, the value of expressions like \sin^2(x_1) + \cos^2(x_2) will be 1 but the error will not be zero.
Value of x = 1.00000 +/- 0.10000 Value of y = 2.00000 +/- 0.20000 Value of x1 = 1.00000 +/- 0.10000 Value of x2 = 1.00000 +/- 0.10000 sin(x) = 0.84147 +/- 0.05403 sqrt(1-sin(x)^2) = 0.54030 +/- 0.08415 cos(x) = 0.54030 +/- 0.08415 tan(x) = 1.55741 +/- 0.34255 sin(x)/cos(x) = 1.55741 +/- 0.34255 sinh(x) = 1.17520 +/- 0.15431 (exp(x)-exp(-x))/2 = 1.17520 +/- 0.15431 sin(x1)*sin(x1) = 0.70807 +/- 0.09093 sin(x1)*sin(x2) = 0.70807 +/- 0.06430 /* Expressions that evaluate to just x */ asin(sin(x)) = 1.00000 +/- 0.10000 asinh(sinh(x)) = 1.00000 +/- 0.10000 atanh(tanh(x)) = 1.00000 +/- 0.10000 exp(ln(x)) = 1.00000 +/- 0.10000 /* Complicated ways of computing 1.0! */ sinh(x)/((exp(x)-exp(-x))/2) = 1.00000 x/exp(ln(x)) = 1.00000 /* Complicated ways of computing 1.0 with single and multiple variates! */ sin(x1)^2+cos(x1)^2 = 1.00000 +/- 0.00000 sin(x1)^2+cos(x2)^2 = 1.00000 +/- 0.12859