As a simple example of the transformational approach, consider writing a
program to solve the quadratic equation, 
.
Mathematically, it is most natural to write the solutions as:
From this mathematical specification, we can produce two different types of solvers: numerical solvers where the coefficients of the polynomial are expressed as floating point numbers and semi-numerical solvers for use when b and c are expressed exactly or in terms of symbolic expressions.
In the first case, we know that direct use of (1)
is unacceptable for three reasons. First, the computation of 
can
overflow even when the value of 
and 
can be
expressed as floating point numbers. To deal with this
(1) should be coded as:
Second, if 
is small then the subtraction case above
will involve subtracting two nearly equal quantities. Thus, a better
way to compute these two roots is
If the computer's arithmetic does not deal with underflow gracefully
then an additional modification may be needed. Third, if 
is
negative, then the square root will return a complex number which must
be dealt with properly.
For each of the problems that arose, a suitable transformation could be defined to convert the original formula into one that does not have the indicated problem. Furthermore, these transforms could be reused to deal with numeric errors that arise in other calculations. These transformation, however, cannot be easily encoded in a subroutine library.
In the exact case, the formula (1) is satisfactory, however the square root must be dealt with carefully to avoid running afoul of branch cut problems.
From the perspective of creating a comprehensive problem solving
environment, the transition from the numeric
solvers to semi-numeric solvers needs to be seamless, since, for
instance, the mathematician rarely feels that introducing into a
computation a small (symbolic) 
increases the difficulty
significantly. By encouraging the development of mathematical programs
via program transformations and by providing a library of transforms,
like those discussed above, we expect to ameliorate the problem.
