Generally when I'm doing symbolic math, it's as a supplement to computation, not as a replacement. In such cases, speed doesn't matter so much.
To take a specific example, think about solving a set of differential equations. The Sundials suite provides an excellent, fast numerical implementation of the popular BDF algorithm. So why would you ever want to do a slower symbolic manipulation of the problem?
First, to get insight into how the BDF solver works when applied to a class of problems, you might express the problem symbolically, apply the algorithm symbolically, and play around with the resulting expressions. This might help you choose tolerances, decide how to express your problem (e.g., to avoid catastrophic roundoff errors), or design a new algorithm geared to your particular problem.
Second, you might use symbolic algebra to derive the Jacobian matrix of derivatives for your problem, then code that Jacobian up for Sundials to use during numerical solution. This would speed up the numerics by giving it a faster, more accurate way to find the Jacobian than doing so numerically.
In both cases, symbolic algebra is not used to find a particular numerical result, but to improve the use of numerical code for solving particular problems.
To take a specific example, think about solving a set of differential equations. The Sundials suite provides an excellent, fast numerical implementation of the popular BDF algorithm. So why would you ever want to do a slower symbolic manipulation of the problem?
First, to get insight into how the BDF solver works when applied to a class of problems, you might express the problem symbolically, apply the algorithm symbolically, and play around with the resulting expressions. This might help you choose tolerances, decide how to express your problem (e.g., to avoid catastrophic roundoff errors), or design a new algorithm geared to your particular problem.
Second, you might use symbolic algebra to derive the Jacobian matrix of derivatives for your problem, then code that Jacobian up for Sundials to use during numerical solution. This would speed up the numerics by giving it a faster, more accurate way to find the Jacobian than doing so numerically.
In both cases, symbolic algebra is not used to find a particular numerical result, but to improve the use of numerical code for solving particular problems.