The Jargon File (version 4.4.7, 29 Dec 2003):

register dancing
 n.

    Many older processor architectures suffer from a serious shortage of
    general-purpose registers. This is especially a problem for
    compiler-writers, because their generated code needs places to store
    temporaries for things like intermediate values in expression evaluation.
    Some designs with this problem, like the Intel 80x86, do have a handful of
    special-purpose registers that can be pressed into service, providing
    suitable care is taken to avoid unpleasant side effects on the state of the
    processor: while the special-purpose register is being used to hold an
    intermediate value, a delicate minuet is required in which the previous
    value of the register is saved and then restored just before the official
    function (and value) of the special-purpose register is again needed.


The Free On-line Dictionary of Computing (30 December 2018):

register dancing

   Many older processor architectures suffer from a serious
   shortage of general-purpose registers.  This is especially a
   problem for compiler-writers, because their generated code
   needs places to store temporaries for things like intermediate
   values in expression evaluation.  Some designs with this
   problem, like the Intel 80x86, do have a handful of
   special-purpose registers that can be pressed into service,
   providing suitable care is taken to avoid unpleasant side
   effects on the state of the processor: while the
   special-purpose register is being used to hold an intermediate
   value, a delicate minuet is required in which the previous
   value of the register is saved and then restored just before
   the official function (and value) of the special-purpose
   register is again needed.

   [Jargon File]