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calc: Simplification Modes

 
 7.5 Simplification Modes
 ========================
 
 The current “simplification mode” controls how numbers and formulas are
 “normalized” when being taken from or pushed onto the stack.  Some
 normalizations are unavoidable, such as rounding floating-point results
 to the current precision, and reducing fractions to simplest form.
 Others, such as simplifying a formula like ‘a+a’ (or ‘2+3’), are done
 automatically but can be turned off when necessary.
 
    When you press a key like ‘+’ when ‘2’ and ‘3’ are on the stack, Calc
 pops these numbers, normalizes them, creates the formula ‘2+3’,
 normalizes it, and pushes the result.  Of course the standard rules for
 normalizing ‘2+3’ will produce the result ‘5’.
 
    Simplification mode commands consist of the lower-case ‘m’ prefix key
 followed by a shifted letter.
 
    The ‘m O’ (‘calc-no-simplify-mode’) command turns off all optional
 simplifications.  These would leave a formula like ‘2+3’ alone.  In
 fact, nothing except simple numbers are ever affected by normalization
 in this mode.  Explicit simplification commands, such as ‘=’ or ‘a s’,
 can still be given to simplify any formulas.  Algebraic
 Definitions, for a sample use of No-Simplification mode.
 
    The ‘m N’ (‘calc-num-simplify-mode’) command turns off simplification
 of any formulas except those for which all arguments are constants.  For
 example, ‘1+2’ is simplified to ‘3’, and ‘a+(2-2)’ is simplified to
 ‘a+0’ but no further, since one argument of the sum is not a constant.
 Unfortunately, ‘(a+2)-2’ is _not_ simplified because the top-level ‘-’
 operator’s arguments are not both constant numbers (one of them is the
 formula ‘a+2’).  A constant is a number or other numeric object (such as
 a constant error form or modulo form), or a vector all of whose elements
 are constant.
 
    The ‘m I’ (‘calc-basic-simplify-mode’) command does some basic
 simplifications for all formulas.  This includes many easy and fast
 algebraic simplifications such as ‘a+0’ to ‘a’, and ‘a + 2 a’ to ‘3 a’,
 as well as evaluating functions like ‘deriv(x^2, x)’ to ‘2 x’.
 
    The ‘m B’ (‘calc-bin-simplify-mode’) mode applies the basic
 simplifications to a result and then, if the result is an integer, uses
 the ‘b c’ (‘calc-clip’) command to clip the integer according to the
 current binary word size.  Binary Functions.  Real numbers are
 rounded to the nearest integer and then clipped; other kinds of results
 (after the basic simplifications) are left alone.
 
    The ‘m A’ (‘calc-alg-simplify-mode’) mode does standard algebraic
 simplifications.  Algebraic Simplifications.
 
    The ‘m E’ (‘calc-ext-simplify-mode’) mode does “extended”, or
 “unsafe”, algebraic simplification.  Unsafe Simplifications.
 
    The ‘m U’ (‘calc-units-simplify-mode’) mode does units
 simplification.  Simplification of Units.  These include the
 algebraic simplifications, plus variable names which are identifiable as
 unit names (like ‘mm’ for “millimeters”) are simplified with their unit
 definitions in mind.
 
    A common technique is to set the simplification mode down to the
 lowest amount of simplification you will allow to be applied
 automatically, then use manual commands like ‘a s’ and ‘c c’
 (‘calc-clean’) to perform higher types of simplifications on demand.
 

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