Interpret StackyMath!

Ruby, 412 410 404 392 380 377 characters

def e m,x='Exception';warn m+x+?!*3;exit;end
def o;e'StackUnderflow'if$*==[];$*.pop;end
u=->n{e'DivisionByZero'if n.infinite?;e'NumberOverflow'if n>2**64;$*<<n}
f=->n{e'NegativeFactorial'if n<0;e'FloatingFactorial'if n.to_i<n;n<2?1:f[n-1]*n}
gets.gsub(/(\d+)|([+*\/%^-])|(!)|D/){$1?u[$1.to_f]:$2?u[eval"o#{$2>?A?:**:$2}o"]:$3?u[f[o]]:u[x=o]+u[x]}
e'EmptyProgram',''if$*==[]
p o

This is regular precision version using Float. The result precision is as in the sample code, but numeric overflow detection is not exact.

Sample run:

bash-4.3$ ruby StackyMath.rb <<< '654,489,48,43/5*7D+-*%'
77.68749999999909

Ruby, 378 377 characters

def e m,x='Exception';warn m+x+?!*3;exit;end
def o;e'StackUnderflow'if$*==[];$*.pop;end
u=->n{e'NumberOverflow'if n>2**64;$*<<n}
f=->n{e'NegativeFactorial'if n<0;e'FloatingFactorial'if n.to_i<n;n<2?1:f[n-1]*n}
gets.gsub(/(\d+)|([+*\/%^-])|(!)|D/){$1?u[Rational$1]:$2?u[eval"o#{$2>?A?:**:$2}o"]:$3?u[f[o]]:u[x=o]+u[x]}rescue e'DivisionByZero'
e'EmptyProgram',''if$*==[]
p o.to_f

This is high precision version using Rational. The result precision is not always the same as in the sample code, but numeric overflow detection is exact.

Sample run:

bash-4.3$ ruby StackyMath-hi.rb <<< '654,489,48,43/5*7D+-*%'
77.6875

JavaScript (ES6), 430 bytes

422 bytes with ES7 by changing Math.pow(2,2) to 2**2

e=m=>{throw alert(m)};u=prompt();u?alert(eval('u.match(/\\d+|[^,]/g).map(o=>s.push(t=o=="/"?(b=p(a=2))?a/b:e`DivisionByZero43*"?2*23-"?2-23+"?2+23%"?2%23^"?Math.pow(2,2)3D"?s.push(r=2)&&r3!"?eval("for(r=i=2;i<0?e`Negative54:i%1?e`Floating54:--i;)r*=i;r"):+o)&&t==Infinity&&e`NumberOverflow4,s=[],p=_=>s.length?s.pop():e`StackUnderflow4);t'.replace(/[2-5]/g,x=>[,,'p()',':o=="','Exception!!!`','Factorial'][x]))):e`EmptyProgram!!!`

Explanation

Uses eval to replace certain common phrases. Ungolfed and without the eval it looks like this:

e=m=>{throw alert(m)};                           // e = throw error, alert displays
                                                 //     message, throw stops execution
u=prompt();                                      // u = received input
u?alert(                                         // display the result
  u.match(/\d+|[^,]/g)                           // get array of numbers and operators
    .map(o=>                                     // iterate over operators
      s.push(t=                                  // t = last pushed value

        // Execute operator
        o=="/"?(b=p(a=p()))?a/b:                 // make sure argument B is not 0
          e`DivisionByZeroException!!!`:
        o=="*"?p()*p():
        o=="-"?p()-p():
        o=="+"?p()+p():
        o=="%"?p()%p():
        o=="^"?Math.pow(p(),p()):
        o=="D"?s.push(r=p())&&r:
        o=="!"?eval("                            // eval to enable for loop in ternary
          for(                                   // no factorial in JS so do this manually
            r=i=p();
            i<0?e`NegativeFactorialException!!!` // check for errors
              :i%1?
                e`FloatingFactorialException!!!`
                :--i;
          )
            r*=i;
          r"):                                   // return r
        +o                                       // if not an operator cast as a number
      )&&t==Infinity&&                           // JS turns anything over 64 bit float
        e`NumberOverflowException!!!`,           //     max value into Infinity
      s=[],                                      // s = stack array
      p=_=>s.length?s.pop():                     // p = check stack then pop
        e`StackUnderflowException!!!`
    )&&t                                         // return top stack element
  ):e`EmptyProgram!!!`                           // error if input length is 0