C, 123 bytes

float d(char*s){int n=*s++,m=(n*12+(n<67?90:6))/7,o=*s++,a=o^35?o^98?0:-1:1;return exp((m+(a?*s++:o)*12+a)/17.3123-37.12);}

Usage:

#include <stdio.h>
#include <math.h>

float d(char*s){int n=*s++,m=(n*12+(n<67?90:6))/7,o=*s++,a=o^35?o^98?0:-1:1;return exp((m+(a?*s++:o)*12+a)/17.3123-37.12);}

int main()
{
    printf("%f\n", d("A4"));
}

The calculated value is always about 0.8 cents less than the exact value, because I cut as many digits as possible from the floating-point numbers.

Overview of the code:

float d(char*s){
{
    int n=*s++,        // read the letter
        m=(n*12+       // multiply by 12/7 to convert from A...G to 0...11
        (n<67?90:6)    // if A or B, add 1 octave; also add some fix-up rounding value
        )/7,

        o=*s++,        // read next char: the octave digit or accidental

        a=o^35?o^98?0:-1:1; // if accidental, convert it into +1 or -1; else 0

        return exp((m+ // I adjusted the factors to use exp instead of pow
            (a?*s++:o) // if was accidental, now read the octave digit
            *12+a)/
            17.3123-   // a more exact value is 17.3123404447
            37.12);    // a more exact value is 37.1193996632
}