How to retrieve the first decimal digit of number efficiently

Question

One obvious solution is:

int n = 2134;
while(n > 9)
    n /= 10;

which takes linear time. Could we do any faster?

Is this any faster than linear time:

char s[100];
sprintf(s, "%d", n);
n = s[0]-'0';

Which are the other ways (efficiency is primary concern)?
I've seen this, except that I need to find only the first digit. (Also, I don't understand the answer).

Answer 1

Some processors have instructions that calculate "how big" a number is, very quickly (see http://en.wikipedia.org/wiki/Leading_zero_count). This can be used to quickly choose a power of 10, and divide by it, instead of dividing by 10 repeatedly.

Suppose you are given a function clz that calculates the number of leading zero bits in a number's binary representation (0...32). Then, you can use a lookup table that gives the proper power of 10 for each number of leading zeros.

uint32_t powers_of_10[33] = {
    1000000000, 1000000000,
    100000000, 100000000, 100000000,
    10000000, 10000000, 10000000,
    1000000, 1000000, 1000000, 1000000,
    100000, 100000, 100000,
    10000, 10000, 10000,
    1000, 1000, 1000, 1000,
    100, 100, 100,
    10, 10, 10,
    1, 1, 1, 1, 1
};

int CalcFirstDecimalDigit(uint32_t x)
{
    int leading_zeros = clz(x);
    x /= powers_of_10[leading_zeros];
    if (x >= 10)
        return 1;
    else
        return x;
}

Answer 2

e.g. for in 32 bit unsigned:

Step 1: determine (by binary search) in which of the following intervals the value is:

0 .. 9
10 .. 99
100 .. 999
1000 .. 9999
10000 .. 99999
100000 .. 999999
1000000 .. 9999999
10000000 .. 99999999
100000000 .. 999999999
1000000000 .. 4294967295

takes max 4 compares

Step 2:

Than calculate the leading digit by one division.