Track first digit during a long multiplication in c

Question

I have an array of integer element range up to 10^5, and I have to find the first element after the total multiplication.

Example:

Array : 2,4,6,7 
multiplication result: 336 and the first element is 3.

Obviously I cannot multiply the elements with the range up to 10^5.

How can I track only the first digit during multiplication?

Answer 1

In c or c++ make integer data type as long double such that first digit of number is before decimal point and rest are after decimal point.

Above can be done as follows:-

long double GetFraction(int number){
    int length = (int) log(number) + 1; // this will give number of digits in given number. And log is log base 10.
    long double fraction = (long double) number / (10^(length - 1);
    return fraction;
}

Example :-

Let number = 12345

length = log(12345) + 1 = 5;
fraction = (long double) 12345 / (10^4) = 1.2345

Now for all integers in array find fraction as mention above and multiply them as follow:-

int GetFirstDigit(int arr[] , int size){
    if(size == 0)
        return 0;
    long double firstDigit = 1.0;
    for(int i = 0 ; i < size ; i++){
        firstDigit = firstDigit*GetFraction(arr[i]);
        if(firstDigit >= 10.00) // You have to shorten your number otherwise it will same as large multiplication and will overflow.
            firstDigit/=10;
    }
    return (int) firstDigit;
}

Disclaimer:- This is my approach and I don't have any formal proof about accuracy of result. But I have verified result for integer up to 10^9 and array size up to 10^5