Printf a double

Question

I have a problem in printing a double in C.

My code:

#include <stdio.h>

main() {
   double n;
   scanf("%lf", &n);
   printf("%f", n);
}

input: 446486416781684178

output: 446486416781684160

Why does the number change?

Answer 1

The number you entered can't be represented exactly in a double .

Typically, a double is represented using IEEE754 double precision format. This format can hold up to 53 bits of precision. The value you entered requires 58 bits of precision. So what is stored is either the next or the previous representable value.

Answer 2

Why does the number change?

It got rounded off, because type double has finite precision.

We're used to roundoff happening to the right of the decimal point. If we write

double d = 0.123456789012345678;
printf("%.18f\n", d);

we are not too surprised if it prints

0.123456789012345677

Type double has the equivalent of about 16 decimal digit's worth of precision (actually it's more complicated than that), so it definitely can't represent all 18 digits of that number 0.123456789012345678.

But your number 446486416781684178 also has 18 significant digits, so we can't be too surprised that it can't be represented exactly, either. In other words, roundoff can happen to the left of the decimal point, also.

Internally, type double can represent numbers with 53 bits of precision. That means it can represent integers up to 2 ⁵³ , or 9007199254740992, with perfect accuracy. But bigger than that — it just can't. It can represent 900..,992. and it can represent 900..,994. but if you try to do 900..,993. it gets rounded back down to 900...992, If we look at the binary representations of these and nearby numbers: we can see why:

Decimal	Binary	Rep?
9007199254740990	11111111111111111111111111111111111111111111111111110	yes
9007199254740991	11111111111111111111111111111111111111111111111111111	yes
9007199254740992	100000000000000000000000000000000000000000000000000000	yes
9007199254740993	100000000000000000000000000000000000000000000000000001	no
9007199254740994	100000000000000000000000000000000000000000000000000010	yes

Since we only have 53 bits of significance, for a 54-bit number like 9007199254740992 or 9007199254740994, the 54th bit has to be 0, which basically means we can only represent even numbers in that range. The 54-bit number 9007199254740993 ends with a 1 bit, so it can't be exactly represented, which I've indicated with a "no" in the "Rep" ("exactly representable") column.

When we get up to a 59-bit number like 446486416781684178, the last six bits have to be 0, which means we can only represent numbers which are a multiple of 2 ⁶ , or 64:

Decimal	Binary	Rep?
446486416781684160	1100011001000111101…001101010011110010110111000000	yes
446486416781684161	1100011001000111101…001101010011110010110111000001	no
...	...	...
446486416781684177	1100011001000111101…001101010011110010110111010001	no
446486416781684178	1100011001000111101…001101010011110010110111010010	no
446486416781684179	1100011001000111101…001101010011110010110111010011	no
...	...	...
446486416781684223	1100011001000111101…001101010011110010110111111111	no
446486416781684224	1100011001000111101…001101010011110010111000000000	yes

Answer 3

The number changes because of the limited precision of the double type. If you don't need floating point values, you could use type long long which is defined to have at least 63 value bits and can represent the number in the question.

Note that it is highly unlikely that the output be 446486416781684160 , the printf("%f", n) call should produce 446486416781684160.000000 . If your program does not produce 6 decimal places, your compiler and/or standard library are not conforming.

Also note that the main function should be defined with an explicit return type of int . The implicit int style has been deprecated more than 20 years ago.

Here is a modified version:

#include <stdio.h>

int main() {
    long long n;
    if (scanf("%lld", &n) == 1) {
        printf("%lld\n", n);
    }
    return 0;
}

Input: 446486416781684178
Output: 446486416781684178