How big of a number can you store in double and float in c?

Question

I am trying to figure out exactly how big number I can use as floating point number and double. But it does not store the way I expected except integer value. double should hold 8 bytes of information which is enough to hold variable a, but it does not hold it right. It shows 1234567890123456768 in which last 2 digits are different. And when I stored 214783648 or any digit in the last digit in float variable b, it shows the same value 214783648. which is supposed to be the limit. So what's going on?

double a;
float b;
int c;
a = 1234567890123456789;
b = 2147483648;
c = 2147483647;
printf("Bytes of double: %d\n", sizeof(double));
printf("Bytes of integer: %d\n", sizeof(int));
printf("Bytes of float: %d\n", sizeof(float));

printf("\n");

printf("You can count up to %.0f in 4 bytes\n", pow(2,32));
printf("You can count up to %.0f with + or - sign in 4 bytes\n", pow(2,31));
printf("You can count up to %.0f in 4 bytes\n", pow(2,64));
printf("You can count up to %.0f with + or - sign in in 8 bytes\n", pow(2,63));

printf("\n");

printf("double number: %.0f\n", a);
printf("floating point: %.0f\n", b);
printf("integer: %d\n", c);

return 0;

Answer 1

The answer to the question of what is the largest (finite) number that can be stored in a floating point type would be FLT_MAX or DBL_MAX for float and double, respectively.

However, that doesn't mean that the type can precisely represent every smaller number or integer (in fact, not even close).

First you need to understand that not all bits of a floating point number are “equal”. A floating point number has an exponent (8 bits in IEEE-754 standard float, 11 bits in double), and a mantissa (23 and 52 bits in float, and double respectively). The number is obtained by multiplying the mantissa (which has an implied leading 1-bit and binary point) by 2^exponent (after normalizing the exponent; its binary value is not used directly). There is also a separate sign bit, so the following applies to negative numbers as well.

As the exponent changes, the distance between consecutive values of the mantissa changes as well, i.e., the greater the exponent, the further apart consecutive representable values of the floating point number are. Thus you may be able to store one number of a given magnitude precisely, but not the “next” number. One should also remember that some seemingly simple fractions can not be represented precisely with any number of binary digits (e.g., 1/10, one tenth, is an infinitely repeating sequence in binary, like 1/3, one third, is in decimal).

When it comes to integers, you can precisely represent every integer up to 2^{mantissa_bits + 1} magnitude. Thus an IEEE-754 float can represent all integers up to 2²⁴ and a double up to 2⁵³ (in the last half of these ranges the consecutive floating point values are exactly one integer apart, since the entire mantissa is used for the integer part only). There are individual larger integers that can be represented, but they are spaced more than one integer apart, i.e., you can represent some integers greater than 2^{mantissa_bits + 1} but every integer only up to that magnitude.

For example:

float f = powf(2.0f, 24.0f);
float f1 = f + 1.0f, f2 = f1 + 2.0f;
double d = pow(2.0, 53.0);
double d1 = d + 1.0, d2 = d + 2.0;
(void) printf("2**24 float = %.0f, +1 = %.0f, +2 = %.0f\n", f, f1, f2);
(void) printf("2**53 double = %.0f, +1 = %.0f, +2 = %.0f\n", d, d1, d2);

Outputs:

2**24 float = 16777216, +1 = 16777216, +2 = 16777218
2**53 double = 9007199254740992, +1 = 9007199254740992, +2 = 9007199254740994

As you can see, adding 1 to 2^{mantissa_bits + 1} makes no difference since the result is not representable, but adding 2 does produce the correct answer (as it happens, at this magnitude the representable numbers are two integers apart since the multiplier has doubled).

 

TL;DR An IEE-754 float can precisely represent all integers up to 2²⁴ and double up to 2⁵³, but only some integers of greater magnitude (the spacing of representable values depends on the magnitude).

Answer 2

sizeof(double) is 8, true, but double needs some bits to store the exponent part as well.

Assuming IEEE-754 is used, double can represent integers at most 2⁵³ precisely, which is less than 1234567890123456789.

See also Double-precision floating-point format.