What is the difference between float and double?
c, c++, floating-point, ieee-754, precision
Solution
Huge difference.
As the name implies, a `double` has 2x the precision of `float`[1]. In general a `double` has 15 decimal digits of precision, while `float` has 7.
Here's how the number of digits are calculated:
`double` has 52 mantissa bits + 1 hidden bit: log(253)÷log(10) = 15.95 digits
`float` has 23 mantissa bits + 1 hidden bit: log(224)÷log(10) = 7.22 digits
This precision loss could lead to greater truncation errors being accumulated when repeated calculations are done, e.g.
float a = 1.f / 81;
float b = 0;
for (int i = 0; i < 729; ++ i)
b += a;
printf("%.7g\n", b); // prints 9.000023
while
double a = 1.0 / 81;
double b = 0;
for (int i = 0; i < 729; ++ i)
b += a;
printf("%.15g\n", b); // prints 8.99999999999996
Also, the maximum value of float is about `3e38`, but double is about `1.7e308`, so using `float` can hit "infinity" (i.e. a special floating-point number) much more easily than `double` for something simple, e.g. computing the factorial of 60.
During testing, maybe a few test cases contain these huge numbers, which may cause your programs to fail if you use floats.
Of course, sometimes, even `double` isn't accurate enough, hence we sometimes have `long double`[1] (the above example gives 9.000000000000000066 on Mac), but all floating point types suffer from round-off errors, so if precision is very important (e.g. money processing) you should use `int` or a fraction class.
Furthermore, don't use `+=` to sum lots of floating point numbers, as the errors accumulate quickly. If you're using Python, use `fsum`. Otherwise, try to implement the Kahan summation algorithm.
[1]: The C and C++ standards do not specify the representation of `float`, `double` and `long double`. It is possible that all three are implemented as IEEE double-precision. Nevertheless, for most architectures (gcc, MSVC; x86, x64, ARM) `float` is indeed a IEEE single-precision floating point number (binary32), and `double` is a IEEE double-precision floating point number (binary64).
Problem
I've read about the difference between double precision and single precision. However, in most cases, `float` and `double` seem to be interchangeable, i.e. using one or the other does not seem to affect the results. Is this really the case? When are floats and doubles interchangeable? What are the differences between them?