Posted By: Anonymous
I’ve read about the difference between double precision and single precision. However, in most cases,
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?
Here’s how the number of digits are calculated:
doublehas 52 mantissa bits + 1 hidden bit: log(253)÷log(10) = 15.95 digits
floathas 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("%.7gn", b); // prints 9.000023
double a = 1.0 / 81; double b = 0; for (int i = 0; i < 729; ++ i) b += a; printf("%.15gn", 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 (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.
: The C and C++ standards do not specify the representation of
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).