In-depth: Floating-point complexities

Adjacent floats (of the same sign) have adjacent integer representations, which makes generating the next (or all) floats trivial

FLT_MIN is not the smallest positive float (FLT_MIN is the smallest positive normalized float)

The smallest positive float is 8,388,608 times smaller than FLT_MIN

FLT_MAX is not the largest positive float (it's the largest finite float, but the special value infinity is larger)

0.1 cannot be exactly represented in a float

All floats can be exactly represented in decimal

Over a hundred decimal digits of mantissa are required to exactly show the value of some floats

9 decimal digits of mantissa (plus sign and exponent) are sufficient to uniquely identify any float

The Visual C++ debugger displays floats with 8 mantissa digits

The integer representation of a float is a piecewise linear approximation of the base-2 logarithm of that float

You can calculate the base-2 log of an integer by assigning it to a float

Most float math gives inexact results due to rounding

The basic IEEE math operations guarantee perfect rounding

Subtraction of floats with similar values (f2 * 0.5 <= f1 <= f2 * 2.0) gives an exact result, with no rounding

Subtraction of floats with similar values can result in a loss of virtually all significant figures (even if the result is exact)

Minor rearrangements in a calculation can take it from catastrophic cancellation to 100% accurate

Storing elapsed game time in a float is a bad idea

Comparing floats requires care, especially around zero

sin(float(pi)) calculates a very accurate approximation to pi-float(pi)

From 2^24 to 2^31, an int32_t has more precision than a float – in that range an int32_t can hold every value that a float can hold, and millions more

pow(2.0f, -149) should calculate the smallest denormal float, but with VC++ it generates zero. pow(0.5f, 149) works.

IEEE float arithmetic guarantees that "if (x != y) return z / (x-y);" will never cause a divide by zero, but this guarantee only applies if denormals are supported

Denormals have horrible performance on some older hardware, which leads to some developers disabling them

If x is a floating-point number then "x == x" may return false – if x is a NaN

Calculations done with higher-precision intermediate values sometimes give more accurate results, sometimes less accurate results, and sometimes just inconsistent results

Double rounding can lead to inaccurate results, even when doing something as simple as assigning a constant to a float

You can printf and scanf every positive float in less than fifteen minutes