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Floating Point

Floating point arithmetic (normally just float) is a method of computer representation of fractional numbers and approximating real numbers, i.e. numbers with higher than integer precision (such as 5.13), which is more complex than e.g. fixed point. The core idea of it is to use a radix ("decimal") point that's not fixed but can move around so as to allow representation of both very small and very big values. Nowadays floating point is the standard way of approximating real numbers in computers (floating point types are called real in some programming languages, even though they represent only rational numbers, floats can't e.g. represent pi exactly), basically all of the popular programming languages have a floating point data type that adheres to the IEEE 754 standard, all personal computers also have the floating point hardware unit (FPU) and so it is widely used in all modern programs. However most of the time a simpler representation of fractional numbers, such as the mentioned fixed point, suffices, and weaker computers (e.g. embedded) may lack the hardware support so floating point operations are emulated in software and therefore slow -- for these reasons we consider floating point bloat and recommend the preference of fixed point.

Floating point can also get tricky, it works most of the time but a danger lies in programmers relying on this kind of magic too much, some new generation programmers may not even be very aware of how float works. One possible pitfall is working with big and small numbers at the same time -- due to differing precision at different scales small values simply get lost when mixed with big numbers and sometimes this has to be worked around with tricks (see e.g. this devlog of The Witness where a float time variable sent into shader is periodically reset so as to not grow too large and cause the mentioned issue).

Is floating point literal evil? Well, of course not, but it is extremely overused. You may need it for precise scientific simulations, e.g. numerical integration, but as our small3dlib shows, you can comfortably do even 3D rendering without it. So always consider whether you REALLY need float. You mostly do not.

How It Works

The very basic idea is following: we have digits in memory and in addition we have a position of the radix point among these digits, i.e. both digits and position of the radix point can change. The fact that the radix point can move is reflected in the name floating point. In the end any number stored in float can be written with a finite number of digits with a radix point, e.g. 12.34. Notice that any such number can also always be written as a simple fraction of two integers (e.g. 12.34 = 1 * 10 + 2 * 1 + 3 * 1/10 + 4 * 1/100 = 617/50), i.e. any such number is always a rational number. This is why we say that floats represent fractional numbers and not true real numbers (real numbers such as pi, e or square root of 2 can only be approximated).

More precisely floats represent numbers by representing two main parts: the base -- actual encoded digits, called mantissa (or significand etc.) -- and the position of the radix point. The position of radix point is called the exponent because mathematically the floating point works similarly to the scientific notation of extreme numbers that use exponentiation. For example instead of writing 0.0000123 scientists write 123 * 10^-7 -- here 123 would be the mantissa and -7 the exponent.

Though various numeric bases can be used, in computers we normally use base 2, so let's consider it from now on. So our numbers will be of format:

mantissa * 2^exponent

Note that besides mantissa and exponent there may also be other parts, typically there is also a sign bit that says whether the number is positive or negative.

Let's now consider an extremely simple floating point format based on the above. Keep in mind this is an EXTREMELY NAIVE inefficient format that wastes values. We won't consider negative numbers. We will use 6 bits for our numbers:

So for example the binary representation 110011 stores mantissa 110 (6) and exponent 011 (3), so the number it represents is 6 * 2^3 = 48. Similarly 001101 represents 1 * 2^-3 = 1/8 = 0.125.

Note a few things: firstly our format is shit because some numbers have multiple representations, e.g. 0 can be represented as 000000, 000001, 000010, 000011 etc., in fact we have 8 zeros! That's unforgivable and formats used in practice address this (usually by prepending an implicit 1 to mantissa).

Secondly notice the non-uniform distribution of our numbers: while we have a nice resolution close to 0 (we can represent 1/16, 2/16, 3/16, ...) but low resolution in higher numbers (the highest number we can represent is 56 but the second highest is 48, we can NOT represent e.g. 50 exactly). Realize that obviously with 6 bits we can still represent only 64 numbers at most! So float is NOT a magical way to get more numbers, with integers on 6 bits we can represent numbers from 0 to 63 spaced exactly by 1 and with our floating point we can represent numbers spaced as close as 1/16th but only in the region near 0, we pay the price of having big gaps in higher numbers.

Also notice that thing like simple addition of numbers become more difficult and time consuming, you have to include conversions and rounding -- while with fixed point addition is a single machine instruction, same as integer addition, here with software implementation we might end up with dozens of instructions (specialized hardware can perform addition fast but still, not all computer have that hardware).

Rounding errors will appear and accumulate during computations: imagine the operation 48 + 1/8. Both numbers can be represented in our system but not the result (48.125). We have to round the result and end up with 48 again. Imagine you perform 64 such additions in succession (e.g. in a loop): mathematically the result should be 48 + 64 * 1/8 = 56, which is a result we can represent in our system, but we will nevertheless get the wrong result (48) due to rounding errors in each addition. So the behavior of float can be non intuitive and dangerous, at least for those who don't know how it works.

Standard Float Format: IEEE 754

IEEE 754 is THE standard that basically all computers use for floating point nowadays -- it specifies the exact representation of floating point numbers as well as rounding rules, required operations applications should implement etc. However note that the standard is kind of shitty -- even if we want to use floating point numbers there exist better ways such as posits that outperform this standard. Nevertheless IEEE 754 has been established in the industry to the point that it's unlikely to go anytime soon. So it's good to know how it works.

Numbers in this standard are signed, have positive and negative zero (oops), can represent plus and minus infinity and different NaNs (not a number). In fact there are thousands to billions of different NaNs which are basically wasted values. These inefficiencies are addressed by the mentioned posits.

Briefly the representation is following (hold on to your chair): leftmost bit is the sign bit, then exponent follows (the number of bits depends on the specific format), the rest of bits is mantissa. In mantissa implicit 1. is considered (except when exponent is all 0s), i.e. we "imagine" 1. in front of the mantissa bits but this 1 is not physically stored. Exponent is in so called biased format, i.e. we have to subtract half (rounded down) of the maximum possible value to get the real value (e.g. if we have 8 bits for exponent and the directly stored value is 120, we have to subtract 255 / 2 = 127 to get the real exponent value, in this case we get -7). However two values of exponent have special meaning; all 0s signify so called denormalized (also subnormal) number in which we consider exponent to be that which is otherwise lowest possible (e.g. -126 in case of 8 bit exponent) but we do NOT consider the implicit 1 in front of mantissa (we instead consider 0.), i.e. this allows storing zero (positive and negative) and very small numbers. All 1s in exponent signify either infinity (positive and negative) in case mantissa is all 0s, or a NaN otherwise -- considering here we have the whole mantissa plus sign bit unused, we actually have many different NaNs (WTF), but usually we only distinguish two kinds of NaNs: quiet (qNaN) and signaling (sNaN, throws and exception) that are distinguished by the leftmost bit in mantissa (1 for qNaN, 0 for sNaN).

The standard specifies many formats that are either binary or decimal and use various numbers of bits. The most relevant ones are the following:

name M bits E bits smallest and biggest number precision <= 1 up to
binary16 (half precision) 10 5 2^(-24), 65504 2048
binary32 (single precision, float) 23 8 2^(-149), 2127 * (2 - 2^-23) ~= 3 * 10^38 16777216
binary64 (double precision, double) 52 11 2^(-1074), ~10^308 9007199254740992
binary128 (quadruple precision) 112 15 2^(-16494), ~10^4932 ~10^34

Example? Let's say we have float (binary34) value 11000000111100000000000000000000: first bit (sign) is 1 so the number is negative. Then we have 8 bits of exponent: 10000001 (129) which converted from the biased format (subtracting 127) gives exponent value of 2. Then mantissa bits follow: 11100000000000000000000. As we're dealing with a normal number (exponent bits are neither all 1s nor all 0s), we have to imagine the implicit 1. in front of mantissa, i.e. our actual mantissa is 1.11100000000000000000000 = 1.875. The final number is therefore -1 * 1.875 * 2^2 = -7.5.

See Also

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