In modern days, programming languages tend to be as high-level as possible to make programmer’s life a little bit easier. However, no matter how advanced programming language is, the code still has to be converted down to the machine code, via compilation, interpretation or even virtual machine such as JVM. Of course, at this stage, rules are different: CPU works with addresses and registers without any classes, even «if» branches look like conditional jumps. One of the most important aspects of this execution is the arithmetic operation, and today we will be talking about one of these «cornerstones»: floating-point numbers and how they may affect your code.

A brief introduction to history

The need for processing large or small values was present since the very first days of computers: even first designs of Charles Babbage’s Analytical Engine sometimes included floating-point arithmetic along with usual integer arithmetic. For a long time, the floating-point format was used primarily for scientific research, especially in physics, due to the large variety of data. It is extremely convenient that distance between Earth and Sun can be expressed in the same amount of bits as the distance between hydrogen and oxygen atoms in water molecules with the same relative precision and, even better, values of different magnitudes may be freely multiplied without large losses in precision.

Almost all the early implementations of floating-point numbers were software due to the complexity of the hardware implementations. Without this common standard, everybody had to come up with their own formats: this is how Microsoft Binary Format and IBM Floating Point Architecture were born; the latter is still used in some fields such as weather forecasting, although it is extremely rare by now.

Intel 8087 coprocessor, announced in 1980, also used its own format called «x87». It was the first coprocessor specifically dedicated for floating-point arithmetic with aims to replace slow library calls with the machine code. Then, based on x87 format, IEEE 754 was born as the first and successful attempt to create a universal standard for floating-point calculations. Soon, Intel started to integrate IEEE 754 into their CPUs, and nowadays almost every system except some embedded ones support the floating-point format.

Theory and experiments

In IEEE 754 single-precision binary floating-point format, 32 bits are split into 1-bit sign flag, 8-bit exponent flag and 23-bit fraction part, in that order (bit sign is the leftmost bit). This information should be enough for us to start some experiments! Let us see how number 1.0 looks like in this format using this simple C code:

union {
   float in;
   unsigned out;
} converter; 
converter.in = float_number; 
unsigned bits = converter.out;

Of course, after getting the bits variable, we only need to print it. For instance, this way:

    1.0 |     1 | S: 0 E: 01111111 M: 00000000000000000000000

Common sense tells that 1 can be expressed in binary fluting-point form as 1.0 * 2<sup>0</sup>, so exponent is 0 and significand is 1, while in IEEE 754 exponent is 1111111 (127 in decimal) and significand is 0.

The mystery behind exponent is simple: the exponent is actually shifted. A zero exponent is represented as 127; exponent of 1 is represented as 128 and so on. Maximum value of exponent should be 255 – 127 = 128, and minimum value should be 0 – 127 = -127. However, values 255 and 0 are reserved, so actual range is -126…127. We will talk about those reserved values later.

Significand is even simpler to explain. Binary significand has one unique property: every significand in normalized form, except for zero, starts with 1 (this is only true for binary numbers). Next, if a number starts with zero, then it is not normalized. For instance, non-normalized 0.000101 * 10<sup>101</sup> is the same as normalized 1.01 * 10<sup>1</sup>. Due to that, there is no need to write initial 1 for normalized numbers: we can just keep it in mind, saving space for one more significant bit. In our case, actual significand is 1 and 23 zeroes, but because 1 is skipped, it is only 23 zeroes.

Let us try some different numbers in comparison with 1.

    1.0 |     1 | S: 0 E: 01111111 M: 00000000000000000000000

   -1.0 |    -1 | S: 1 E: 01111111 M: 00000000000000000000000

    2.0 |     2 | S: 0 E: 10000000 M: 00000000000000000000000

    4.0 |     4 | S: 0 E: 10000001 M: 00000000000000000000000

  1 / 8 | 0.125 | S: 0 E: 01111100 M: 00000000000000000000000

As we can see, a negative sign just inverts sign flag without touching the rest (this seems obvious, but it is not always the case in computer science: for integers, a negative sign is much more complex than just flipping one bit!). Changing the exponent by trying different powers of two works as expected.

    1.0 |     1 | S: 0 E: 01111111 M: 00000000000000000000000

    3.0 |     3 | S: 0 E: 10000000 M: 10000000000000000000000

    5.0 |     5 | S: 0 E: 10000001 M: 01000000000000000000000

    0.2 |   0.2 | S: 0 E: 01111100 M: 10011001100110011001101

It is easy to see that numbers 3 and 5 are represented as 1.1 and 1.01 with a proper exponent. 0.2 should not differ much from them, but it is. What happened?

It is easier to explain on decimals. 0.2 is the same number as 1/5. In the same time, not each radical can be represented as a decimal floating-point number: for example, 2/3 is 0.666666… It happens because 3 does not have any non-trivial common divisors with 10 (10 = 2 * 5, neither of them is 3). In the same time, 2/3 can be easily represented in base 12 as 0.8 (12 = 2 * 2 * 3). The same trick is true for a binary system: 2 does not have any common divisors with 5, so 0.2 can only be represented as infinitely long 0.00110011001100… In the same time, we only have 23 significant bits! So, we are inevitably losing precision.

Let us try with some multiplications.

    1.0 |     1 | S: 0 E: 01111111 M: 00000000000000000000000

0.2^2*25 |     1 | S: 0 E: 01111111 M: 00000000000000000000001

25*0.2^2 |     1 | S: 0 E: 01111111 M: 00000000000000000000000

Both 1 and 0.2 * 0.2 * 25 are printed as 1, but they are actually different! Due to the precision loss, 0.2 * 0.2 * 25 is not the same as 1, and the expression (0.2f * 0.2f * 25.0f == 1.0f) is actually false. In the same time, if we execute 25 * 0.2 first, then the result is actually correct. It means that the rule (a * b) * c = a * (b * c) is not always true for floating point numbers!

Special numbers

Remember about the fact that zero can never be written in the normalized form because it does not contain any 1s in its binary representation? Zero is the special number.

      0 |     0 | S: 0 E: 00000000 M: 00000000000000000000000

     -0 |    -0 | S: 1 E: 00000000 M: 00000000000000000000000

For zero, IEEE 754 uses exponent value of 0 and significand value of 0. In addition, as you can see, there are actually two zero values: +0 and -0. In terms of comparison, (0.0f == -0.0f), is actually true, sign just does not count. +0 and -0 loosely correspond to the mathematical concept of infinitesimal, positive and negative.

Are there any special numbers with exponent value of 0? Yes. They are called «denormalized numbers». Those numbers can represent extremely small values, lesser than the minimum normalized number (which should be a little larger than 1 * 2<sup>-127</sup>). Examples:

 2^-126 | 1.17549e-38 | S: 0 E: 00000001 M: 00000000000000000000000

 2^-127 | 5.87747e-39 | S: 0 E: 00000000 M: 10000000000000000000000

 2^-128 | 2.93874e-39 | S: 0 E: 00000000 M: 01000000000000000000000

 2^-149 |  1.4013e-45 | S: 0 E: 00000000 M: 00000000000000000000001

 2^-150 |           0 | S: 0 E: 00000000 M: 00000000000000000000000

A denormalized number have the virtual exponent value of 1, but, at the same time, they do not have omitted 1 as their first omitted digit. The only consequence is that denormalized numbers quickly lose precision: to store numbers between 2<sup>-128</sup> and 2<sup>-127</sup>, we are only using 21 digits of information instead of 23.

It is easy to see that zero is the special case of denormalized number. Moreover, as we can see, the least possible single-precision floating-point number is actually 2<sup>-149</sup>, or approximately 1.4013 * 10<sup>-45</sup>.

Numbers with the exponent of 11111111 are reserved for the «other end» of the number scale: Infinity and the special value called «Not a Number».

 1 /  0 |   inf | S: 0 E: 11111111 M: 00000000000000000000000

 1 / -0 |  -inf | S: 1 E: 11111111 M: 00000000000000000000000

  2^128 |   inf | S: 0 E: 11111111 M: 00000000000000000000000

As with zeroes, infinity can be either positive or negative. It can be achieved by dividing any non-zero number to zero or by getting any number larger than maximum allowed (which is a little less than 2<sup>128</sup>). Infinity is processed as follows:

Infinity > Any number

Infinity = Infinity

Infinity > -Infinity

Any number / Infinity = 0 (sign is set properly)

Infinity * Infinity = Infinity (again, sign is set properly)

Infinity / Infinity = NaN

Infinity * 0 = NaN

Not a Number, or NaN, is, perhaps, the most interesting floating-point value. It can be obtained in multiple ways. First, it is the result of any indeterminate form:

Infinity * 0

0 / 0 or Infinity / Infinity

Infinity – Infinity or –Infinity + Infinity

Secondly, it can be the result of some non-trivial operations. Power function may return NaN in case of any of those indeterminate forms: 0<sup>0</sup>, 1<sup>Infinity</sup>, Infinity<sup>0</sup>. Any operation which can result in complex number may return NaN in this case: log(-1.0f), sqrt(-1.0f), sin(2.0f) are the examples.

Lastly, any operation involving NaN as any of the operands always return NaN. Because of that, NaN can sometimes quickly “spread” through data like a computer virus. The only exception is min or max: those functions should return non-NaN argument. NaN is never equal to any other number, even itself (it can be used to test numbers against NaN).

Actual contents of NaN are implementation-defined; IEEE 754 only requires that exponent should be 11111111, significand should be non-zero (zero is reserved for the infinity) and the sign does not matter.

    0/0 |  -nan | S: 1 E: 11111111 M: 10000000000000000000000

IEEE 754 differentiates two types of NaN: quiet NaN and signaling NaN. Their only difference is that signaling NaN generates interruption while quiet NaN does not. Again, the application decides if it generates quiet NaN or signaling NaN. For instance, GCC C compiler always generates quiet NaN unless explicitly specified to behave the other way around.

Conclusions

What can we learn from all the facts and experiments above? In any language operating with the floating-point data type, beware of the following:

– You should almost never directly compare two floating-point numbers unless you know what you are doing! A better way to do it is to compare with some precision.

if (a == b) – wrong!

if (fabsf(a – b) < epsilon) – correct!

Floating-point numbers lose precision even when you are just working with such seemingly harmless numbers as 0.2 or 71.3. You should be extra careful when working with a large amount of floating-point operations over the same data: errors may build up rather quickly. If you are getting unexpected results and you suspect rounding errors, try to use a different approach and minimize errors.

– In the world of floating-point arithmetic, multiplication is not associative: a * (b * c) is not always equal to (a * b) * c.

Additional measures should be taken if you are working with either extremely large values, extremely small numbers and/or numbers close to zero: in case of overflow or underflow those values will be transformed into +Infinty, -Infinity or 0. Numeric limits for single-precision floating-point numbers are approximately 1.175494e-38 to 3.402823e+38 (1.4013e-45 to 3.402823e+38 if we also count denormalized numbers)а.

Beware if your system generates «quiet NaN». Sometimes, it may help you to not crash the application. Sometimes, it may spoil program execution beyond the recognition.

Nowadays, floating-point numbers operations are extremely fast, with speed comparable to the usual integer arithmetic: a number of floating-point operations per second, or FLOPS, is perhaps the most well-known measure of computer performance. The only downside is that the programmer should be aware of all the pitfalls regarding the precision and special floating-point values.