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A number is, depending on scope, either a {{ | A number is, depending on scope, either a {{Link|https://en.wikipedia.org/wiki/Single-precision_floating-point_format|single precision floating-point number}} (when talking about scripting) or a range of numerical types when talking about config context. | ||
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In [[SQF Syntax|SQF]], there are multiple accepted number formats. | In [[SQF Syntax|SQF]], there are multiple accepted number formats. | ||
However, all of them will result in the same {{hl|SCALAR}} ([[typeName]]) value type | However, all of them will result in the same {{hl|SCALAR}} ([[typeName]]) value type '''as long as they remain within float range'''; | ||
The '''{{Color|# | The '''{{Color|#080|largest positive}}''' number that can be achieved is {{hl|3.4028235e38}} and the '''{{Color|#800|largest negative}}''' is {{hl|-3.4028235e38}}.<br> | ||
It also is possible to generate positive or negative infinity values using either {{ | It also is possible to generate positive or negative infinity values using either {{hl|1e39}} (string representation: {{hl|1.#INF}}) or {{hl|-1e39}} (string representation: {{hl|-1.#INF}}), both being [[typeName]] {{hl|"NaN"}}. | ||
To check if a number is finite one can use the [[finite]] operator. | To check if a number is finite one can use the [[finite]] operator. | ||
{{Feature|informative|Regex to match all numbers in [[SQF Syntax|SQF]] is {{ | {{Feature|informative|Regex to match all numbers in [[SQF Syntax|SQF]] is {{hl|<nowiki>(((\$|0x)[0-9a-fA-F]+)|(\.[0-9]+))|(\b[0-9]+(\.[0-9]+|[eE][-+]?[0-9]+)?)\b</nowiki>}}}} | ||
{{Feature|important|Due to technical limitations, the precision of floating-point numbers is limited. Please refer to {{ | {{Feature|important|Due to technical limitations, the precision of floating-point numbers is limited. Please refer to {{Link|https://en.wikipedia.org/wiki/IEEE_754|IEEE 754}} for more info about the technical details.}} | ||
=== Decimal (Base 10) === | === Decimal (Base 10) === | ||
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Some Examples would be: | Some Examples would be: | ||
* {{hl|5.197}} | * {{hl|5.197}} | ||
* {{hl| 0.47}} | * {{hl| 0.47}} | ||
* {{hl| 16.0}} | * {{hl| 16.0}} | ||
* {{hl|.8314}} | * {{hl|.8314}} | ||
* {{hl|12345}} | * {{hl|12345}} | ||
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==== Scientific Notation ==== | ==== Scientific Notation ==== | ||
The {{ | The {{Link|https://en.wikipedia.org/wiki/Scientific_notation|Scientific Notation}} is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. | ||
It starts of like a normal decimal and then gets expressed by an {{hl|E}} (Not case-sensitive, thus {{hl|e}} is also valid) followed by an '''optional''' {{hl|+}} or {{hl|-}} sign and ends with a range of digits. | It starts of like a normal decimal and then gets expressed by an {{hl|E}} (Not case-sensitive, thus {{hl|e}} is also valid) followed by an '''optional''' {{hl|+}} or {{hl|-}} sign and ends with a range of digits. | ||
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=== Hexadecimal (Base 16) === | === Hexadecimal (Base 16) === | ||
In [[SQF Syntax|SQF]], {{ | In [[SQF Syntax|SQF]], {{Link|https://en.wikipedia.org/wiki/Hexadecimal|hexadecimal}} (also base 16, or hex) is a positional numeral system with a base of 16. | ||
They start either with {{hl|0x}} or with a single {{hl|$}}. | They start either with {{hl|0x}} or with a single {{hl|$}}. | ||
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== Floating-point Precision == | == Floating-point Precision == | ||
Humans are typically used to performing mathematical calculations on the set of ''real numbers'' (ℝ) | Humans are typically used to performing mathematical calculations on the set of ''real numbers'' (ℝ). There are no gaps between real numbers, i.e. there is always another real number between two real numbers. However, as mentioned above, every number in scripting is a ''floating-point number''. Because any given floating-point number can only consist of a finite amount of bits (a computer's memory is finite after all), any given floating-point format can only represent a finite amount of distinct numbers. Consequently, there are gaps between floating-point numbers. | ||
This induces the following problem: There are real numbers that the computer does not have a binary representation for (and that it therefore can not work with) because there is no representation for these numbers in the specific floating-point format used by the computer. Of course, by extension, this also applies to the [[Real Virtuality]] Engine - there are real numbers that it does not have a representation for. | This induces the following problem: There are real numbers that the computer does not have a binary representation for (and that it therefore can not work with) because there is no representation for these numbers in the specific floating-point format used by the computer. Of course, by extension, this also applies to the [[Real Virtuality]] Engine - there are real numbers that it does not have a representation for. | ||
This issue is dealt with by rounding to the nearest number that can actually be represented as a floating-point number. | This issue is dealt with by rounding to the nearest number that can actually be represented as a floating-point number. | ||
{| class="wikitable" style="margin: auto" | {| class="wikitable" style="margin: auto" | ||
|- | |- | ||
! <var>n</var> ∈ ℝ !! <var>n</var> in {{arma3}}<br>(value of <sqf inline>n toFixed 20</sqf>) | ! <var>n</var> ∈ ℝ !! <var>n</var> in {{arma3}}<br><span style="font-weight: normal">(value of <sqf inline>n toFixed 20</sqf>)</span> | ||
|- style="background-color: lightgreen" | |- style="background-color: lightgreen" | ||
| 0 || 0.00000000000000000000 | | 0 || 0.00000000000000000000 | ||
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| 2.51 || 2.50999999046325680000 | | 2.51 || 2.50999999046325680000 | ||
|} | |} | ||
Unfortunately, this rounding introduces a new problem in the form of unexpected, seemingly erroneous behaviour: For instance, despite being entirely correct mathematically, the expression <sqf inline>0.3 + 0.4 == 0.7</sqf> surprisingly returns [[false]] in {{arma3}}. In order to understand why this happens, examine the evaluation of the expression step by step: | Unfortunately, this rounding introduces a new problem in the form of unexpected, seemingly erroneous behaviour: For instance, despite being entirely correct mathematically, the expression <sqf inline>0.3 + 0.4 == 0.7</sqf> surprisingly returns [[false]] in {{arma3}}. In order to understand why this happens, examine the evaluation of the expression step by step: | ||
# The 0.3 from the source code has to be rounded to 0.30000001192092896 | # The 0.3 from the source code has to be rounded to 0.30000001192092896 | ||
# The 0.4 from the source code has to be rounded to 0.40000000596046448 | # The 0.4 from the source code has to be rounded to 0.40000000596046448 | ||
# Now the addition can be performed: 0.30000001192092896 + 0.40000000596046448 = 0.70000004768371582 (that is the result with floating-point arithmetic; the result of the same addition in ℝ is 0.70000001788139344) | # Now the addition can be performed: 0.30000001192092896 + 0.40000000596046448 = 0.70000004768371582<br>(that is the result with floating-point arithmetic; the result of the same addition in ℝ is 0.70000001788139344) | ||
# The 0.7 from the source code has to be rounded to 0.69999998807907104 | # The 0.7 from the source code has to be rounded to 0.69999998807907104 | ||
# Finally, the comparison <sqf inline>0.70000004768371582 == 0.69999998807907104</sqf> returns [[false]] | # Finally, the comparison <sqf inline>0.70000004768371582 == 0.69999998807907104</sqf> returns [[false]] | ||
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To add to the confusion, <sqf inline>0.3 + 0.4</sqf> will actually output {{hl|0.7}} in the [[Arma 3: Debug Console|Debug Console]] or when used with commands like [[hint]] or [[systemChat]] - that is because numbers are rounded yet again when they are converted to [[String]] for the output. If this needs to be avoided, the [[toFixed]] command can be used to obtain the accurate value of a number. | To add to the confusion, <sqf inline>0.3 + 0.4</sqf> will actually output {{hl|0.7}} in the [[Arma 3: Debug Console|Debug Console]] or when used with commands like [[hint]] or [[systemChat]] - that is because numbers are rounded yet again when they are converted to [[String]] for the output. If this needs to be avoided, the [[toFixed]] command can be used to obtain the accurate value of a number. | ||
Floating-point formats are designed so that | Floating-point formats are designed so that the density of floating-point numbers is highest around the number zero. Accordingly, there are more floating-point numbers between 0 and 1 than there are between 100,000 and 100,001. As such, the gaps between consecutive floating-point numbers with small absolute values are also small, while the gaps between consecutive floating-point numbers with large absolute values become increasingly larger. For instance, the next larger floating-point number after 87,654,320 is 87,654,328 in {{arma3}}. | ||
=== Consequences === | === Consequences === | ||
* Fractional numbers should not be compared for equality. Instead | * Fractional floating-point numbers should not be compared for exact equality. Instead, one should check the distance (i.e. absolute difference) between the numbers. Numbers can then be considered equal if the distance between them is smaller than some threshold (e.g. 0.000001). Consequently, expressions like <sqf inline>0.3 + 0.4 == 0.7</sqf> should be replaced with expressions such as <sqf inline>_res = 0.3 + 0.4; abs (_res - 0.7) < 0.000001</sqf>.<br>Note that this is only necessary if fractional numbers are involved; if only whole numbers are involved, this procedure is not necessary. | ||
* Large numbers can not always be incremented / decremented. For example, <sqf inline>87654320 + 1</sqf> returns | * Large numbers can not always be incremented / decremented. For example, <sqf inline>87654320 + 1</sqf> returns {{hl|87,654,320}} because any real number between 87,654,316 and 87,654,324 has to be rounded to 87,654,320 in {{arma3}}. | ||
[[Category: Data Types]] | [[Category: Data Types]] |
Revision as of 11:54, 9 March 2024
A number is, depending on scope, either a single precision floating-point number (when talking about scripting) or a range of numerical types when talking about config context.
Scripting
In SQF, there are multiple accepted number formats. However, all of them will result in the same SCALAR (typeName) value type as long as they remain within float range;
The largest positive number that can be achieved is 3.4028235e38 and the largest negative is -3.4028235e38.
It also is possible to generate positive or negative infinity values using either 1e39 (string representation: 1.#INF) or -1e39 (string representation: -1.#INF), both being typeName "NaN".
To check if a number is finite one can use the finite operator.
Decimal (Base 10)
A decimal number is your normal 0.5 syntax stuff with one extra: You may omit the initial pack of digits.
Some Examples would be:
- 5.197
- 0.47
- 16.0
- .8314
- 12345
A regex catching these kind of numbers could look like this ((\.[0-9]+)|(\b[0-9]+(\.[0-9]+)?))\b
Scientific Notation
The Scientific Notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.
It starts of like a normal decimal and then gets expressed by an E (Not case-sensitive, thus e is also valid) followed by an optional + or - sign and ends with a range of digits.
Some examples would be:
- 1.23E4 ⇔ 1.23 ⋅ 104 ⇔ 12300
- 5e-2 ⇔ 5 ⋅ 10-2 ⇔ 0.05
A regex catching these kind of numbers could look like this ([0-9]+.)?[0-9]+[eE][+-]?[0-9]+
Hexadecimal (Base 16)
In SQF, hexadecimal (also base 16, or hex) is a positional numeral system with a base of 16. They start either with 0x or with a single $.
This gets followed by one of the following characters: 0 1 2 3 4 5 6 7 8 9 A B C D E F.
Note that casing does not matter, thus both 0xa and 0xA are valid.
Some Examples would be:
- 0xa5
- $5C
- $FFFFFF
- 0x123ABC
A regex catching these kind of numbers could look like this ((\$|0x)[0-9a-fA-F]+)\b
Config
Unlike scripting, configs actually do allow for multiple number types (Integer, Float, etc). They are properly stored with their corresponding type in mind.
class MyClass
{
myInt = 1; // integer
myFloat = 0.01; // float
};
// description.ext example
disabledAI = 1; // boolean: 0 = false, everything else = true (usually 1)
Floating-point Precision
Humans are typically used to performing mathematical calculations on the set of real numbers (ℝ). There are no gaps between real numbers, i.e. there is always another real number between two real numbers. However, as mentioned above, every number in scripting is a floating-point number. Because any given floating-point number can only consist of a finite amount of bits (a computer's memory is finite after all), any given floating-point format can only represent a finite amount of distinct numbers. Consequently, there are gaps between floating-point numbers.
This induces the following problem: There are real numbers that the computer does not have a binary representation for (and that it therefore can not work with) because there is no representation for these numbers in the specific floating-point format used by the computer. Of course, by extension, this also applies to the Real Virtuality Engine - there are real numbers that it does not have a representation for.
This issue is dealt with by rounding to the nearest number that can actually be represented as a floating-point number.
n ∈ ℝ | n in Arma 3 (value of n toFixed 20) |
---|---|
0 | 0.00000000000000000000 |
1 | 1.00000000000000000000 |
1.1 | 1.10000002384185790000 |
2.5 | 2.50000000000000000000 |
2.51 | 2.50999999046325680000 |
Unfortunately, this rounding introduces a new problem in the form of unexpected, seemingly erroneous behaviour: For instance, despite being entirely correct mathematically, the expression 0.3 + 0.4 == 0.7 surprisingly returns false in Arma 3. In order to understand why this happens, examine the evaluation of the expression step by step:
- The 0.3 from the source code has to be rounded to 0.30000001192092896
- The 0.4 from the source code has to be rounded to 0.40000000596046448
- Now the addition can be performed: 0.30000001192092896 + 0.40000000596046448 = 0.70000004768371582
(that is the result with floating-point arithmetic; the result of the same addition in ℝ is 0.70000001788139344) - The 0.7 from the source code has to be rounded to 0.69999998807907104
- Finally, the comparison 0.70000004768371582 == 0.69999998807907104 returns false
To add to the confusion, 0.3 + 0.4 will actually output 0.7 in the Debug Console or when used with commands like hint or systemChat - that is because numbers are rounded yet again when they are converted to String for the output. If this needs to be avoided, the toFixed command can be used to obtain the accurate value of a number.
Floating-point formats are designed so that the density of floating-point numbers is highest around the number zero. Accordingly, there are more floating-point numbers between 0 and 1 than there are between 100,000 and 100,001. As such, the gaps between consecutive floating-point numbers with small absolute values are also small, while the gaps between consecutive floating-point numbers with large absolute values become increasingly larger. For instance, the next larger floating-point number after 87,654,320 is 87,654,328 in Arma 3.
Consequences
- Fractional floating-point numbers should not be compared for exact equality. Instead, one should check the distance (i.e. absolute difference) between the numbers. Numbers can then be considered equal if the distance between them is smaller than some threshold (e.g. 0.000001). Consequently, expressions like 0.3 + 0.4 == 0.7 should be replaced with expressions such as _res = 0.3 + 0.4; abs (_res - 0.7) < 0.000001.
Note that this is only necessary if fractional numbers are involved; if only whole numbers are involved, this procedure is not necessary. - Large numbers can not always be incremented / decremented. For example, 87654320 + 1 returns 87,654,320 because any real number between 87,654,316 and 87,654,324 has to be rounded to 87,654,320 in Arma 3.