# Flt

This library provides arbitrary precision floating-point types for Ruby. All types and functions are within a namespace called Flt. Decimal and Binary floating point numbers are implemented in classes Flt::DecNum and Flt::BinNum. These types are completely written in Ruby using the multiple precision native integers. The performance could be improved in the future by using a C extension based on the decNumber library.

The `Flt::Tolerance`

classes and the `Flt.Tolerance()`

constructor handle floating
point tolerances defined in flexible ways.

Context classes are defined in the files flt/float.rb and flt/bigdecimal.rb for Float and BigDecimal numbers that aid to the interchangeability of floating point types. This represent the only definition of identifiers outside the Flt namespace: the methods Float.context() and BigDecimal.context() and some contants in Float.

This library is the successor of the ruby-decimal gem, that defined the
Decimal class for decimal floating point; that class has been renamed to
`Flt::DecNum`

and support has been added for binary floating point and
tolerances.

The documentation for this package is available at http://www.rubydoc.info/github/jgoizueta/flt/master

The code is at http://github.com/jgoizueta/flt/

## DecNum

`Flt::DecNum`

is a standards-compliant arbitrary precision decimal
floating-point type for Ruby. It is based on the Python Decimal class.

## Standars compliance.

DecNum is designed to be conformant to the General Decimal Arithmetic Specification and the revised IEEE 754 standard (IEEE 754-2008).

# Examples of use

To install the library use gem from the command line:

```
gem install flt
```

Then require the library in your code (if it fails you may need to `require 'rubygems'`

first)

```
require 'flt'
include Flt
```

Now we can use the DecNum class simply like this:

`puts DecNum(1)/DecNum(3) # -> 0.3333333333333333333333333333`

DecNum() is a constructor that can be used instead of DecNum.new()

## Contexts

Contexts are environments for arithmetic operations. They govern precision, set rules for rounding, determine which signals are treated as exceptions, and limit the range for exponents.

Each thread has an active context that can be accessed like this:

`puts DecNum.context.precision # -> 28`

The active context can be modified globally for the current thread:

```
DecNum.context.precision = 2
puts DecNum.context.precision # -> 2
puts DecNum(1)/DecNum(3) # -> 0.33
DecNum.context.precision += 7
puts DecNum.context.precision # -> 9
puts DecNum(1)/DecNum(3) # -> 0.333333333
```

Or it can be altered locally inside a block:

```
DecNum.context do
DecNum.context.precision = 5
puts DecNum.context.precision # -> 5
end
puts DecNum.context.precision # -> 9
```

The block for a local context can be passed the current context as an argument:

```
DecNum.context do |local_context|
local_context.precision = 5
puts DecNum.context.precision # -> 5
end
puts DecNum.context.precision # -> 9
```

A context object can also be used to define the local context:

```
my_context = DecNum::Context(precision: 20)
DecNum.context(my_context) do |context|
puts context.precision
end # -> 20
```

And individual parameters can be assigned like this:

```
puts DecNum.context.precision # -> 9
puts DecNum.context.rounding # -> half_even
DecNum.context(rounding: :down) do |context|
puts context.precision # -> 9
puts context.rounding # -> down
end
```

Contexts created with the `DecNum::Context()`

constructor inherit from
DecNum::DefaultContext. Default context attributes can be established by
modifying that object:

```
DecNum::DefaultContext.precision = 10
DecNum.context = DecNum::Context(rounding: :half_up)
puts DecNum.context.precision # -> 10
```

Note that a context object assigned to DecNum.context is copied, so it is not altered through DecNum.context:

```
puts my_context.precision # -> 20
DecNum.context = my_context
DecNum.context.precision = 2
puts my_context.precision # -> 20
```

So, DefaultContext is not altered when modifying DecNum.context.

Methods that use a context have an optional parameter to override the active
context (`DecNum.context`

) :

```
DecNum.context.precision = 3
puts DecNum(1).divide(3) # -> 0.333
puts DecNum(1).divide(3, my_context) # -> 0.33333333333333333333
```

Individual context parameters can also be overriden:

`puts DecNum(1).divide(3, precision: 6) # -> 0.333333`

There are two additional predefined contexts `DecNum::ExtendedContext`

and
DecNum::BasicContext that are not meant to be modified; they can be used to
achieve reproducible results. We will use `DecNum::ExtendedContext`

in the
following examples:

`DecNum.context = DecNum::ExtendedContext`

Most decimal operations can be executed by using either Context or DecNum methods:

```
puts DecNum.context.exp(1) # -> 2.71828183
puts DecNum(1).exp # -> 2.71828183
```

If using Context methods, values are automatically converted as if the DecNum() constructor was used.

## Rounding

Results are normally rounded using the precision (number of significant digits) and rounding mode defined in the context.

```
DecNum.context.precision = 4
puts DecNum(1)/DecNum(3) # -> 0.3333
puts DecNum('1E20')-DecNum('1E-20') # -> 1.000E+20
DecNum.context.rounding = :half_up
puts +DecNum('100.05') # -> 100.1
DecNum.context.rounding = :half_even
puts +DecNum('100.05') # -> 100.0
```

Note that input values are not rounded, only results; we use the plus operator to force rounding here:

```
DecNum.context.precision = 4
x = DecNum('123.45678')
puts x # -> 123.45678
puts +x # -> 123.5
```

Precision can be also set to 'exact' to avoid rounding, by using the exact
property or using a 0 precision. In exact mode results are never rounded and
results that have an infinite number of digits trigger the `DecNum::Inexact`

exception.

```
DecNum.context.exact = true
puts DecNum('1E20')-DecNum('1E-20') # -> 99999999999999999999.99999999999999999999
puts DecNum(16).sqrt # -> 4
puts DecNum(16)/DecNum(4) # -> 4
puts DecNum(1)/DecNum(3) # -> Exception : Flt::Num::Inexact
DecNum.context.precision = 5
puts DecNum('1E20')-DecNum('1E-20') # -> 1.0000E+20
puts DecNum(16).sqrt # -> 4
puts DecNum(16)/DecNum(4) # -> 4
puts DecNum(1)/DecNum(3) # -> 0.33333
```

There are also some methods for explicit rounding that provide an interface
compatible with that of the Ruby `Float`

class:

```
puts DecNum('101.5').round # -> 102
puts DecNum('101.5').round(0) # -> 102
puts DecNum('101.12345').round(2) # -> 101.12
puts DecNum('101.12345').round(-1) # -> 1.0E+2
puts DecNum('101.12345').round(places: 2) # -> 101.12
puts DecNum('101.12345').round(precision: 2) # -> 1.0E+2
puts DecNum('101.5').round(rounding: :half_up) # -> 102
puts DecNum('101.5').ceil # -> 102
puts DecNum('101.5').floor # -> 101
puts DecNum('101.5').truncate # -> 101
```

## Special values

In addition to finite numbers, a DecNum object can represent some special values:

- Infinity (+Infinity, -Infinity). The method DecNum#infinite? returns true for these to values. DecNum.infinity DecNum.infinity(-1) can be used to get these values.
- NaN (not a number) represents undefined results. The method DecNum#nan? returns true for it and DecNum.nan can be used to obtain it. There is a variant, sNaN (signaling NaN) that causes an invalid operation condition if used; it can be detected with DecNum.snan?. A NaN can also include diagnostic information in its sign and coefficient.

Any of the special values can be detected with DecNum#special? Finite numbers can be clasified with these methods:

- DecNum#zero? detects a zero value (note that there are two zero values: +0 and -0)
- DecNum#normal? detects normal values: those whose adjusted exponents are not less than @emin@.
- DecNum#subnormal? detects subnormal values: those whose adjusted exponents are less than @emin@.

## Exceptions

Exceptional conditions that may arise during operations have corresponding classes that represent them:

- DecNum::InvalidOperation
- DecNum::DivisionByZero
- DecNum::DivisionImpossible
- DecNum::DivisionUndefined
- DecNum::Inexact
- DecNum::Overflow
- DecNum::Underflow
- DecNum::Clamped
- DecNum::InvalidContext
- DecNum::Rounded
- DecNum::Subnormal
- DecNum::ConversionSyntax

For each condition, a flag and a trap (boolean values) exist in the context. When a condition occurs, the corresponding flag in the context takes the value true (and remains set until cleared) and a exception is raised if the corresponding trap has the value true.

```
DecNum.context.traps[DecNum::DivisionByZero] = false
DecNum.context.flags[DecNum::DivisionByZero] = false
puts DecNum(1)/DecNum(0) # -> Infinity
puts DecNum.context.flags[DecNum::DivisionByZero] # -> true
DecNum.context.traps[DecNum::DivisionByZero] = true
puts DecNum(1)/DecNum(0) # -> Exception : Flt::Num::DivisionByZero
```

## Numerical conversion

By default, `DecNum`

is interoperable with `Integer`

and `Rational`

. Conversion
happens automatically to operands:

```
puts DecNum('0.1') + 1 # -> 1.1
puts 7 + DecNum('0.2') # -> 7.2
puts Rational(5,2) + DecNum('3') # -> 5.5
```

Conversion can also be done explicitly with the DecNum constructor:

```
puts DecNum(7) # -> 7
puts DecNum(Rational(1,10)) # -> 0.1
```

Converting a `DecNum`

to other numerical types can be done with specific
Ruby-style methods. (note the truncated result of `to_i`

)

```
puts DecNum('1.1').to_i # -> 1
puts DecNum('1.1').to_r # -> 11/10
```

Or with a generic method:

```
puts DecNum('1.1').convert_to(Integer) # -> 1
puts DecNum('1.1').convert_to(Rational) # -> 11/10
```

Thera are also GDAS style conversion operations:

`puts DecNum('1.1').to_integral_value # -> 1`

And conversion is also possible to `Float`

:

```
puts DecNum('1.1').to_f # -> 1.1
puts DecNum('1.1').convert_to(Float) # -> 1.1
puts Float(DecNum('1.1')) # -> 1.1
```

Types with predefined bidirectional conversion (`Integer`

and `Rational`

) can be
operated with `DecNum`

on either side of an operator, and the result will be a
`DecNum`

. For `Float`

there is no predefined bidirectional conversion (see below
how to define it) and the result of an operation between `DecNum`

and `Float`

will
be of type `Float`

.

```
puts (DecNum('1.1') + 2.0).class # -> Float
puts (2.0 + DecNum('1.1')).class # -> Float
```

The conversion system is extensible. For example, we can include BigDecimal into it by defining suitable conversion procedures:

```
DecNum.context.define_conversion_from(BigDecimal) do |x, context|
DecNum(x.to_s)
end
DecNum.context.define_conversion_to(BigDecimal) do |x|
BigDecimal(x.to_s)
end
```

Now we can mix `BigDecimals`

and `Decimals`

in expressions and convert from `DecNum`

to `BigDecimal`

:

```
puts BigDecimal('1.1') + DecNum('2.2') # -> 3.3
puts DecNum('1.1').convert_to(BigDecimal) # -> 0.11E1
```

Note that the conversions are defined in a `Context`

object and will be
available only when that context applies. That way we can define conversions
for specific purposes without affecting a program globally.

As another example consider conversion from Float to DecNum, which is not defined by default because it can be defined in different ways depending on the purpose.

A `Float`

constant such as 0.1 defines a `Float`

object which has a numerical
value close to, but not exactly 1/10. When converting that `Float`

to `DecNum`

we
could decide to preserve the exact numerical value of the number or try to
find a simple decimal expression within a given tolerance. If we take the
first approach we can define this conversion:

```
DecNum.context.define_conversion_from(Float) do |x, context|
s,e = Math.frexp(x)
s = Math.ldexp(s, Float::MANT_DIG).to_i
e -= Float::MANT_DIG
DecNum(s*(Float::RADIX**e))
end
```

Note that the conversion we've defined depends on the context precision:

```
DecNum.local_context(precision: 20) { puts DecNum(0.1) } # -> 0.10000000000000000555
DecNum.local_context(precision: 12) { puts DecNum(0.1) } # -> 0.100000000000
DecNum.local_context(exact: true) { puts DecNum(0.1) } # -> 0.1000000000000000055511151231257827021181583404541015625
```

A different approach for Float to DecNum conversion is to find the shortest (fewer digits) DecNum that rounds to the Float with the binary precision that the Float has. We will assume that the DecNum to Float conversion done with the rounding mode of the DecNum context. The BinNum class has a method to perform this kind of conversion, so we will use it.

```
DecNum.context.define_conversion_from(Float) do |x, dec_context|
BinNum.context(:rounding=>dec_context.rounding) do |bin_context|
BinNum(x).to_decimal
end
end
```

The result is independent of the context precision.

```
puts DecNum(0.1) # -> 0.1
puts DecNum(1.0/3) # -> 0.3333333333333333
```

This conversion gives the results expected most of the time, but it must be
noticed that there must be some compromise, because different decimal literals
convert to the same `Float`

value:

`puts DecNum(0.10000000000000001) # -> 0.1`

There's also some uncertainty because the way the Ruby interpreter parses Float literals may not be well specified; in the usual case (IEEE Double Floats and round-to-even) the results will be as expected (correctly rounded Floats), but some platforms may behave differently.

The `BinNum`

also a instance method `to_decimal_exact`

to perform the previous
'exact' conversion, that could have be written:

```
DecNum.context.define_conversion_from(Float) do |x, context|
DecNum.context(context) do
BinNum(x).to_decimal_exact
end
end
```

## Abbreviation

The use of DecNum can be made less verbose by requiring:

`require 'flt/d'`

This file defines `D`

as a synonym for `DecNum`

:

```
D.context.precision = 3
puts +D('1.234') # -> 1.23
```

Some convenient methods are added to numeric classes by requiring the optional flt/sugar.rb. This must be explicitely required because it could cause conflicts with other extensions of these classes.

```
require 'flt/sugar'
puts 34.odd? # -> false
puts 34.even? # -> true
puts 0.1.split.inspect # -> [1, 7205759403792794, -56]
puts (-0.1).sign # -> -1
```

A shortcut notation for DecNum is defined in this file (based on an idea by coderrr which allows exact definitions with almost literal decimal syntax (note the underscore after the dot.)

`puts 10._123456789123456789 # -> 10.123456789123456789`

Additional underscores can be used to separate digits at any place except before the decimal point:

```
puts 100_000._000_001 # -> 100000.000001
puts 100_000._000_001.class # -> Flt::DecNum
```

But note that `100_000.000_001`

is a valid `Float`

(it's not a `DecNum`

because
there isn't an underscore just after the decimal point):

```
puts 100_000.000_001 # -> 100000.000001
puts 100_000.000_001.class # -> Float
```

There's also one important caveat with this notation: negative numbers with a zero integral part must be parenthesed (otherwise the minus has no effect because it affects only the null integer part):

```
puts -0._5 # -> 0.5
puts -(0._5) # -> -0.5
```

## Error analysis

The DecNum#ulp() method returns the value of a "unit in the last place" for a given number under the current context.

```
D.context.precision = 4
puts D('1.5').ulp # -> 0.001
puts D('1.5E10').ulp # -> 1E+7
```

Whe can compute the error in ulps of an approximation `aprx`

to correclty
rounded value `exct`

with:

```
def ulps(exct, aprx)
(aprx-exct).abs/exct.ulp
end
puts ulps(DecNum('0.5000'), DecNum('0.5003')) # -> 3
puts ulps(DecNum('0.5000'), DecNum('0.4997')) # -> 3
puts ulps(DecNum('0.1000'), DecNum('0.1003')) # -> 3E+1
puts ulps(DecNum('0.1000'), DecNum('0.0997')) # -> 3E+1
puts ulps(DecNum(1), DecNum(10).next_minus) # -> 8.999E+4
puts ulps(DecNum(1), DecNum(10).next_plus) # -> 9.01E+4
```

Note that in the definition of ulps we use exct.ulp. If we had use aprx.ulp DecNum(10).next_plus would seem to be a better approximation to DecNum(1) than DecNum(10).next_minus. (Admittedly, such bad approximations should not be common.)

## BinNum Input/Output

BinNum can be defined with a decimal string literal and converted to one with to_s, as DecNum, but since this involves a change of base these are inexact operations subject to some specific precision limits.

If we define the number with a binary literal, no base conversion is involved and the result is exactly defined; here we define a number with just one bit of precision:

```
x = BinNum('0.001', base: 2)
puts x.number_of_digits # -> 1
puts x.to_s(base: 2) # -> 0.001
```

Note that we could have defined it with more precision, e.g.

```
y = BinNum('0.001000', base: 2)
puts y.number_of_digits # -> 4
puts y.to_s(base: 2) # -> 0.001000
```

But let's get back to our one bit quantity, x, and convert it to a decimal string. Since the internal precision is only one bit it contains very little information:

`puts x # -> 0.1`

We can obtain more digits with the :all_digits option which show all the decimal digits that are significative for the given precision of 1 bit:

`puts x.to_s(all_digits: true) # -> 0.12`

We can also obtain the exact value of x by using the Num.convert_exact method to convert it to decimal (base 10):

`puts Num.convert_exact(x,10) # -> 0.125`

Let's convert the default decimal output back to another BinNum which will preserve its precision:

`y = BinNum(x.to_s)`

The result may seem ok:

`puts y # -> 0.1`

But is not exactly what we originally had:

```
puts y==x # -> false
puts y # -> 0.1
puts y.number_of_digits # -> 5
puts y.to_s(base: 2) # -> 0.00011010
```

The new value y has gained some digits because of the intermediate conversion to decimal: one decimal digit contains more information than one bit, and the result shows that.

If we wanted to preserve exactly the number we should have done this:

```
y = BinNum(x.to_s, :fixed, precision: x.number_of_digits)
puts y==x # -> true
```

To preserve the value we had to explicitly determine how many bits should y have.

With the :fixed options the number produced by BinNum is rounded to the context precision (which can be overriden as in the example by other options):

```
puts BinNum(x.to_s, :fixed, precision: 32).to_s(base: 2) # -> 0.00011001100110011001100110011001101
puts BinNum(x.to_s, :fixed, precision: 1).to_s(base: 2) # -> 0.001
```

Note also that if we normalize a value we will change it's precision to that of the context:

```
puts x.number_of_digits # -> 1
puts x.normalize.number_of_digits # -> 53
puts x.normalize.to_s # -> 0.125
```

## Mathematical functions

There are two mathematical functions modules analogous to Ruby's Math for Float, Flt::DecNum::Math and Flt::BinNum::Math. Currently they consist of basic trigonometric functions, including hypot, logarithms and the exponential function, and the constants e and pi.

Its functions can be accessed in a number of ways:

```
require 'flt/math'
DecNum.context(precision: 10) do |context|
# As module functions, using the current context for the enclosing Num class:
puts DecNum::Math.sin(1)*DecNum::Math.pi # -> 2.643559064
# As functions of a context object:
puts context.sin(1)*context.pi # -> 2.643559064
# Through a math block:
puts DecNum.context.math{sin(1)*pi} # -> 2.643559064
puts DecNum.math{sin(1)*pi} # -> 2.643559064
# And can be *included* to be used as private instance methods:
include DecNum::Math
puts sin(1)*pi # -> 2.643559064
end
```

## More Information

- Decimal Floating point type: see the base Flt::Num class and the Flt::DecNum class
- Binary Floating point type: see the base Flt::Num class and the Flt::BinNum class
- Floating Point Contexts: see documentation for classes Flt::Num::ContextBase, Flt::DecNum::Context and Flt::BinNum::Context
- Floating Point Tolerance: see the flt/tolerance.rb file and the Flt::Tolerance class
- Constructors: see Flt.DecNum(), Flt.BinNum() and Flt.Tolerance().
- Trigonometry functions: see Flt::Trigonometry.
- Complex number support: see the flt/complex.rb file

# DecNum vs BigDecimal

DecNum solves some of the difficulties of using BigDecimal.

One of the major problems with BigDecimal is that it's not easy to control the number of significant digits of the results. While addition, subtraction and multiplication are exact (unless a limit is used), divisions will need to be passed precision explicitly or else an indeterminate number of significant digits will be lost. Part of the problem is that numbers don't keep track of its precision (0.1000 is not distinguishable from 0.1.)

With DecNum, Context objects are used to specify the exact number of digits to be used for all operations making the code cleaner and the results more easily predictable.

```
DecNum.context.precision = 10
puts DecNum(1)/DecNum(3)
```

Contexts are thread-safe and can be used for individual operations:

`puts DecNum(1).divide(DecNum(e), DecNum::Context(precision: 4))`

Which can be abbreviated:

`puts DecNum(1).divide(DecNum(e), precision: 4)`

Or use locally in a block without affecting other code:

```
DecNum.context {
DecNum.context.precision = 3
puts DecNum(1)/DecNum(3)
}
puts DecNum.context.precision
```

Which can also be abbreviated:

`DecNum.context(precision: 3) { puts DecNum(1)/DecNum(3) }`

This allows in general to write simpler code; e.g. this is an exponential function, adapted from the 'recipes' in Python's Decimal:

```
def exp(x, c=nil)
i, lasts, s, fact, num = 0, 0, 1, 1, 1
DecNum.context(c) do |context|
context.precision += 2
while s != lasts
lasts = s
i += 1
fact *= i
num *= x
s += num / fact
end
end
return +s
end
```

The final unary + applied to the result forces it to be rounded to the current precision (because we have computed it with two extra digits) The result of this method does not have trailing non-significant digits, as is common with BigDecimal (e.g. in the exp implementation available in the standard Ruby library, in bigdecimal/math)

# Roadmap

- Trigonometry optimizations
- Implement the missing GDA functions: rotate, shift, trim, and, or, xor, invert, max, min, maxmag, minmag, comparetotal, comparetotmag