Class: Flt::Num

Inherits:
Numeric show all
Extended by:
AuxiliarFunctions, Support
Includes:
Comparable, AuxiliarFunctions, Support::AuxiliarFunctions
Defined in:
lib/flt/num.rb,
lib/flt/complex.rb

Overview

ComplexContext

Direct Known Subclasses

BinNum, DecNum

Defined Under Namespace

Modules: AuxiliarFunctions Classes: Clamped, ContextBase, ConversionSyntax, DivisionByZero, DivisionImpossible, DivisionUndefined, Error, Exception, Inexact, InvalidContext, InvalidOperation, Overflow, Rounded, Subnormal, Underflow

Constant Summary collapse

ROUND_HALF_EVEN = 
:half_even
ROUND_HALF_DOWN = 
:half_down
ROUND_HALF_UP = 
:half_up
ROUND_FLOOR = 
:floor
ROUND_CEILING = 
:ceiling
ROUND_DOWN = 
:down
ROUND_UP = 
:up
ROUND_05UP = 
:up05
EXCEPTIONS = 
FlagValues(Clamped, InvalidOperation, DivisionByZero, Inexact, Overflow, Underflow,
Rounded, Subnormal, DivisionImpossible, ConversionSyntax)

Constants included from AuxiliarFunctions

AuxiliarFunctions::EXP_INC, AuxiliarFunctions::LOG10_LB_CORRECTION, AuxiliarFunctions::LOG10_MULT, AuxiliarFunctions::LOG2_LB_CORRECTION, AuxiliarFunctions::LOG2_MULT, AuxiliarFunctions::LOG_PREC_INC, AuxiliarFunctions::LOG_RADIX_EXTRA, AuxiliarFunctions::LOG_RADIX_INC

Constants included from Support::AuxiliarFunctions

Support::AuxiliarFunctions::MAXIMUM_SMALLISH_INTEGER, Support::AuxiliarFunctions::NBITS_BLOCK, Support::AuxiliarFunctions::NBITS_LIMIT, Support::AuxiliarFunctions::NDIGITS_BLOCK, Support::AuxiliarFunctions::NDIGITS_LIMIT

Class Attribute Summary collapse

Class Method Summary collapse

Instance Method Summary collapse

Methods included from Support

FlagValues, adjust_digits, simplified_round_mode

Methods included from AuxiliarFunctions

_convert, _div_nearest, _exp, _iexp, _ilog, _log, _log_radix_digits, _log_radix_lb, _log_radix_mult, _normalize, _number_of_digits, _parser, _power, _rshift_nearest, _sqrt_nearest, log10_lb, log2_lb

Methods included from Support::AuxiliarFunctions

_nbits, _ndigits, detect_float_rounding

Constructor Details

#initialize(*args) ⇒ Num

A floating point-number value can be defined by:

  • A String containing a text representation of the number

  • An Integer

  • A Rational

  • For binary floating point: a Float

  • A Value of a type for which conversion is defined in the context.

  • Another floating-point value of the same type.

  • A sign, coefficient and exponent (either as separate arguments, as an array or as a Hash with symbolic keys), or a signed coefficient and an exponent. This is the internal representation of Num, as returned by Num#split. The sign is +1 for plus and -1 for minus; the coefficient and exponent are integers, except for special values which are defined by :inf, :nan or :snan for the exponent.

An optional Context can be passed after the value-definint argument to override the current context and options can be passed in a last hash argument; alternatively context options can be overriden by options of the hash argument.

When the number is defined by a numeric literal (a String), it can be followed by a symbol that specifies the mode used to convert the literal to a floating-point value:

  • :free is currently the default for all cases. The precision of the input literal (including trailing zeros) is preserved and the precision of the context is ignored. When the literal is in the same base as the floating-point radix, (which, by default, is the case for DecNum only), the literal is preserved exactly in floating-point. Otherwise, all significative digits that can be derived from the literal are generanted, significative meaning here that if the digit is changed and the value converted back to a literal of the same base and precision, the original literal will not be obtained.

  • :short is a variation of :free in which only the minimun number of digits that are necessary to produce the original literal when the value is converted back with the same original precision; namely, given an input in base b1, its :short representation in base 2 is the shortest number in base b2 such that when converted back to base b2 with the same precision that the input had, the result is identical to the input:

    short = Num[b2].new(input, :short, base: b1)
Num[b1].context.precision = precision_of_inpu
Num[b1].new(short.to_s(base: b2), :fixed, base: b2)) == Num[b1].new(input, :free, base: b1)

  • :fixed will round and normalize the value to the precision specified by the context (normalize meaning that exaclty the number of digits specified by the precision will be generated, even if the original literal has fewer digits.) This may fail returning NaN (and raising Inexact) if the context precision is :exact, but not if the floating-point radix is a multiple of the input base.

Options that can be passed for construction from literal:

  • :base is the numeric base of the input, 10 by default.



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# File 'lib/flt/num.rb', line 1485

def initialize(*args)
  options = args.pop if args.last.is_a?(Hash)
  options ||= {}
  context = args.pop if args.size > 0 && (args.last.kind_of?(ContextBase) || args.last.nil?)
  context ||= options.delete(:context)
  mode = args.pop if args.last.is_a?(Symbol) && ![:inf, :nan, :snan].include?(args.last)
  args = args.first if args.size==1 && args.first.is_a?(Array)
  if args.empty? && !options.empty?
    args = [options.delete(:sign)||+1,
            options.delete(:coefficient) || 0,
            options.delete(:exponent) || 0]
  end
  mode ||= options.delete(:mode)
  base = options.delete(:base)
  context = options if context.nil? && !options.empty?
  context = define_context(context)

  case args.size
  when 3
    # internal representation
    @sign, @coeff, @exp = args
    # TO DO: validate

  when 2
    # signed integer and scale
    @coeff, @exp = args
    if @coeff < 0
      @sign = -1
      @coeff = -@coeff
    else
      @sign = +1
    end

  when 1
    arg = args.first
    case arg

    when num_class
      @sign, @coeff, @exp = arg.split

    when *context.coercible_types
      v = context._coerce(arg)
      @sign, @coeff, @exp = v.is_a?(Num) ? v.split : v

    when String
      if arg.strip != arg
        @sign,@coeff,@exp = context.exception(ConversionSyntax, "no trailing or leading whitespace is permitted").split
        return
      end
      m = _parser(arg, :base => base)
      if m.nil?
        @sign,@coeff,@exp = context.exception(ConversionSyntax, "Invalid literal for DecNum: #{arg.inspect}").split
        return
      end
      @sign = (m.sign == '-') ? -1 : +1
      if m.int || m.onlyfrac
        sign = @sign
        if m.int
          intpart = m.int
          fracpart = m.frac
        else
          intpart = ''
          fracpart = m.onlyfrac
        end
        fracpart ||= ''
        base = m.base
        exp = m.exp.to_i
        coeff = (intpart+fracpart).to_i(base)
        if m.exp_base && m.exp_base != base
          # The exponent uses a different base;
          # compute exponent in base; assume base = exp_base**k
          k = Math.log(base, m.exp_base).round
          exp -= fracpart.size*k
          base = m.exp_base
        else
          exp -= fracpart.size
        end

        if false
          # Old behaviour: use :fixed format when num_class.radix != base
          # Advantages:
          # * Behaviour similar to Float: BinFloat(txt) == Float(txt)
          mode ||= ((num_class.radix == base) ? :free : :fixed)
        else
          # New behaviour: the default is always :free
          # Advantages:
          # * Is coherent with construction of DecNum from decimal literal:
          #   preserve precision of the literal with independence of context.
          mode ||= :free
        end

        if mode == :free && base == num_class.radix
          # simple case, the job is already done
          #
          # :free mode with same base must not be handled by the Reader;
          # note that if we used the Reader for the same base case in :free mode,
          # an extra 'significative' digit would be added, because that digit
          # is significative in the sense that (under non-directed rounding,
          # and with the significance interpretation of Reader wit the all-digits option)
          # it's not free to take any value without affecting the value of
          # the other digits: e.g. input: '0.1', the result of :free
          # conversion with the Reader is '0.10' because de last digit is not free;
          # if it was 9 for example the actual value would round to '0.2' with the input
          # precision given here.
          #
          # On the other hand, :short, should be handled by the Reader even when
          # the input and output bases are the same because we want to find the shortest
          # number that can be converted back to the input with the same input precision.
        else
          rounding = context.rounding
          reader = Support::Reader.new(:mode=>mode)
          ans = reader.read(context, rounding, sign, coeff, exp, base)
          context.exception(Inexact,"Inexact decimal to radix #{num_class.radix} conversion") if !reader.exact?
          if !reader.exact? && context.exact?
            sign, coeff, exp =  num_class.nan.split
          else
            sign, coeff, exp = ans.split
          end
        end
        @sign, @coeff, @exp = sign, coeff, exp
      else
        if m.diag
          # NaN
          @coeff = (m.diag.nil? || m.diag.empty?) ? nil : m.diag.to_i
          @coeff = nil if @coeff==0
           if @coeff
             max_diag_len = context.maximum_nan_diagnostic_digits
             if max_diag_len && @coeff >= context.int_radix_power(max_diag_len)
                @sign,@coeff,@exp = context.exception(ConversionSyntax, "diagnostic info too long in NaN").split
               return
             end
           end
          @exp = m.signal ? :snan : :nan
        else
          # Infinity
          @coeff = 0
          @exp = :inf
        end
      end
    else
      raise TypeError, "invalid argument #{arg.inspect}"
    end
  else
    raise ArgumentError, "wrong number of arguments (#{args.size} for 1, 2 or 3)"
  end
end

Class Attribute Details

._base_coercible_types ⇒ Object (readonly)

Returns the value of attribute _base_coercible_types.



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# File 'lib/flt/num.rb', line 173

def _base_coercible_types
  @_base_coercible_types
end

._base_conversions ⇒ Object (readonly)

Returns the value of attribute _base_conversions.



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# File 'lib/flt/num.rb', line 174

def _base_conversions
  @_base_conversions
end

Class Method Details

.[](*args) ⇒ Object

Num can be use to obtain a floating-point numeric class with radix base, so that, for example, Num is equivalent to BinNum and Num to DecNum.

If the base does not correspond to one of the predefined classes (DecNum, BinNum), a new class is dynamically generated.

The [] operator can also be applied to classes derived from Num to act as a constructor (short hand for .new):

Flt::Num[10]['0.1'] # same as FLt::DecNum['0.1'] or Flt.DecNum('0.1') or Flt::DecNum.new('0.1')

Raises:

  • (RuntimeError) 


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# File 'lib/flt/num.rb', line 4734

def [](*args)
  return self.Num(*args) if self!=Num # && self.ancestors.include?(Num)
  raise RuntimeError, "Invalid number of arguments (#{args.size}) for Num.[]; 1 expected." unless args.size==1
  base = args.first

  case base
  when 10
    DecNum
  when 2
    BinNum
  else
    class_name = "Base#{base}Num"
    unless Flt.const_defined?(class_name)
      cls = Flt.const_set class_name, Class.new(Num) {
        def initialize(*args)
          super(*args)
        end
      }
      meta_cls = class <<cls;self;end
      meta_cls.send :define_method, :radix do
        base
      end

      cls.const_set :Context, Class.new(Num::ContextBase)
      cls::Context.send :define_method, :initialize do |*options|
        super(cls, *options)
      end

      default_digits = 10
      default_elimit = 100

      cls.const_set :DefaultContext, cls::Context.new(
        :exact=>false, :precision=>default_digits, :rounding=>:half_even,
        :elimit=>default_elimit,
        :flags=>[],
        :traps=>[DivisionByZero, Overflow, InvalidOperation],
        :ignored_flags=>[],
        :capitals=>true,
        :clamp=>true,
        :angle=>:rad
      )

    end
    Flt.const_get class_name

  end
end

.base_coercible_types ⇒ Object 



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# File 'lib/flt/num.rb', line 175

def base_coercible_types
  Num._base_coercible_types
end

.base_conversions ⇒ Object 



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# File 'lib/flt/num.rb', line 178

def base_conversions
  Num._base_conversions
end

.ccontext(*args) ⇒ Object 



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# File 'lib/flt/complex.rb', line 275

def self.ccontext(*args)
  ComplexContext(self.context(*args))
end

.Context(*args) ⇒ Object

Context constructor; if an options hash is passed, the options are applied to the default context; if a Context is passed as the first argument, it is used as the base instead of the default context.

Note that this method should be called on concrete floating point types such as Flt::DecNum and Flt::BinNum, and not in the abstract base class Flt::Num.

See Flt::Num::ContextBase#new() for the valid options



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# File 'lib/flt/num.rb', line 1278

def self.Context(*args)
  case args.size
    when 0
      base = self::DefaultContext
    when 1
      arg = args.first
      if arg.instance_of?(self::Context)
        base = arg
        options = nil
      elsif arg.instance_of?(Hash)
        base = self::DefaultContext
        options = arg
      else
        raise TypeError,"invalid argument for #{num_class}.Context"
      end
    when 2
      base = args.first
      options = args.last
    else
      raise ArgumentError,"wrong number of arguments (#{args.size} for 0, 1 or 2)"
  end

  if options.nil? || options.empty?
    base
  else
    self::Context.new(base, options)
  end

end

.context(*args, &blk) ⇒ Object

The current context (thread-local). If arguments are passed they are interpreted as in Num.define_context() and an altered copy of the current context is returned. If a block is given, this method is a synonym for Num.local_context().



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# File 'lib/flt/num.rb', line 1332

def self.context(*args, &blk)
  if blk
    # setup a local context
    local_context(*args, &blk)
  elsif args.empty?
    # return the current context
    ctxt = self._context
    self._context = ctxt = self::DefaultContext.dup if ctxt.nil?
    ctxt
  else
    # Return a modified copy of the current context
    if args.first.kind_of?(ContextBase)
      self.define_context(*args)
    else
      self.define_context(self.context, *args)
    end
  end
end

.context=(c) ⇒ Object

Change the current context (thread-local).



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# File 'lib/flt/num.rb', line 1352

def self.context=(c)
  self._context = c.dup
end

.define_context(*options) ⇒ Object

Define a context by passing either of:

  • A Context object (of the same type)

  • A hash of options (or nothing) to alter a copy of the current context.

  • A Context object and a hash of options to alter a copy of it



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# File 'lib/flt/num.rb', line 1312

def self.define_context(*options)
  context = options.shift if options.first.instance_of?(self::Context)
  if context && options.empty?
    context
  else
    context ||= self.context
    self.Context(context, *options)
  end
end

.Flags(*values) ⇒ Object 



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# File 'lib/flt/num.rb', line 396

def self.Flags(*values)
  Flt::Support::Flags(EXCEPTIONS,*values)
end

.infinity(sign = +1) ⇒ Object

A floating-point infinite number with the specified sign



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# File 'lib/flt/num.rb', line 1412

def infinity(sign=+1)
  new [sign, 0, :inf]
end

.int_div_radix_power(x, n) ⇒ Object 



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# File 'lib/flt/num.rb', line 1434

def int_div_radix_power(x,n)
  n < 0 ? (x * self.radix**(-n) ) : (x / self.radix**n)
end

.int_mult_radix_power(x, n) ⇒ Object 



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# File 'lib/flt/num.rb', line 1430

def int_mult_radix_power(x,n)
  n < 0 ? (x / self.radix**(-n)) : (x * self.radix**n)
end

.int_radix_power(n) ⇒ Object 



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# File 'lib/flt/num.rb', line 1426

def int_radix_power(n)
  self.radix**n
end

.local_context(*args) ⇒ Object

Defines a scope with a local context. A context can be passed which will be set a the current context for the scope; also a hash can be passed with options to apply to the local scope. Changes done to the current context are reversed when the scope is exited.



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# File 'lib/flt/num.rb', line 1365

def self.local_context(*args)
  begin
    keep = self.context # use this so _context is initialized if necessary
    self.context = define_context(*args) # this dups the assigned context
    result = yield _context
  ensure
    # TODO: consider the convenience of copying the flags from DecNum.context to keep
    # This way a local context does not affect the settings of the previous context,
    # but flags are transferred.
    # (this could be done always or be controlled by some option)
    #   keep.flags = DecNum.context.flags
    # Another alternative to consider: logically or the flags:
    #   keep.flags ||= DecNum.context.flags # (this requires implementing || in Flags)
    self._context = keep
    result
  end
end

.math(*args, &blk) ⇒ Object 



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# File 'lib/flt/num.rb', line 1438

def math(*args, &blk)
  self.context.math(*args, &blk)
end

.nan ⇒ Object

A floating-point NaN (not a number)



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# File 'lib/flt/num.rb', line 1417

def nan()
  new [+1, nil, :nan]
end

.Num(*args) ⇒ Object

Num is the general constructor that can be invoked on specific Flt::Num-derived classes.



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# File 'lib/flt/num.rb', line 1640

def Num(*args)
  if args.size==1 && args.first.instance_of?(self)
    args.first
  else
    new(*args)
  end
end

.num_class ⇒ Object 



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# File 'lib/flt/num.rb', line 1400

def num_class
  self
end

.one_half ⇒ Object

One half: 1/2



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# File 'lib/flt/num.rb', line 1422

def one_half
  new '0.5'
end

.set_context(*args) ⇒ Object

Modify the current context, e.g. DecNum.set_context(:precision=>10)



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# File 'lib/flt/num.rb', line 1357

def self.set_context(*args)
  self.context = define_context(*args)
end

.zero(sign = +1) ⇒ Object

A floating-point number with value zero and the specified sign



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# File 'lib/flt/num.rb', line 1407

def zero(sign=+1)
  new [sign, 0, 0]
end

Instance Method Details

#%(other, context = nil) ⇒ Object

Modulo of two decimal numbers



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# File 'lib/flt/num.rb', line 1793

def %(other, context=nil)
  _bin_op :%, :modulo, other, context
end

#*(other, context = nil) ⇒ Object

Multiplication of two decimal numbers



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# File 'lib/flt/num.rb', line 1783

def *(other, context=nil)
  _bin_op :*, :multiply, other, context
end

#**(other, context = nil) ⇒ Object

Power



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# File 'lib/flt/num.rb', line 1798

def **(other, context=nil)
  _bin_op :**, :power, other, context
end

#+(other, context = nil) ⇒ Object

Addition of two decimal numbers



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# File 'lib/flt/num.rb', line 1773

def +(other, context=nil)
  _bin_op :+, :add, other, context
end

#+@(context = nil) ⇒ Object

Unary plus operator



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# File 'lib/flt/num.rb', line 1767

def +@(context=nil)
  #(context || num_class.context).plus(self)
  _pos(context)
end

#-(other, context = nil) ⇒ Object

Subtraction of two decimal numbers



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# File 'lib/flt/num.rb', line 1778

def -(other, context=nil)
  _bin_op :-, :subtract, other, context
end

#-@(context = nil) ⇒ Object

Unary minus operator



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# File 'lib/flt/num.rb', line 1761

def -@(context=nil)
  #(context || num_class.context).minus(self)
  _neg(context)
end

#/(other, context = nil) ⇒ Object

Division of two decimal numbers



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# File 'lib/flt/num.rb', line 1788

def /(other, context=nil)
  _bin_op :/, :divide, other, context
end

#<(other) ⇒ Object 



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# File 'lib/flt/num.rb', line 2839

def <(other)
  (self<=>other) < 0
end

#<=(other) ⇒ Object

For MRI this is unnecesary, but it is needed for Rubinius because of the coercion done in Numeric#< etc.



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# File 'lib/flt/num.rb', line 2836

def <=(other)
  (self<=>other) <= 0
end

#<=>(other) ⇒ Object

Internal comparison operator: returns -1 if the first number is less than the second, 0 if both are equal or +1 if the first is greater than the secong.



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# File 'lib/flt/num.rb', line 2778

def <=>(other)
  case other
  when *num_class.context.coercible_types_or_num
    other = Num(other)
    if self.special? || other.special?
      if self.nan? || other.nan?
        1
      else
        self_v = self.finite? ? 0 : self.sign
        other_v = other.finite? ? 0 : other.sign
        self_v <=> other_v
      end
    else
      if self.zero?
        if other.zero?
          0
        else
          -other.sign
        end
      elsif other.zero?
        self.sign
      elsif other.sign < self.sign
        +1
      elsif self.sign < other.sign
        -1
      else
        self_adjusted = self.adjusted_exponent
        other_adjusted = other.adjusted_exponent
        if self_adjusted == other_adjusted
          self_padded,other_padded = self.coefficient,other.coefficient
          d = self.exponent - other.exponent
          if d>0
            self_padded *= num_class.int_radix_power(d)
          else
            other_padded *= num_class.int_radix_power(-d)
          end
          (self_padded <=> other_padded)*self.sign
        elsif self_adjusted > other_adjusted
          self.sign
        else
          -self.sign
        end
      end
    end
  else
    if !self.nan? && defined? other.coerce
      x, y = other.coerce(self)
      x <=> y
    else
      nil
    end
  end
end

#==(other) ⇒ Object 



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# File 'lib/flt/num.rb', line 2831

def ==(other)
  (self<=>other) == 0
end

#>(other) ⇒ Object 



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# File 'lib/flt/num.rb', line 2845

def >(other)
  (self<=>other) > 0
end

#>=(other) ⇒ Object 



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# File 'lib/flt/num.rb', line 2842

def >=(other)
  (self<=>other) >= 0
end

#_abs(round = true, context = nil) ⇒ Object

Returns a copy with positive sign



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# File 'lib/flt/num.rb', line 3551

def _abs(round=true, context=nil)
  return copy_abs if not round

  if special?
    ans = _check_nans(context)
    return ans if ans
  end
  if sign>0
    ans = _neg(context)
  else
    ans = _pos(context)
  end
  ans
end

#_check_nans(context = nil, other = nil) ⇒ Object

Check if the number or other is NaN, signal if sNaN or return NaN; return nil if none is NaN.



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# File 'lib/flt/num.rb', line 3447

def _check_nans(context=nil, other=nil)
  #self_is_nan = self.nan?
  #other_is_nan = other.nil? ? false : other.nan?
  if self.nan? || (other && other.nan?)
    context = define_context(context)
    return context.exception(InvalidOperation, 'sNaN', self) if self.snan?
    return context.exception(InvalidOperation, 'sNaN', other) if other && other.snan?
    return self._fix_nan(context) if self.nan?
    return other._fix_nan(context)
  else
    return nil
  end
end

#_fix(context) ⇒ Object

Round if it is necessary to keep within precision.



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# File 'lib/flt/num.rb', line 3567

def _fix(context)
  return self if context.exact?

  if special?
    if nan?
      return _fix_nan(context)
    else
      return Num(self)
    end
  end

  etiny = context.etiny
  etop  = context.etop
  if zero?
    exp_max = context.clamp? ? etop : context.emax
    new_exp = [[@exp, etiny].max, exp_max].min
    if new_exp!=@exp
      context.exception Clamped
      return Num(sign,0,new_exp)
    else
      return Num(self)
    end
  end

  nd = number_of_digits
  exp_min = nd + @exp - context.precision
  if exp_min > etop
    context.exception Inexact
    context.exception Rounded
    return context.exception(Overflow, 'above Emax', sign)
  end

  self_is_subnormal = exp_min < etiny

  if self_is_subnormal
    context.exception Subnormal
    exp_min = etiny
  end

  if @exp < exp_min
    context.exception Rounded
    # dig is the digits number from 0 (MS) to number_of_digits-1 (LS)
    # dg = numberof_digits-dig is from 1 (LS) to number_of_digits (MS)
    dg = exp_min - @exp # dig = number_of_digits + exp - exp_min
    if dg > number_of_digits # dig<0
      d = Num(sign,1,exp_min-1)
      dg = number_of_digits # dig = 0
    else
      d = Num(self)
    end
    changed = d._round(context.rounding, dg)
    coeff = num_class.int_div_radix_power(d.coefficient, dg)
    coeff += 1 if changed==1
    ans = Num(sign, coeff, exp_min)
    if changed!=0
      context.exception Inexact
      if self_is_subnormal
        context.exception Underflow
        if ans.zero?
          context.exception Clamped
        end
      elsif ans.number_of_digits == context.precision+1
        if ans.exponent< etop
          ans = Num(ans.sign, num_class.int_div_radix_power(ans.coefficient,1), ans.exponent+1)
        else
          ans = context.exception(Overflow, 'above Emax', d.sign)
        end
      end
    end
    return ans
  end

  if context.normalized?
    exp = @exp
    coeff = @coeff
    if self_is_subnormal
      if exp > context.etiny
        coeff = num_class.int_mult_radix_power(coeff, exp - context.etiny)
        exp = context.etiny
      end
    else
      min_normal_coeff = context.minimum_normalized_coefficient
      while coeff < min_normal_coeff
        coeff = num_class.int_mult_radix_power(coeff, 1)
        exp -= 1
      end
    end
    if exp != @exp || coeff != @coeff
      return Num(@sign, coeff, exp)
    end
  end

  if context.clamp? &&  @exp>etop
    context.exception Clamped
    self_padded = num_class.int_mult_radix_power(@coeff, @exp-etop)
    return Num(sign,self_padded,etop)
  end

  return Num(self)
end

#_fix_nan(context) ⇒ Object

adjust payload of a NaN to the context



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# File 'lib/flt/num.rb', line 3669

def _fix_nan(context)
  if  !context.exact?
    payload = @coeff
    payload = nil if payload==0

    max_payload_len = context.maximum_nan_diagnostic_digits

    if number_of_digits > max_payload_len
        payload = payload.to_s[-max_payload_len..-1].to_i
        return num_class.Num([@sign, payload, @exp])
    end
  end
  Num(self)
end

#_neg(context = nil) ⇒ Object

Returns copy with sign inverted



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# File 'lib/flt/num.rb', line 3521

def _neg(context=nil)
  if special?
    ans = _check_nans(context)
    return ans if ans
  end
  if zero?
    ans = copy_abs
  else
    ans = copy_negate
  end
  context = define_context(context)
  ans._fix(context)
end

#_pos(context = nil) ⇒ Object

Returns a copy with precision adjusted



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# File 'lib/flt/num.rb', line 3536

def _pos(context=nil)
  if special?
    ans = _check_nans(context)
    return ans if ans
  end
  if zero?
    ans = copy_abs
  else
    ans = Num(self)
  end
  context = define_context(context)
  ans._fix(context)
end

#_rescale(exp, rounding) ⇒ Object

Rescale so that the exponent is exp, either by padding with zeros or by truncating digits, using the given rounding mode.

Specials are returned without change. This operation is quiet: it raises no flags, and uses no information from the context.

exp = exp to scale to (an integer) rounding = rounding mode



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# File 'lib/flt/num.rb', line 3470

def _rescale(exp, rounding)

  return Num(self) if special?
  return Num(sign, 0, exp) if zero?
  return Num(sign, @coeff*num_class.int_radix_power(self.exponent - exp), exp) if self.exponent > exp
  #nd = number_of_digits + self.exponent - exp
  nd = exp - self.exponent
  if number_of_digits < nd
    slf = Num(sign, 1, exp-1)
    nd = number_of_digits
  else
    slf = num_class.new(self)
  end

  changed = slf._round(rounding, nd)
  coeff = num_class.int_div_radix_power(@coeff, nd)
  coeff += 1 if changed==1
  Num(slf.sign, coeff, exp)

end

#_watched_rescale(exp, context, watch_exp) ⇒ Object 



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# File 'lib/flt/num.rb', line 3491

def _watched_rescale(exp, context, watch_exp)
  if !watch_exp
    ans = _rescale(exp, context.rounding)
    context.exception(Rounded) if ans.exponent > self.exponent
    context.exception(Inexact) if ans != self
    return ans
  end

  if exp < context.etiny || exp > context.emax
    return context.exception(InvalidOperation, "target operation out of bounds in quantize/rescale")
  end

  return Num(@sign, 0, exp)._fix(context) if zero?

  self_adjusted = adjusted_exponent
  return context.exception(InvalidOperation,"exponent of quantize/rescale result too large for current context") if self_adjusted > context.emax
  return context.exception(InvalidOperation,"quantize/rescale has too many digits for current context") if (self_adjusted - exp + 1 > context.precision) && !context.exact?

  ans = _rescale(exp, context.rounding)
  return context.exception(InvalidOperation,"exponent of rescale result too large for current context") if ans.adjusted_exponent > context.emax
  return context.exception(InvalidOperation,"rescale result has too many digits for current context") if (ans.number_of_digits > context.precision) && !context.exact?
  if ans.exponent > self.exponent
    context.exception(Rounded)
    context.exception(Inexact) if ans!=self
  end
  context.exception(Subnormal) if !ans.zero? && (ans.adjusted_exponent < context.emin)
  return ans._fix(context)
end

#abs(context = nil) ⇒ Object

Absolute value



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# File 'lib/flt/num.rb', line 2038

def abs(context=nil)
  if special?
    ans = _check_nans(context)
    return ans if ans
  end
  sign<0 ? _neg(context) : _pos(context)
end

#add(other, context = nil) ⇒ Object

Addition



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# File 'lib/flt/num.rb', line 1803

def add(other, context=nil)

  context = define_context(context)
  other = _convert(other)

  if self.special? || other.special?
    ans = _check_nans(context,other)
    return ans if ans

    if self.infinite?
      if self.sign != other.sign && other.infinite?
        return context.exception(InvalidOperation, '-INF + INF')
      end
      return Num(self)
    end

    return Num(other) if other.infinite?
  end

  exp = [self.exponent, other.exponent].min
  negativezero = (context.rounding == ROUND_FLOOR && self.sign != other.sign)

  if self.zero? && other.zero?
    sign = [self.sign, other.sign].max
    sign = -1 if negativezero
    ans = Num([sign, 0, exp])._fix(context)
    return ans
  end

  if self.zero?
    exp = [exp, other.exponent - context.precision - 1].max unless context.exact?
    return other._rescale(exp, context.rounding)._fix(context)
  end

  if other.zero?
    exp = [exp, self.exponent - context.precision - 1].max unless context.exact?
    return self._rescale(exp, context.rounding)._fix(context)
  end

  op1, op2 = _normalize(self, other, context.precision)

  result_sign = result_coeff = result_exp = nil
  if op1.sign != op2.sign
    return ans = Num(negativezero ? -1 : +1, 0, exp)._fix(context) if op1.coefficient == op2.coefficient
    op1,op2 = op2,op1 if op1.coefficient < op2.coefficient
    result_sign = op1.sign
    op1,op2 = op1.copy_negate, op2.copy_negate if result_sign < 0
  elsif op1.sign < 0
    result_sign = -1
    op1,op2 = op1.copy_negate, op2.copy_negate
  else
    result_sign = +1
  end

  if op2.sign == +1
    result_coeff = op1.coefficient + op2.coefficient
  else
    result_coeff = op1.coefficient - op2.coefficient
  end

  result_exp = op1.exponent

  return Num(result_sign, result_coeff, result_exp)._fix(context)

end

#adjusted_exponent ⇒ Object

Exponent of the magnitude of the most significant digit of the operand



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# File 'lib/flt/num.rb', line 2875

def adjusted_exponent
  if special?
    0
  else
    @exp + number_of_digits - 1
  end
end

#ceil(opt = {}) ⇒ Object

General ceiling operation (as for Float) with same options for precision as Flt::Num#round()



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# File 'lib/flt/num.rb', line 3192

def ceil(opt={})
  opt[:rounding] = :ceiling
  round opt
end

#coefficient ⇒ Object

Significand as an integer, unsigned



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# File 'lib/flt/num.rb', line 2927

def coefficient
  @coeff
end

#coerce(other) ⇒ Object

Used internally to convert numbers to be used in an operation to a suitable numeric type



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# File 'lib/flt/num.rb', line 1736

def coerce(other)
  case other
    when *num_class.context.coercible_types_or_num
      [Num(other),self]
    when Float
      [other, self.to_f]
    else
      super
  end
end

#compare(other, context = nil) ⇒ Object

Compares like <=> but returns a Num value.



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# File 'lib/flt/num.rb', line 2861

def compare(other, context=nil)

  other = _convert(other)

  if self.special? || other.special?
    ans = _check_nans(context, other)
    return ans if ans
  end

  return Num(self <=> other)

end

#convert_to(type, context = nil) ⇒ Object

Convert to other numerical type.



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# File 'lib/flt/num.rb', line 2649

def convert_to(type, context=nil)
  context = define_context(context)
  context.convert_to(type, self)
end

#copy_abs ⇒ Object

Returns a copy of with the sign set to +



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# File 'lib/flt/num.rb', line 2961

def copy_abs
  Num(+1,@coeff,@exp)
end

#copy_negate ⇒ Object

Returns a copy of with the sign inverted



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# File 'lib/flt/num.rb', line 2966

def copy_negate
  Num(-@sign,@coeff,@exp)
end

#copy_sign(other) ⇒ Object

Returns a copy of with the sign of other



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# File 'lib/flt/num.rb', line 2971

def copy_sign(other)
  sign = other.respond_to?(:sign) ? other.sign : ((other < 0) ? -1 : +1)
  Num(sign, @coeff, @exp)
end

#digits ⇒ Object 



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# File 'lib/flt/num.rb', line 2901

def digits
  @coeff.is_a?(Integer) ? @coeff.digits(num_class.radix).reverse! : []
end

#div(other, context = nil) ⇒ Object

Ruby-style integer division: (x/y).floor



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# File 'lib/flt/num.rb', line 2250

def div(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return [ans,ans] if ans

  sign = self.sign * other.sign

  if self.infinite?
    return context.exception(InvalidOperation, 'INF // INF') if other.infinite?
    return num_class.infinity(sign)
  end

  if other.zero?
    if self.zero?
      return context.exception(DivisionUndefined, '0 // 0')
    else
      return context.exception(DivisionByZero, 'x // 0', sign)
    end
  end
  return self._divide_floor(other, context).first
end

#divide(other, context = nil) ⇒ Object

Division



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# File 'lib/flt/num.rb', line 1912

def divide(other, context=nil)
  context = define_context(context)
  other = _convert(other)
  resultsign = self.sign * other.sign
  if self.special? || other.special?
    ans = _check_nans(context,other)
    return ans if ans
    if self.infinite?
      return context.exception(InvalidOperation,"(+-)INF/(+-)INF") if other.infinite?
      return num_class.infinity(resultsign)
    end
    if other.infinite?
      context.exception(Clamped,"Division by infinity")
      return num_class.new([resultsign, 0, context.etiny])
    end
  end

  if other.zero?
    return context.exception(DivisionUndefined, '0 / 0') if self.zero?
    return context.exception(DivisionByZero, 'x / 0', resultsign)
  end

  if self.zero?
    exp = self.exponent - other.exponent
    coeff = 0
  else
    prec = context.exact? ? self.number_of_digits + 4*other.number_of_digits : context.precision
    shift = other.number_of_digits - self.number_of_digits + prec
    shift += 1
    exp = self.exponent - other.exponent - shift
    if shift >= 0
      coeff, remainder = (self.coefficient*num_class.int_radix_power(shift)).divmod(other.coefficient)
    else
      coeff, remainder = self.coefficient.divmod(other.coefficient*num_class.int_radix_power(-shift))
    end
    if remainder != 0
      return context.exception(Inexact) if context.exact?
      # result is not exact; adjust to ensure correct rounding
      if num_class.radix == 10
        # perform 05up rounding so the the final rounding will be correct
        coeff += 1 if (coeff%5) == 0
      else
        # since we will round to less digits and there is a remainder, we just need
        # to append some nonzero digit; but we must avoid producing a tie (adding a single
        # digit whose value is radix/2), so we append two digits, 01, that will be rounded away
        coeff = num_class.int_mult_radix_power(coeff, 2) + 1
        exp -= 2
      end
    else
      # result is exact; get as close to idaal exponent as possible
      ideal_exp = self.exponent - other.exponent
      while (exp < ideal_exp) && ((coeff % num_class.radix)==0)
        coeff /= num_class.radix
        exp += 1
      end
    end

  end
  return Num(resultsign, coeff, exp)._fix(context)

end

#divide_int(other, context = nil) ⇒ Object

General Decimal Arithmetic Specification integer division: (x/y).truncate



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# File 'lib/flt/num.rb', line 2225

def divide_int(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return ans if ans

  sign = self.sign * other.sign

  if self.infinite?
    return context.exception(InvalidOperation, 'INF // INF') if other.infinite?
    return num_class.infinity(sign)
  end

  if other.zero?
    if self.zero?
      return context.exception(DivisionUndefined, '0 // 0')
    else
      return context.exception(DivisionByZero, 'x // 0', sign)
    end
  end
  return self._divide_truncate(other, context).first
end

#divmod(other, context = nil) ⇒ Object

Ruby-style integer division and modulo: (x/y).floor, x - y*(x/y).floor



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# File 'lib/flt/num.rb', line 2191

def divmod(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return [ans,ans] if ans

  sign = self.sign * other.sign

  if self.infinite?
    if other.infinite?
      ans = context.exception(InvalidOperation, 'divmod(INF,INF)')
      return [ans,ans]
    else
      return [num_class.infinity(sign), context.exception(InvalidOperation, 'INF % x')]
    end
  end

  if other.zero?
    if self.zero?
      ans = context.exception(DivisionUndefined, 'divmod(0,0)')
      return [ans,ans]
    else
      return [context.exception(DivisionByZero, 'x // 0', sign),
               context.exception(InvalidOperation, 'x % 0')]
    end
  end

  quotient, remainder = self._divide_floor(other, context)
  return [quotient, remainder._fix(context)]
end

#divrem(other, context = nil) ⇒ Object

General Decimal Arithmetic Specification integer division and remainder:

(x/y).truncate, x - y*(x/y).truncate



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# File 'lib/flt/num.rb', line 2158

def divrem(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return [ans,ans] if ans

  sign = self.sign * other.sign

  if self.infinite?
    if other.infinite?
      ans = context.exception(InvalidOperation, 'divmod(INF,INF)')
      return [ans,ans]
    else
      return [num_class.infinity(sign), context.exception(InvalidOperation, 'INF % x')]
    end
  end

  if other.zero?
    if self.zero?
      ans = context.exception(DivisionUndefined, 'divmod(0,0)')
      return [ans,ans]
    else
      return [context.exception(DivisionByZero, 'x // 0', sign),
               context.exception(InvalidOperation, 'x % 0')]
    end
  end

  quotient, remainder = self._divide_truncate(other, context)
  return [quotient, remainder._fix(context)]
end

#eql?(other) ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 2855

def eql?(other)
  return false unless other.is_a?(num_class)
  reduce.split == other.reduce.split
end

#even? ⇒ Boolean

returns true if is an even integer

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 2995

def even?
  # integral? && ((to_i%2)==0)
  if finite?
    if @exp>0 || @coeff==0
      true
    else
      if @exp <= -number_of_digits
        false
      else
        m = num_class.int_radix_power(-@exp)
        if (@coeff % m) == 0
          # ((@coeff / m) % 2) == 0
          ((@coeff / m) & 1) == 0
        else
          false
        end
      end
    end
  else
    false
  end
end

#exp(context = nil) ⇒ Object

Exponential function



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# File 'lib/flt/num.rb', line 2494

def exp(context=nil)
  context = num_class.define_context(context)

  # exp(NaN) = NaN
  ans = _check_nans(context)
  return ans if ans

  # exp(-Infinity) = 0
  return num_class.zero if self.infinite? && (self.sign == -1)

  # exp(0) = 1
  return Num(1) if self.zero?

  # exp(Infinity) = Infinity
  return Num(self) if self.infinite?

  # the result is now guaranteed to be inexact (the true
  # mathematical result is transcendental). There's no need to
  # raise Rounded and Inexact here---they'll always be raised as
  # a result of the call to _fix.
  return context.exception(Inexact, 'Inexact exp') if context.exact?
  p = context.precision
  adj = self.adjusted_exponent

  if self.sign == +1 and adj > _number_of_digits((context.emax+1)*3)
    # overflow
    ans = Num(+1, 1, context.emax+1)
  elsif self.sign == -1 and adj > _number_of_digits((-context.etiny+1)*3)
    # underflow to 0
    ans = Num(+1, 1, context.etiny-1)
  elsif self.sign == +1 and adj < -p
    # p+1 digits; final round will raise correct flags
    ans = Num(+1, num_clas.int_radix_power(p)+1, -p)
  elsif self.sign == -1 and adj < -p-1
    # p+1 digits; final round will raise correct flags
    ans = Num(+1, num_clas.int_radix_power(p+1)-1, -p-1)
  else
    # general case
    x_sign = self.sign
    x = self.copy_sign(+1)
    i, lasts, s, fact, num = 0, 0, 1, 1, 1
    elim = [context.emax, -context.emin, 10000].max
    xprec = num_class.radix==10 ? 3 : 4
    num_class.local_context(context, :extra_precision=>xprec, :rounding=>:half_even, :elimit=>elim) do
      while s != lasts
        lasts = s
        i += 1
        fact *= i
        num *= x
        s += num / fact
      end
      s = num_class.Num(1)/s if x_sign<0
    end
    ans = s
  end

  # at this stage, ans should round correctly with *any*
  # rounding mode, not just with ROUND_HALF_EVEN
  num_class.context(context, :rounding=>:half_even) do |local_context|
    ans = ans._fix(local_context)
    context.flags = local_context.flags
  end

  return ans
end

#exponent ⇒ Object

Exponent of the significand as an integer.



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# File 'lib/flt/num.rb', line 2932

def exponent
  @exp
end

#finite? ⇒ Boolean

Returns whether the number is finite

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1684

def finite?
  !special?
end

#floor(opt = {}) ⇒ Object

General floor operation (as for Float) with same options for precision as Flt::Num#round()



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# File 'lib/flt/num.rb', line 3199

def floor(opt={})
  opt[:rounding] = :floor
  round opt
end

#fma(other, third, context = nil) ⇒ Object

Fused multiply-add.

Computes (self*other+third) with no rounding of the intermediate product self*other.



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# File 'lib/flt/num.rb', line 3214

def fma(other, third, context=nil)
  context =define_context(context)
  other = _convert(other)
  third = _convert(third)
  if self.special? || other.special?
    return context.exception(InvalidOperation, 'sNaN', self) if self.snan?
    return context.exception(InvalidOperation, 'sNaN', other) if other.snan?
    if self.nan?
      product = self
    elsif other.nan?
      product = other
    elsif self.infinite?
      return context.exception(InvalidOperation, 'INF * 0 in fma') if other.zero?
      product = num_class.infinity(self.sign*other.sign)
    elsif other.infinite?
      return context.exception(InvalidOperation, '0 * INF  in fma') if self.zero?
      product = num_class.infinity(self.sign*other.sign)
    end
  else
    product = Num(self.sign*other.sign,self.coefficient*other.coefficient, self.exponent+other.exponent)
  end
  return product.add(third, context)
end

#fraction_part ⇒ Object

Fraction part (as a Num)



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# File 'lib/flt/num.rb', line 2945

def fraction_part
  ans = _check_nans
  return ans if ans
  self - self.integer_part
end

#fractional_exponent ⇒ Object

Exponent as though the significand were a fraction (the decimal point before its first digit)



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# File 'lib/flt/num.rb', line 2889

def fractional_exponent
  scientific_exponent + 1
end

#hash ⇒ Object 



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# File 'lib/flt/num.rb', line 2851

def hash
  ([num_class]+reduce.split).hash # TODO: optimize
end

#infinite? ⇒ Boolean

Returns whether the number is infinite

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1679

def infinite?
  @exp == :inf
end

#inspect ⇒ Object 



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# File 'lib/flt/num.rb', line 2767

def inspect
  class_name = num_class.to_s.split('::').last
  if $DEBUG
    "#{class_name}('#{self}') [coeff:#{@coeff.inspect} exp:#{@exp.inspect} s:#{@sign.inspect} radix:#{num_class.radix}]"
  else
    "#{class_name}('#{self}')"
  end
end

#integer_part ⇒ Object

Integer part (as a Num)



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# File 'lib/flt/num.rb', line 2937

def integer_part
  ans = _check_nans
  return ans if ans
  return_as_num = {:places=>0}
  self.sign < 0 ? self.ceil(return_as_num) : self.floor(return_as_num)
end

#integral? ⇒ Boolean

Returns true if the value is an integer

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 2977

def integral?
  if finite?
    if @exp>=0 || @coeff==0
      true
    else
      if @exp <= -number_of_digits
        false
      else
        m = num_class.int_radix_power(-@exp)
        (@coeff % m) == 0
      end
    end
  else
    false
  end
end

#integral_exponent ⇒ Object

Exponent of the significand as an integer. Synonym of exponent



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# File 'lib/flt/num.rb', line 2916

def integral_exponent
  # fractional_exponent - number_of_digits
  @exp
end

#integral_significand ⇒ Object

Significand as an integer, unsigned. Synonym of coefficient



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# File 'lib/flt/num.rb', line 2911

def integral_significand
  @coeff
end

#ln(context = nil) ⇒ Object

Returns the natural (base e) logarithm



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# File 'lib/flt/num.rb', line 2561

def ln(context=nil)
  context = num_class.define_context(context)

  # ln(NaN) = NaN
  ans = _check_nans(context)
  return ans if ans

  # ln(0.0) == -Infinity
  return num_class.infinity(-1) if self.zero?

  # ln(Infinity) = Infinity
  return num_class.infinity if self.infinite? && self.sign == +1

  # ln(1.0) == 0.0
  return num_class.zero if self == Num(1)

  # ln(negative) raises InvalidOperation
  return context.exception(InvalidOperation, 'ln of a negative value') if self.sign==-1

  # result is irrational, so necessarily inexact
  return context.exception(Inexact, 'Inexact exp') if context.exact?

  elim = [context.emax, -context.emin, 10000].max
  xprec = num_class.radix==10 ? 3 : 4
  num_class.local_context(context, :extra_precision=>xprec, :rounding=>:half_even, :elimit=>elim) do

    one = num_class.Num(1)

    x = self
    if (expo = x.adjusted_exponent)<-1 || expo>=2
      x = x.scaleb(-expo)
    else
      expo = nil
    end

    x = (x-one)/(x+one)
    x2 = x*x
    ans = x
    d = ans
    i = one
    last_ans = nil
    while ans != last_ans
      last_ans = ans
      x = x2*x
      i += 2
      d = x/i
      ans += d
    end
    ans *= 2
    if expo
      ans += num_class.Num(num_class.radix).ln*expo
    end
  end

  num_class.context(context, :rounding=>:half_even) do |local_context|
    ans = ans._fix(local_context)
    context.flags = local_context.flags
  end
  return ans
end

#log(b = nil, context = nil) ⇒ Object

Ruby-style logarithm of arbitrary base, e (natural base) by default



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# File 'lib/flt/num.rb', line 2623

def log(b=nil, context=nil)
  if b.nil?
    self.ln(context)
  elsif b==10
    self.log10(context)
  elsif b==2
    self.log2(context)
  else
    context = num_class.define_context(context)
    +num_class.context(:extra_precision=>3){self.ln(context)/num_class[b].ln(context)}
  end
end

#log10(context = nil) ⇒ Object

Returns the base 10 logarithm



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# File 'lib/flt/num.rb', line 2637

def log10(context=nil)
  context = num_class.define_context(context)
  num_class.context(:extra_precision=>3){self.ln/num_class.Num(10).ln}
end

#log2(context = nil) ⇒ Object

Returns the base 2 logarithm



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# File 'lib/flt/num.rb', line 2643

def log2(context=nil)
  context = num_class.define_context(context)
  num_class.context(context, :extra_precision=>3){self.ln()/num_class.Num(2).ln}
end

#logb(context = nil) ⇒ Object

Returns the exponent of the magnitude of the most significant digit.

The result is the integer which is the exponent of the magnitude of the most significant digit of the number (as though it were truncated to a single digit while maintaining the value of that digit and without limiting the resulting exponent).



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# File 'lib/flt/num.rb', line 2462

def logb(context=nil)
  context = define_context(context)
  ans = _check_nans(context)
  return ans if ans
  return num_class.infinity if infinite?
  return context.exception(DivisionByZero,'logb(0)',-1) if zero?
  Num(adjusted_exponent)._fix(context)
end

#minus(context = nil) ⇒ Object

Unary prefix minus operator



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# File 'lib/flt/num.rb', line 2052

def minus(context=nil)
  _neg(context)
end

#modulo(other, context = nil) ⇒ Object

Ruby-style modulo: x - y*div(x,y)



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# File 'lib/flt/num.rb', line 2276

def modulo(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return ans if ans

  #sign = self.sign * other.sign

  if self.infinite?
    return context.exception(InvalidOperation, 'INF % x')
  elsif other.zero?
    if self.zero?
      return context.exception(DivisionUndefined, '0 % 0')
    else
      return context.exception(InvalidOperation, 'x % 0')
    end
  end

  return self._divide_floor(other, context).last._fix(context)
end

#multiply(other, context = nil) ⇒ Object

Multiplication



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# File 'lib/flt/num.rb', line 1883

def multiply(other, context=nil)
  context = define_context(context)
  other = _convert(other)
  resultsign = self.sign * other.sign
  if self.special? || other.special?
    ans = _check_nans(context,other)
    return ans if ans

    if self.infinite?
      return context.exception(InvalidOperation,"(+-)INF * 0") if other.zero?
      return num_class.infinity(resultsign)
    end
    if other.infinite?
      return context.exception(InvalidOperation,"0 * (+-)INF") if self.zero?
      return num_class.infinity(resultsign)
    end
  end

  resultexp = self.exponent + other.exponent

  return Num(resultsign, 0, resultexp)._fix(context) if self.zero? || other.zero?
  #return Num(resultsign, other.coefficient, resultexp)._fix(context) if self.coefficient==1
  #return Num(resultsign, self.coefficient, resultexp)._fix(context) if other.coefficient==1

  return Num(resultsign, other.coefficient*self.coefficient, resultexp)._fix(context)

end

#nan? ⇒ Boolean

Returns whether the number is not actualy one (NaN, not a number).

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1664

def nan?
  @exp==:nan || @exp==:snan
end

#next_minus(context = nil) ⇒ Object

Largest representable number smaller than itself



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# File 'lib/flt/num.rb', line 2057

def next_minus(context=nil)
  context = define_context(context)
  if special?
    ans = _check_nans(context)
    return ans if ans
    if infinite?
      return Num(self) if @sign == -1
      # @sign == +1
      if context.exact?
         return context.exception(InvalidOperation, 'Exact +INF next minus')
      else
        return Num(+1, context.maximum_coefficient, context.etop)
      end
    end
  end

  return context.exception(InvalidOperation, 'Exact next minus') if context.exact?

  result = nil
  num_class.local_context(context) do |local|
    local.rounding = :floor
    local.ignore_all_flags
    result = self._fix(local)
    if result == self
      result = self - Num(+1, 1, local.etiny-1)
    end
  end
  result
end

#next_plus(context = nil) ⇒ Object

Smallest representable number larger than itself



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# File 'lib/flt/num.rb', line 2088

def next_plus(context=nil)
  context = define_context(context)

  if special?
    ans = _check_nans(context)
    return ans if ans
    if infinite?
      return Num(self) if @sign == +1
      # @sign == -1
      if context.exact?
         return context.exception(InvalidOperation, 'Exact -INF next plus')
      else
        return Num(-1, context.maximum_coefficient, context.etop)
      end
    end
  end

  return context.exception(InvalidOperation, 'Exact next plus') if context.exact?

  result = nil
  num_class.local_context(context) do |local|
    local.rounding = :ceiling
    local.ignore_all_flags
    result = self._fix(local)
    if result == self
      result = self + Num(+1, 1, local.etiny-1)
    end
  end
  result

end

#next_toward(other, context = nil) ⇒ Object

Returns the number closest to self, in the direction towards other.



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# File 'lib/flt/num.rb', line 2121

def next_toward(other, context=nil)
  context = define_context(context)
  other = _convert(other)
  ans = _check_nans(context,other)
  return ans if ans

  return context.exception(InvalidOperation, 'Exact next_toward') if context.exact?

  comparison = self <=> other
  return self.copy_sign(other) if comparison == 0

  if comparison == -1
    result = self.next_plus(context)
  else # comparison == 1
    result = self.next_minus(context)
  end

  # decide which flags to raise using value of ans
  if result.infinite?
    context.exception Overflow, 'Infinite result from next_toward', result.sign
    context.exception Rounded
    context.exception Inexact
  elsif result.adjusted_exponent < context.emin
    context.exception Underflow
    context.exception Subnormal
    context.exception Rounded
    context.exception Inexact
    # if precision == 1 then we don't raise Clamped for a
    # result 0E-etiny.
    context.exception Clamped if result.zero?
  end

  result
end

#nonzero? ⇒ Boolean

Returns whether the number not zero

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1694

def nonzero?
  special? || @coeff>0
end

#normal?(context = nil) ⇒ Boolean

Returns whether the number is normal

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1706

def normal?(context=nil)
  return false if special? || zero?
  context = define_context(context)
  (context.emin <= self.adjusted_exponent) &&  (self.adjusted_exponent <= context.emax)
end

#normalize(context = nil) ⇒ Object

Normalizes (changes quantum) so that the coefficient has precision digits, unless it is subnormal. For surnormal numbers the Subnormal flag is raised an a subnormal is returned with the smallest possible exponent.

This is different from reduce GDAS function which was formerly called normalize, and corresponds to the classic meaning of floating-point normalization.

Note that the number is also rounded (precision is reduced) if it had more precision than the context.



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# File 'lib/flt/num.rb', line 2436

def normalize(context=nil)
  context = define_context(context)
  return Num(self) if self.special? || self.zero? || context.exact?
  sign, coeff, exp = self._fix(context).split
  if self.subnormal?(context)
    context.exception Subnormal
    if exp > context.etiny
      coeff = num_class.int_mult_radix_power(coeff, exp - context.etiny)
      exp = context.etiny
    end
  else
    min_normal_coeff = context.minimum_normalized_coefficient
    while coeff < min_normal_coeff
      coeff = num_class.int_mult_radix_power(coeff, 1)
      exp -= 1
    end
  end
  Num(sign, coeff, exp)
end

#num_class ⇒ Object 



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# File 'lib/flt/num.rb', line 1395

def num_class
  self.class
end

#number_class(context = nil) ⇒ Object

Classifies a number as one of ‘sNaN’, ‘NaN’, ‘-Infinity’, ‘-Normal’, ‘-Subnormal’, ‘-Zero’,

'+Zero', '+Subnormal', '+Normal', '+Infinity'



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# File 'lib/flt/num.rb', line 1715

def number_class(context=nil)
  return "sNaN" if snan?
  return "NaN" if nan?
  if infinite?
    return '+Infinity' if @sign==+1
    return '-Infinity' # if @sign==-1
  end
  if zero?
    return '+Zero' if @sign==+1
    return '-Zero' # if @sign==-1
  end
  define_context(context)
  if subnormal?(context)
    return '+Subnormal' if @sign==+1
    return '-Subnormal' # if @sign==-1
  end
  return '+Normal' if @sign==+1
  return '-Normal' if @sign==-1
end

#number_of_digits ⇒ Object

Number of digits in the significand



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# File 'lib/flt/num.rb', line 2894

def number_of_digits
  # digits.size
  @coeff.is_a?(Integer) ? @coeff.to_s(num_class.radix).size : 0
end

#odd? ⇒ Boolean

returns true if is an odd integer

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 3019

def odd?
  # integral? && ((to_i%2)==1)
  # integral? && !even?
  if finite?
    if @exp>0 || @coeff==0
      false
    else
      if @exp <= -number_of_digits
        false
      else
        m = num_class.int_radix_power(-@exp)
        if (@coeff % m) == 0
          # ((@coeff / m) % 2) == 1
          ((@coeff / m) & 1) == 1
        else
          false
        end
      end
    end
  else
    false
  end
end

#plus(context = nil) ⇒ Object

Unary prefix plus operator



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# File 'lib/flt/num.rb', line 2047

def plus(context=nil)
  _pos(context)
end

#power(other, modulo = nil, context = nil) ⇒ Object

Raises to the power of x, to modulo if given.

With two arguments, compute self**other. If self is negative then other must be integral. The result will be inexact unless other is integral and the result is finite and can be expressed exactly in ‘precision’ digits.

With three arguments, compute (self**other) % modulo. For the three argument form, the following restrictions on the arguments hold:

- all three arguments must be integral
- other must be nonnegative
- at least one of self or other must be nonzero
- modulo must be nonzero and have at most 'precision' digits

The result of a.power(b, modulo) is identical to the result that would be obtained by computing (a**b) % modulo with unbounded precision, but may be computed more efficiently. It is always exact.



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# File 'lib/flt/num.rb', line 3263

def power(other, modulo=nil, context=nil)
  if context.nil? && (modulo.kind_of?(ContextBase) || modulo.is_a?(Hash))
    context = modulo
    modulo = nil
  end

  context = num_class.define_context(context)
  other = _convert(other)

  ans = _check_nans(context, other)
  return ans if ans

  # 0**0 = NaN (!), x**0 = 1 for nonzero x (including +/-Infinity)
  if other.zero?
    if self.zero?
      return context.exception(InvalidOperation, '0 ** 0')
    else
      return Num(1)
    end
  end

  # result has sign -1 iff self.sign is -1 and other is an odd integer
  result_sign = +1
  _self = self
  if _self.sign == -1
    if other.integral?
      result_sign = -1 if !other.even?
    else
      # -ve**noninteger = NaN
      # (-0)**noninteger = 0**noninteger
      unless self.zero?
        return context.exception(InvalidOperation, 'x ** y with x negative and y not an integer')
      end
    end
    # negate self, without doing any unwanted rounding
    _self = self.copy_negate
  end

  # 0**(+ve or Inf)= 0; 0**(-ve or -Inf) = Infinity
  if _self.zero?
    return (other.sign == +1) ? Num(result_sign, 0, 0) : num_class.infinity(result_sign)
  end

  # Inf**(+ve or Inf) = Inf; Inf**(-ve or -Inf) = 0
  if _self.infinite?
    return (other.sign == +1) ? num_class.infinity(result_sign) : Num(result_sign, 0, 0)
  end

  # 1**other = 1, but the choice of exponent and the flags
  # depend on the exponent of self, and on whether other is a
  # positive integer, a negative integer, or neither
  if _self == Num(1)
    return _self if context.exact?
    if other.integral?
      # exp = max(self._exp*max(int(other), 0),
      # 1-context.prec) but evaluating int(other) directly
      # is dangerous until we know other is small (other
      # could be 1e999999999)
      if other.sign == -1
        multiplier = 0
      elsif other > context.precision
        multiplier = context.precision
      else
        multiplier = other.to_i
      end

      exp = _self.exponent * multiplier
      if exp < 1-context.precision
        exp = 1-context.precision
        context.exception Rounded
      end
    else
      context.exception Rounded
      context.exception Inexact
      exp = 1-context.precision
    end

    return Num(result_sign, num_class.int_radix_power(-exp), exp)
  end

  # compute adjusted exponent of self
  self_adj = _self.adjusted_exponent

  # self ** infinity is infinity if self > 1, 0 if self < 1
  # self ** -infinity is infinity if self < 1, 0 if self > 1
  if other.infinite?
    if (other.sign == +1) == (self_adj < 0)
      return Num(result_sign, 0, 0)
    else
      return num_class.infinity(result_sign)
    end
  end

  # from here on, the result always goes through the call
  # to _fix at the end of this function.
  ans = nil

  # crude test to catch cases of extreme overflow/underflow.  If
  # log_radix(self)*other >= radix**bound and bound >= len(str(Emax))
  # then radixs**bound >= radix**len(str(Emax)) >= Emax+1 and hence
  # self**other >= radix**(Emax+1), so overflow occurs.  The test
  # for underflow is similar.
  bound = _self._log_radix_exp_bound + other.adjusted_exponent
  if (self_adj >= 0) == (other.sign == +1)
    # self > 1 and other +ve, or self < 1 and other -ve
    # possibility of overflow
    if bound >= _number_of_digits(context.emax)
      ans = Num(result_sign, 1, context.emax+1)
    end
  else
    # self > 1 and other -ve, or self < 1 and other +ve
    # possibility of underflow to 0
    etiny = context.etiny
    if bound >= _number_of_digits(-etiny)
      ans = Num(result_sign, 1, etiny-1)
    end
  end

  # try for an exact result with precision +1
  if ans.nil?
    if context.exact?
      if other.adjusted_exponent < 100 # ???? 4 ? ...
        test_precision = _self.number_of_digits*other.to_i+1
      else
        test_precision = _self.number_of_digits+1
      end
    else
      test_precision = context.precision + 1
    end
    ans = _self._power_exact(other, test_precision)
    if !ans.nil? && (result_sign == -1)
      ans = Num(-1, ans.coefficient, ans.exponent)
    end
  end

  # usual case: inexact result, x**y computed directly as exp(y*log(x))
  if !ans.nil?
    return ans if context.exact?
  else
    return context.exception(Inexact, "Inexact power") if context.exact?

    p = context.precision
    xc = _self.coefficient
    xe = _self.exponent
    yc = other.coefficient
    ye = other.exponent
    yc = -yc if other.sign == -1

    # compute correctly rounded result:  start with precision +3,
    # then increase precision until result is unambiguously roundable
    extra = 3
    coeff, exp = nil, nil
    loop do
      coeff, exp = _power(xc, xe, yc, ye, p+extra)
      break if (coeff % (num_class.int_radix_power(_number_of_digits(coeff)-p)/2)) != 0 # base 2: (coeff % (10**(_number_of_digits(coeff)-p-1))) != 0
      extra += 3
    end
    ans = Num(result_sign, coeff, exp)
  end

  # the specification says that for non-integer other we need to
  # raise Inexact, even when the result is actually exact.  In
  # the same way, we need to raise Underflow here if the result
  # is subnormal.  (The call to _fix will take care of raising
  # Rounded and Subnormal, as usual.)
  if !other.integral?
    context.exception Inexact
    # pad with zeros up to length context.precision+1 if necessary
    if ans.number_of_digits <= context.precision
      expdiff = context.precision+1 - ans.number_of_digits
      ans = Num(ans.sign, num_class.int_mult_radix_power(ans.coefficient, expdiff), ans.exponent-expdiff)
    end
    context.exception Underflow if ans.adjusted_exponent < context.emin
  end

  ans = ans % modulo if modulo

  # unlike exp, ln and log10, the power function respects the
  # rounding mode; no need to use ROUND_HALF_EVEN here
  ans._fix(context)
end

#qnan? ⇒ Boolean

Returns whether the number is a quite NaN (non-signaling)

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1669

def qnan?
  @exp == :nan
end

#quantize(exp, context = nil, watch_exp = true) ⇒ Object

Quantize so its exponent is the same as that of y.



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# File 'lib/flt/num.rb', line 3062

def quantize(exp, context=nil, watch_exp=true)
  exp = _convert(exp)
  context = define_context(context)
  if self.special? || exp.special?
    ans = _check_nans(context, exp)
    return ans if ans
    if exp.infinite? || self.infinite?
      return Num(self) if exp.infinite? && self.infinite?
      return context.exception(InvalidOperation, 'quantize with one INF')
    end
  end
  exp = exp.exponent
  _watched_rescale(exp, context, watch_exp)
end

#rationalize(tol = nil) ⇒ Object

Approximate conversion to Rational within given tolerance



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# File 'lib/flt/num.rb', line 2687

def rationalize(tol=nil)
  tol ||= Flt.Tolerance(Rational(1,2),:ulps)
  case tol
  when Integer
    Rational(*Support::Rationalizer.max_denominator(self, tol, num_class))
  else
    Rational(*Support::Rationalizer[tol].rationalize(self))
  end
end

#reduce(context = nil) ⇒ Object

Reduces an operand to its simplest form by removing trailing 0s and incrementing the exponent. (formerly called normalize in GDAS)



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# File 'lib/flt/num.rb', line 2387

def reduce(context=nil)
  context = define_context(context)
  if special?
    ans = _check_nans(context)
    return ans if ans
  end
  dup = _fix(context)
  return dup if dup.infinite?

  return Num(dup.sign, 0, 0) if dup.zero?

  exp_max = context.clamp? ? context.etop : context.emax
  end_d = nd = dup.number_of_digits
  exp = dup.exponent
  coeff = dup.coefficient
  dgs = dup.digits
  while (dgs[end_d-1]==0) && (exp < exp_max)
    exp += 1
    end_d -= 1
  end
  return Num(dup.sign, coeff/num_class.int_radix_power(nd-end_d), exp)
end

#reduced_exponent ⇒ Object

Exponent corresponding to the integral significand with all trailing digits removed. Does not use any context; equals the value of self.reduce.exponent (but as an integer rather than a Num) except for special values and when the number is rounded under the context or exceeds its limits.



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# File 'lib/flt/num.rb', line 2413

def reduced_exponent
  if self.special? || self.zero?
    0
  else
    exp = self.exponent
    dgs = self.digits
    nd = dgs.size # self.number_of_digits
    while dgs[nd-1]==0
      exp += 1
      nd -= 1
    end
    exp
  end
end

#remainder(other, context = nil) ⇒ Object

General Decimal Arithmetic Specification remainder: x - y*divide_int(x,y)



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# File 'lib/flt/num.rb', line 2299

def remainder(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return ans if ans

  #sign = self.sign * other.sign

  if self.infinite?
    return context.exception(InvalidOperation, 'INF % x')
  elsif other.zero?
    if self.zero?
      return context.exception(DivisionUndefined, '0 % 0')
    else
      return context.exception(InvalidOperation, 'x % 0')
    end
  end

  return self._divide_truncate(other, context).last._fix(context)
end

#remainder_near(other, context = nil) ⇒ Object

General Decimal Arithmetic Specification remainder-near:

x - y*round_half_even(x/y)



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# File 'lib/flt/num.rb', line 2323

def remainder_near(other, context=nil)
  context = define_context(context)
  other = _convert(other)

  ans = _check_nans(context,other)
  return ans if ans

  sign = self.sign * other.sign

  if self.infinite?
    return context.exception(InvalidOperation, 'remainder_near(INF,x)')
  elsif other.zero?
    if self.zero?
      return context.exception(DivisionUndefined, 'remainder_near(0,0)')
    else
      return context.exception(InvalidOperation, 'remainder_near(x,0)')
    end
  end

  if other.infinite?
    return Num(self)._fix(context)
  end

  ideal_exp = [self.exponent, other.exponent].min
  if self.zero?
    return Num(self.sign, 0, ideal_exp)._fix(context)
  end

  expdiff = self.adjusted_exponent - other.adjusted_exponent
  if (expdiff >= context.precision+1) && !context.exact?
    return context.exception(DivisionImpossible)
  elsif expdiff <= -2
    return self._rescale(ideal_exp, context.rounding)._fix(context)
  end

    self_coeff = self.coefficient
    other_coeff = other.coefficient
    de = self.exponent - other.exponent
    if de >= 0
      self_coeff = num_class.int_mult_radix_power(self_coeff, de)
    else
      other_coeff = num_class.int_mult_radix_power(other_coeff, -de)
    end
    q, r = self_coeff.divmod(other_coeff)
    if 2*r + (q&1) > other_coeff
      r -= other_coeff
      q += 1
    end

    return context.exception(DivisionImpossible) if q >= num_class.int_radix_power(context.precision) && !context.exact?

    sign = self.sign
    if r < 0
      sign = -sign
      r = -r
    end

  return Num(sign, r, ideal_exp)._fix(context)

end

#rescale(exp, context = nil, watch_exp = true) ⇒ Object

Rescale so that the exponent is exp, either by padding with zeros or by truncating digits.



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# File 'lib/flt/num.rb', line 3045

def rescale(exp, context=nil, watch_exp=true)
  context = define_context(context)
  exp = _convert(exp)
  if self.special? || exp.special?
    ans = _check_nans(context, exp)
    return ans if ans
    if exp.infinite? || self.infinite?
      return Num(self) if exp.infinite? && self.infinite?
      return context.exception(InvalidOperation, 'rescale with one INF')
    end
  end
  return context.exception(InvalidOperation,"exponent of rescale is not integral") unless exp.integral?
  exp = exp.to_i
  _watched_rescale(exp, context, watch_exp)
end

#round(*args) ⇒ Object

General rounding.

With an integer argument this acts like Float#round: the parameter specifies the number of fractional digits (or digits to the left of the decimal point if negative).

Options can be passed as a Hash instead; valid options are:

  • :rounding method for rounding (see Context#new())

The precision can be specified as:

  • :places number of fractional digits as above.

  • :exponent specifies the exponent corresponding to the digit to be rounded (exponent == -places)

  • :precision or :significan_digits is the number of digits

  • :power 10^exponent, value of the digit to be rounded, should be passed as a type convertible to Num.

  • :index 0-based index of the digit to be rounded

  • :rindex right 0-based index of the digit to be rounded

The default is :places=>0 (round to integer).

Example: ways of specifiying the rounding position

number:     1   2   3   4  .  5    6    7    8
:places    -3  -2  -1   0     1    2    3    4
:exponent   3   2   1   0    -1   -2   -3   -4
:precision  1   2   3   4     5    6    7    8
:power    1E3 1E2  10   1   0.1 1E-2 1E-3 1E-4
:index      0   1   2   3     4    5    6    7
:rindex     7   6   5   4     3    2    1    0



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# File 'lib/flt/num.rb', line 3146

def round(*args)
  # Admit first integer argument
  places = args.shift if args.first.is_a?(Integer)
  opt = args.shift || {}
  raise "Invalid arguments for round"  unless args.empty?

  # Support Ruby 2.4-style parameters:
  half_rounding = opt.delete(:half)
  case half_rounding
  when :even
    opt[:rounding] = :half_even
  when :up
    opt[:rounding] = :half_up
  when :down
    opt[:rounding] = :half_down
  end

  r = opt[:rounding] || :half_up
  as_int = false
  if v = opt[:precision] || opt[:significant_digits]
    prec = v
  elsif v = opt[:places] || places
    prec = adjusted_exponent + 1 + v
  elsif v = opt[:exponent]
    prec = adjusted_exponent + 1 - v
  elsif v = opt[:power]
    prec = adjusted_exponent + 1 - num_class.Num(v).adjusted_exponent
  elsif v = opt[:index]
    prec = i+1
  elsif v = opt[:rindex]
    prec = number_of_digits - v
  else
    prec = adjusted_exponent + 1
    as_int = true
  end
  dg = number_of_digits-prec
  changed = _round(r, dg)
  coeff = num_class.int_div_radix_power(@coeff, dg)
  exp = @exp + dg
  coeff += 1 if changed==1
  result = Num(@sign, coeff, exp)
  return as_int ? result.to_i : result
end

#same_quantum?(other) ⇒ Boolean

Return true if has the same exponent as other.

If either operand is a special value, the following rules are used:

  • return true if both operands are infinities

  • return true if both operands are NaNs

  • otherwise, return false.

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 3083

def same_quantum?(other)
  other = _convert(other)
  if self.special? || other.special?
    return (self.nan? && other.nan?) || (self.infinite? && other.infinite?)
  end
  return self.exponent == other.exponent
end

#scaleb(other, context = nil) ⇒ Object

Adds a value to the exponent.



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# File 'lib/flt/num.rb', line 2472

def scaleb(other, context=nil)

  context = define_context(context)
  other = _convert(other)
  ans = _check_nans(context, other)
  return ans if ans
  return context.exception(InvalidOperation) if other.infinite? || other.exponent != 0
  unless context.exact?
    liminf = -2 * (context.emax + context.precision)
    limsup =  2 * (context.emax + context.precision)
    i = other.to_i
    return context.exception(InvalidOperation) if !((liminf <= i) && (i <= limsup))
  end
  return Num(self) if infinite?
  return Num(@sign, @coeff, @exp+i)._fix(context)

end

#scientific_exponent ⇒ Object

Synonym for Num#adjusted_exponent()



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# File 'lib/flt/num.rb', line 2884

def scientific_exponent
  adjusted_exponent
end

#sign ⇒ Object

Sign of the number: +1 for plus / -1 for minus.



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# File 'lib/flt/num.rb', line 2922

def sign
  @sign
end

#snan? ⇒ Boolean

Returns whether the number is a signaling NaN

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1674

def snan?
  @exp == :snan
end

#special? ⇒ Boolean

Returns whether the number is a special value (NaN or Infinity).

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1659

def special?
  @exp.instance_of?(Symbol)
end

#split ⇒ Object

Returns the internal representation of the number, composed of:

  • a sign which is +1 for plus and -1 for minus

  • a coefficient (significand) which is a nonnegative integer

  • an exponent (an integer) or :inf, :nan or :snan for special values

The value of non-special numbers is sign*coefficient*10^exponent



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# File 'lib/flt/num.rb', line 1654

def split
  [@sign, @coeff, @exp]
end

#sqrt(context = nil) ⇒ Object

Square root



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# File 'lib/flt/num.rb', line 1975

def sqrt(context=nil)
  context = define_context(context)
  if special?
    ans = _check_nans(context)
    return ans if ans
    return Num(self) if infinite? && @sign==+1
  end
  return Num(@sign, 0, @exp/2)._fix(context) if zero?
  return context.exception(InvalidOperation, 'sqrt(-x), x>0') if @sign<0
  prec = context.precision + 1

  # express the number in radix**2 base
  e = (@exp >> 1)
  if (@exp & 1)!=0
    c = @coeff*num_class.radix
    l = (number_of_digits >> 1) + 1
  else
    c = @coeff
    l = (number_of_digits+1) >> 1
  end
  shift = prec - l
  if shift >= 0
    c = num_class.int_mult_radix_power(c, (shift<<1))
    exact = true
  else
    c, remainder = c.divmod(num_class.int_radix_power((-shift)<<1))
    exact = (remainder==0)
  end
  e -= shift

  n = num_class.int_radix_power(prec)
  while true
    q = c / n
    break if n <= q
    n = ((n + q) >> 1)
  end
  exact = exact && (n*n == c)

  if exact
    if shift >= 0
      n = num_class.int_div_radix_power(n, shift)
    else
      n = num_class.int_mult_radix_power(n, -shift)
    end
    e += shift
  else
    return context.exception(Inexact) if context.exact?
    # result is not exact; adjust to ensure correct rounding
    if num_class.radix == 10
      n += 1 if (n%5)==0
    else
      n = num_class.int_mult_radix_power(n, 2) + 1
      e -= 2
    end
  end
  ans = Num(+1,n,e)
  num_class.local_context(:rounding=>:half_even) do
    ans = ans._fix(context)
  end
  return ans
end

#subnormal?(context = nil) ⇒ Boolean

Returns whether the number is subnormal

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1699

def subnormal?(context=nil)
  return false if special? || zero?
  context = define_context(context)
  self.adjusted_exponent < context.emin
end

#subtract(other, context = nil) ⇒ Object

Subtraction



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# File 'lib/flt/num.rb', line 1870

def subtract(other, context=nil)

  context = define_context(context)
  other = _convert(other)

  if self.special? || other.special?
    ans = _check_nans(context,other)
    return ans if ans
  end
  return add(other.copy_negate, context)
end

#to_f ⇒ Object

Conversion to Float



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# File 'lib/flt/num.rb', line 2698

def to_f
  if special?
    if @exp==:inf
      @sign/0.0
    else
      0.0/0.0
    end
  elsif zero?
    @sign*0.0
  else
    f = nil
    f ||= to_s.to_f if num_class.radix == 10 # very precise, but slow
    unless f
      c = @coeff.to_f
      if c.finite?
        # This is fast to compute but may introduce roundoff error, and overflow or yield NaN
        f = if @exp >= 0
              @sign*c*(num_class.radix.to_f**@exp)
            else
              @sign*c/(num_class.radix.to_f**(-@exp))
            end
      end
      f = nil unless f && f.finite?
    end
    f ||= to_r.to_f # not so slow, also may introduce roundoff error due to arithmetic
  end
end

#to_i ⇒ Object

Ruby-style to integer conversion.



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# File 'lib/flt/num.rb', line 2655

def to_i
  if special?
    if nan?
      #return context.exception(InvalidContext)
      num_class.context.exception InvalidContext
      return nil
    end
    raise Error, "Cannot convert infinity to Integer"
  end
  if @exp >= 0
    return @sign*num_class.int_mult_radix_power(@coeff,@exp)
  else
    return @sign*num_class.int_div_radix_power(@coeff,-@exp)
  end
end

#to_int_scale ⇒ Object

Return the value of the number as an signed integer and a scale.



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# File 'lib/flt/num.rb', line 2952

def to_int_scale
  if special?
    nil
  else
    [@sign*integral_significand, integral_exponent]
  end
end

#to_integral_exact(context = nil) ⇒ Object

Rounds to a nearby integer. May raise Inexact or Rounded.



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# File 'lib/flt/num.rb', line 3092

def to_integral_exact(context=nil)
  context = define_context(context)
  if special?
    ans = _check_nans(context)
    return ans if ans
    return Num(self)
  end
  return Num(self) if @exp >= 0
  return Num(@sign, 0, 0) if zero?
  context.exception Rounded
  ans = _rescale(0, context.rounding)
  context.exception Inexact if ans != self
  return ans
end

#to_integral_value(context = nil) ⇒ Object

Rounds to a nearby integer. Doesn’t raise Inexact or Rounded.



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# File 'lib/flt/num.rb', line 3108

def to_integral_value(context=nil)
  context = define_context(context)
  if special?
    ans = _check_nans(context)
    return ans if ans
    return Num(self)
  end
  return Num(self) if @exp >= 0
  return _rescale(0, context.rounding)
end

#to_r ⇒ Object

Conversion to Rational. Conversion of special values will raise an exception under Ruby 1.9



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# File 'lib/flt/num.rb', line 2673

def to_r
  if special?
    num = (@exp == :inf) ? @sign : 0
    Rational.respond_to?(:new!) ? Rational.new!(num,0) : Rational(num,0)
  else
    if @exp < 0
      Rational(@sign*@coeff, num_class.int_radix_power(-@exp))
    else
      Rational(num_class.int_mult_radix_power(@sign*@coeff,@exp), 1)
    end
  end
end

#to_s(*args) ⇒ Object

Representation as text of a number: this is an alias of Num#format



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# File 'lib/flt/num.rb', line 3239

def to_s(*args)
  format(*args)
end

#truncate(opt = {}) ⇒ Object

General truncate operation (as for Float) with same options for precision as Flt::Num#round()



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# File 'lib/flt/num.rb', line 3206

def truncate(opt={})
  opt[:rounding] = :down
  round opt
end

#ulp(context = nil, mode = :low) ⇒ Object

ulp (unit in the last place) according to the definition proposed by J.M. Muller in “On the definition of ulp(x)” INRIA No. 5504 If the mode parameter has the value :high the Golberg ulp is computed instead; which is different on the powers of the radix (which are the borders between areas of different ulp-magnitude)



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# File 'lib/flt/num.rb', line 2731

def ulp(context = nil, mode=:low)
  context = define_context(context)

  return context.exception(InvalidOperation, "ulp in exact context") if context.exact?

  if self.nan?
    return Num(self)
  elsif self.infinite?
    # The ulp here is context.maximum_finite - context.maximum_finite.next_minus
    return Num(+1, 1, context.etop)
  elsif self.zero? || self.adjusted_exponent <= context.emin
    # This is the ulp value for self.abs <= context.minimum_normal*num_class.context
    # Here we use it for self.abs < context.minimum_normal*num_class.context;
    #  because of the simple exponent check; the remaining cases are handled below.
    return context.minimum_nonzero
  else
    # The next can compute the ulp value for the values that
    #   self.abs > context.minimum_normal && self.abs <= context.maximum_finite
    # The cases self.abs < context.minimum_normal*num_class.context have been handled above.

    # assert self.normal? && self.abs>context.minimum_nonzero
    norm = self.normalize
    exp = norm.integral_exponent
    sig = norm.integral_significand

    # Powers of the radix, r**n, are between areas with different ulp values: r**(n-p-1) and r**(n-p)
    # (p is context.precision).
    # This method and the ulp definitions by Muller, Kahan and Harrison assign the smaller ulp value
    # to r**n; the definition by Goldberg assigns it to the larger ulp (so ulp varies with adjusted_exponent).
    # The next line selects the smaller ulp for powers of the radix:
    exp -= 1 if sig == num_class.int_radix_power(context.precision-1) if mode == :low

    return Num(+1, 1, exp)
  end
end

#zero? ⇒ Boolean

Returns whether the number is zero

Returns:

  • (Boolean) 


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# File 'lib/flt/num.rb', line 1689

def zero?
  @coeff==0 && !special?
end