Class: Flt::Support::Formatter

Inherits:
Object
  • Object
show all
Defined in:
lib/flt/support/formatter.rb

Overview

Burger and Dybvig free formatting algorithm, from their paper: “Printing Floating-Point Numbers Quickly and Accurately” (Robert G. Burger, R. Kent Dybvig)

This algorithm formats arbitrary base floating point numbers as decimal text literals. The floating-point (with fixed precision) is interpreted as an approximate value, representing any value in its ‘rounding-range’ (the interval where all values round to the floating-point value, with the given precision and rounding mode). An alternative approach which is not taken here would be to represent the exact floating-point value with some given precision and rounding mode requirements; that can be achieved with Clinger algorithm (which may fail for exact precision).

The variables used by the algorithm are stored in instance variables: Quotients of integers will be used to hold the magnitudes: All numbers in the randound-range are rounded to @v (with the given precision p) significant digit right after the radix point. @b**@k is the first power of @b >= u

The rounding range of @v is the interval of values that round to @v under the runding-mode. If the rounding mode is one of the round-to-nearest variants (even, up, down), then it is ((v+v-)/2 = (@[email protected]_m)/@s, (v+v+)/2 = (@[email protected]_)/2) whith the boundaries open or closed as explained below. In this case:

@m_m/@s = (@v - (v + v-)/2) where v- = @v.next_minus is the lower adjacent to v floating point value
@m_p/@s = ((v + v+)/2 - @v) where v+ = @v.next_plus is the upper adjacent to v floating point value

If the rounding is directed, then the rounding interval is either (v-, @v] or [@v, v+] if @roundh, then @k is the minimum @k with (@[email protected]_p)/@s <= @output_b**@k

@k = ceil(logB((@[email protected]_p)/2)) with lobB the @output_b base logarithm

if @roundh, then @k is the minimum @k with (@[email protected]_p)/@s < @output_b**@k

@k = 1+floor(logB((@r+@m_p)/2))

p is the input floating point precision

Constant Summary collapse

ITERATIONS_BEFORE_KEEPING_TRACK_OF_REMAINDERS = 
10000
ESTIMATE_FLOAT_LOG_B = 
{2=>1/Math.log(2), 10=>1/Math.log(10), 16=>1/Math.log(16)}

Instance Attribute Summary collapse

Instance Method Summary collapse

Constructor Details

#initialize(input_b, input_min_e, output_b, options = {}) ⇒ Formatter

A Formatted object is created to format floating point numbers given:

  • The input base in which numbers to be formatted are defined

  • The input minimum exponent

  • The output base to which the input is converted.

  • The :raise_on_repeat option, true by default specifies that when an infinite sequence of repeating significant digits is found on the output (which may occur when using the all-digits options and using directed-rounding) an InfiniteLoopError exception is raised. If this option is false, then no exception occurs, and instead of generating an infinite sequence of digits, the formatter object will have a ‘repeat’ property which designs the first digit to be repeated (it is an index into digits). If this equals the size of digits, it is assumend, that the digit to be repeated is a zero which follows the last digit present in digits.



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# File 'lib/flt/support/formatter.rb', line 70

def initialize(input_b, input_min_e, output_b, options={})
  @b = input_b
  @min_e = input_min_e
  @output_b = output_b
  # result of last operation
  @adjusted_digits = @digits = nil
  # for "all-digits" mode results (which are truncated, rather than rounded),
  # round_up contains information to round the result:
  # * it is nil if the rest of digits are zero (the result is exact)
  # * it is :lo if there exist non-zero digits beyond the significant ones (those returned), but
  #   the value is below the tie (the value must be rounded up only for :up rounding mode)
  # * it is :tie if there exists exactly one nonzero digit after the significant and it is radix/2,
  #   for round-to-nearest it is atie.
  # * it is :hi otherwise (the value should be rounded-up except for the :down mode)
  @round_up = nil

  options = { :raise_on_repeat => true }.merge(options)
  # when significant repeating digits occur (+all+ parameter and directed rounding)
  # @repeat is set to the index of the first repeating digit in @digits;
  # (if equal to @digits.size, that would indicate an infinite sequence of significant zeros)
  @repeat = nil
  # the :raise_on_repeat options (by default true) causes exceptions when repeating is found
  @raise_on_repeat = options[:raise_on_repeat]
end

Instance Attribute Details

#repeat ⇒ Object (readonly)

Returns the value of attribute repeat.



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# File 'lib/flt/support/formatter.rb', line 232

def repeat
  @repeat
end

#round_up ⇒ Object (readonly)

Returns the value of attribute round_up.



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# File 'lib/flt/support/formatter.rb', line 232

def round_up
  @round_up
end

Instance Method Details

#adjusted_digits(round_mode) ⇒ Object

Access rounded result of format operation: scaling (position of radix point) and digits



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# File 'lib/flt/support/formatter.rb', line 235

def adjusted_digits(round_mode)
  if @adjusted_digits.nil? && !@digits.nil?
    @adjusted_k, @adjusted_digits = Support.adjust_digits(
                                      @k, @digits,
                                      :round_mode => round_mode,
                                      :negative => @minus,
                                      :round_up => @round_up,
                                      :base => @output_b)
  end
  return @adjusted_k, @adjusted_digits
end

#b_power(n) ⇒ Object 



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# File 'lib/flt/support/formatter.rb', line 294

def b_power(n)
  @b**n
end

#detect_repetitions(r) ⇒ Object

Detect indefinite repetitions in generate_max returns the number of digits that are being repeated (0 indicates the next digit would repeat and it would be a zero)



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# File 'lib/flt/support/formatter.rb', line 313

def detect_repetitions(r)
  return nil unless @may_repeat
  @n_iters += 1
  if r == 0 && @m_p == 0
    repeat_count = 0
  elsif (@n_iters > ITERATIONS_BEFORE_KEEPING_TRACK_OF_REMAINDERS)
    if @rs.include?(r)
      repeat_count = @rs.index(r) - @rs.size
    else
      @rs << r
    end
  end
  if repeat_count
    raise InfiniteLoopError, "Infinite digit sequence." if @raise_on_repeat
    repeat_count
  else
    nil
  end
end

#digits ⇒ Object

Access result of format operation: scaling (position of radix point) and digits



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# File 'lib/flt/support/formatter.rb', line 228

def digits
  return @k, @digits
end

#format(v, f, e, round_mode, p = nil, all = false) ⇒ Object

This method converts v = f*b**e into a sequence of output_b-base digits, so that if the digits are converted back to a floating-point value of precision p (correctly rounded), the result is exactly v.

If round_mode is not nil, then just enough digits to produce v using that rounding is used; otherwise enough digits to produce v with any rounding are delivered.

If the all parameter is true, all significant digits are generated without rounding, Significant digits here are all digits that, if used on input, cannot arbitrarily change while preserving the parsed value of the floating point number. Since the digits are not rounded more digits may be needed to assure round-trip value preservation.

This is useful to reflect the precision of the floating point value in the output; in particular trailing significant zeros are shown. But note that, for directed rounding and base conversion this may need to produce an infinite number of digits, in which case an exception will be raised unless the :raise_on_repeat option has been set to false in the Formatter object. In that case the formatter objetct will have a repeat property that specifies the point in the digit sequence where irepetition starts. The digits from that point to the end to the digits sequence repeat indefinitely.

This digit-repetition is specially frequent for the :up rounding mode, in which any number with a finite numberof nonzero digits equal to or less than the precision will haver and infinite sequence of zero significant digits.

The:down rounding (truncation) could be used to show the exact value of the floating point but beware: if the value has not an exact representation in the output base this will lead to an infinite loop or repeating squence.

When the all parameters is used the result is not rounded (is truncated), and the round_up flag is set to indicate that nonzero digits exists beyond the returned digits; the possible values of the round_up flag are:

  • nil : the rest of digits are zero or repeat (the result is exact)

  • :lo : there exist non-zero digits beyond the significant ones (those returned), but the value is below the tie (the value must be rounded up only for :up rounding mode)

  • :tie : there exists exactly one nonzero digit after the significant and it is radix/2, for round-to-nearest it is atie.

  • :hi : the value is closer to the rounded-up value (incrementing the last significative digit.)

Note that the round_mode here is not the rounding mode applied to the output; it is the rounding mode that applied to input preserves the original floating-point value (with the same precision as input). should be rounded-up.



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# File 'lib/flt/support/formatter.rb', line 139

def format(v, f, e, round_mode, p=nil, all=false)
  context = v.class.context
  # TODO: consider removing parameters f,e and using v.split instead
  @minus = (context.sign(v)==-1)
  @v = context.copy_sign(v, +1) # don't use context.abs(v) because it rounds (and may overflow also)
  @f = f.abs
  @e = e
  @round_mode = round_mode
  @all_digits = all
  p ||= context.precision

  # adjust the rounding mode to work only with positive numbers
  @round_mode = Support.simplified_round_mode(@round_mode, @minus)

  # determine the high,low inclusion flags of the rounding limits
  case @round_mode
    when :half_even
      # rounding rage is (v-m-,v+m+) if v is odd and [v+m-,v+m+] if even
      @round_l = @round_h = ((@f % 2) == 0)
    when :up
      # rounding rage is (v-,v]
      # ceiling is treated here assuming f>0
      @round_l, @round_h = false, true
    when :down
      # rounding rage is [v,v+)
      # floor is treated here assuming f>0
      @round_l, @round_h = true, false
    when :half_up
      # rounding rage is [v+m-,v+m+)
      @round_l, @round_h = true, false
    when :half_down
      # rounding rage is (v+m-,v+m+]
      @round_l, @round_h = false, true
    else # :nearest
      # Here assume only that round-to-nearest will be used, but not which variant of it
      # The result is valid for any rounding (to nearest) but may produce more digits
      # than stricly necessary for specific rounding modes.
      # That is, enough digits are generated so that when the result is
      # converted to floating point with the specified precision and
      # correct rounding (to nearest), the result is the original number.
      # rounding range is (v+m-,v+m+)
      @round_l = @round_h = false
  end

  # TODO: use context.next_minus, next_plus instead of direct computing, don't require min_e & ps
  # Now compute the working quotients @r/@s, @m_p/@s = (v+ - @v), @m_m/@s = (@v - v-) and scale them.
  if @e >= 0
    if @f != b_power(p-1)
      be = b_power(@e)
      @r, @s, @m_p, @m_m = @f*be*2, 2, be, be
    else
      be = b_power(@e)
      be1 = be*@b
      @r, @s, @m_p, @m_m = @f*be1*2, @b*2, be1, be
    end
  else
    if @e == @min_e || @f != b_power(p-1)
      @r, @s, @m_p, @m_m = @f*2, b_power(-@e)*2, 1, 1
    else
      @r, @s, @m_p, @m_m = @f*@b*2, b_power(1-@e)*2, @b, 1
    end
  end
  @k = 0
  @context = context
  scale_optimized!


  # The value to be formatted is @v=@r/@s; m- = @m_m/@s = (@v - v-)/@s; m+ = @m_p/@s = (v+ - @v)/@s
  # Now adjust @m_m, @m_p so that they define the rounding range
  case @round_mode
  when :up
    # ceiling is treated here assuming @f>0
    # rounding range is -v,@v
    @m_m, @m_p = @m_m*2, 0
  when :down
    # floor is treated here assuming #f>0
    # rounding range is @v,v+
    @m_m, @m_p = 0, @m_p*2
  else
    # rounding range is v-,v+
    # @m_m, @m_p = @m_m, @m_p
  end

  # Now m_m, m_p define the rounding range
  all ? generate_max : generate

end

#generate ⇒ Object 



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# File 'lib/flt/support/formatter.rb', line 393

def generate
  list = []
  r, s, m_p, m_m, = @r, @s, @m_p, @m_m
  loop do
    d,r = (r*@output_b).divmod(s)
    m_p *= @output_b
    m_m *= @output_b
    tc1 = @round_l ? (r<=m_m) : (r<m_m)
    tc2 = @round_h ? (r+m_p >= s) : (r+m_p > s)

    if not tc1
      if not tc2
        list << d
      else
        list << d+1
        break
      end
    else
      if not tc2
        list << d
        break
      else
        if r*2 < s
          list << d
          break
        else
          list << d+1
          break
        end
      end
    end

  end
  @digits = list
end

#generate_max ⇒ Object 



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# File 'lib/flt/support/formatter.rb', line 352

def generate_max
  @round_up = false
  list = []
  r, s, m_p, m_m, = @r, @s, @m_p, @m_m

  start_repetition_dectection

  loop do
    if repeat_count = detect_repetitions(r)
      @repeat = list.size + repeat_count
      break
    end

    d,r = (r*@output_b).divmod(s)

    m_p *= @output_b
    m_m *= @output_b

    list << d

    tc1 = @round_l ? (r<=m_m) : (r<m_m)
    tc2 = @round_h ? (r+m_p >= s) : (r+m_p > s)

    if tc1 && tc2
      if r != 0
        r *= 2
        if r > s
          @round_up = :hi
        elsif r == s
          @round_up = :tie
        else
          @rund_up = :lo
        end
      end
      break
    end
  end
  @digits = list
  remove_redundant_repetitions
end

#output_b_power(n) ⇒ Object 



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# File 'lib/flt/support/formatter.rb', line 298

def output_b_power(n)
  @output_b**n
end

#remove_redundant_repetitions ⇒ Object 



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# File 'lib/flt/support/formatter.rb', line 333

def remove_redundant_repetitions
  if ITERATIONS_BEFORE_KEEPING_TRACK_OF_REMAINDERS > 0 && @repeat
    if @repeat < @digits.size
      repeating_digits = @digits[@repeat..-1]
      l = repeating_digits.size
      pos = @repeat - l
      while pos >= 0 && @digits[pos, l] == repeating_digits
        pos -= l
      end
      first_repeat = pos + l
      if first_repeat < @repeat
        @repeat = first_repeat
        @digits = @digits[0, @repeat+l]
      end
    end
  end
  @digits
end

#scale! ⇒ Object

using local vars instead of instance vars: it makes a difference in performance



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# File 'lib/flt/support/formatter.rb', line 272

def scale!
  r, s, m_p, m_m, k,output_b = @r, @s, @m_p, @m_m, @k,@output_b
  loop do
    if (@round_h ? (r+m_p >= s) : (r+m_p > s)) # k is too low
      s *= output_b
      k += 1
    elsif (@round_h ? ((r+m_p)*output_b<s) : ((r+m_p)*output_b<=s)) # k is too high
      r *= output_b
      m_p *= output_b
      m_m *= output_b
      k -= 1
    else
      @s = s
      @r = r
      @m_p = m_p
      @m_m = m_m
      @k = k
      break
    end
  end
end

#scale_fixup! ⇒ Object

fix up scaling (final step): specialized version of scale! This performs a single up scaling step, i.e. behaves like scale2, but the input must be at most one step down from the final result



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# File 'lib/flt/support/formatter.rb', line 503

def scale_fixup!
  if (@round_h ? (@r+@m_p >= @s) : (@r+@m_p > @s)) # too low?
    @s *= @output_b
    @k += 1
  end
end

#scale_optimized! ⇒ Object

scale_o1! is an optimized version of scale!; it requires an additional parameters with the floating-point number v=r/s

It uses a Float estimate of ceil(logB(v)) that may need to adjusted one unit up TODO: find easy to use estimate; determine max distance to correct value and use it for fixing,

or use the general scale! for fixing (but remembar to multiply by exptt(...))
(determine when Math.log is aplicable, etc.)



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# File 'lib/flt/support/formatter.rb', line 437

def scale_optimized!
  context = @context # @v.class.context
  return scale! if context.zero?(@v)

  # 1. compute estimated_scale

  # 1.1. try to use Float logarithms (Math.log)
  v = @v
  v_abs = context.copy_sign(v, +1) # don't use v.abs because it rounds (and may overflow also)
  v_flt = v_abs.to_f
  b = @output_b
  log_b = ESTIMATE_FLOAT_LOG_B[b]
  log_b = ESTIMATE_FLOAT_LOG_B[b] = 1.0/Math.log(b) if log_b.nil?
  estimated_scale = nil
  fixup = false
  begin
    l = ((b==10) ? Math.log10(v_flt) : Math.log(v_flt)*log_b)
    estimated_scale =(l - 1E-10).ceil
    fixup = true
  rescue
    # rescuing errors is more efficient than checking (v_abs < Float::MAX.to_i) && (v_flt > Float::MIN) when v is a Flt
  else
    # estimated_scale = nil
  end

  # 1.2. Use Flt::DecNum logarithm
  if estimated_scale.nil?
    v.to_decimal_exact(:precision=>12) if v.is_a?(BinNum)
    if v.is_a?(DecNum)
      l = nil
      DecNum.context(:precision=>12) do
        case b
        when 10
          l = v_abs.log10
        else
          l = v_abs.ln/Flt.DecNum(b).ln
        end
      end
      l -= Flt.DecNum(+1,1,-10)
      estimated_scale = l.ceil
      fixup = true
    end
  end

  # 1.3 more rough Float aproximation
    # TODO: optimize denominator, correct numerator for more precision with first digit or part
    # of the coefficient (like _log_10_lb)
  estimated_scale ||= (v.adjusted_exponent.to_f * Math.log(v.class.context.radix) * log_b).ceil

  if estimated_scale >= 0
    @k = estimated_scale
    @s *= output_b_power(estimated_scale)
  else
    sc = output_b_power(-estimated_scale)
    @k = estimated_scale
    @r *= sc
    @m_p *= sc
    @m_m *= sc
  end
  fixup ? scale_fixup! : scale!

end

#scale_original!(really = false) ⇒ Object

Given r/s = v (number to convert to text), m_m/s = (v - v-)/s, m_p/s = (v+ - v)/s Scale the fractions so that the first significant digit is right after the radix point, i.e. find k = ceil(logB((r+m_p)/s)), the smallest integer such that (r+m_p)/s <= B^k if k>=0 return:

r=r, s=s*B^k, m_p=m_p, m_m=m_m

if k<0 return:

r=r*B^k, s=s, m_p=m_p*B^k, m_m=m_m*B^k

scale! is a general iterative method using only (multiprecision) integer arithmetic.



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# File 'lib/flt/support/formatter.rb', line 256

def scale_original!(really=false)
  loop do
    if (@round_h ? (@r+@m_p >= @s) : (@r+@m_p > @s)) # k is too low
      @s *= @output_b
      @k += 1
    elsif (@round_h ? ((@r+@m_p)*@output_b<@s) : ((@r+@m_p)*@output_b<=@s)) # k is too high
      @r *= @output_b
      @m_p *= @output_b
      @m_m *= @output_b
      @k -= 1
    else
      break
    end
  end
end

#start_repetition_dectection ⇒ Object 



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# File 'lib/flt/support/formatter.rb', line 302

def start_repetition_dectection
  @may_repeat = (@m_p == 0 || @m_m == 0)
  @n_iters = 0
  @rs = []
end