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Feature #6857

bigdecimal/math BigMath.E/BigMath.exp R. P. Feynman inspired optimization

Added by royaltm (Rafał Michalski) over 8 years ago. Updated about 3 years ago.

Status:
Assigned
Priority:
Normal
Target version:
-
[ruby-core:47130]

Description

The algorythms to calculate E and exp programmed in BigMath module are the very straightforward interpretation of the series 1 + x + x2/2! +
x3/3! + ....
Therefore they are slow.

Try it yourself:

  require 'bigdecimal/math'

  def timer; s=Time.now; yield; puts Time.now-s; end

  timer { BigMath.E(1000) }   #->  0.038848
  timer { BigMath.E(10000) }  #-> 16.526972
  timer { BigMath.E(100000) } #-> lost patience

That's because every iteration divides 1 by n! and the dividend grows extremely fast.

In "Surely You're Joking, Mr. Feynman!" (great book, you should read it if you didn't already) R. P. Feynman said:

"One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for ex, which is 1 + x + x2/2! +
x3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x4/4! you
multiply that term by x and divide by 5. It's very simple."

Yes it's very simple indeed. Why it's not been applied in such a great, modern and popular language? Is it because people just forget about simple solutions today?

Here is a Feynman's optimized version of BigMath.E:

  def E(prec)
    raise ArgumentError, "Zero or negative precision for E" if prec <= 0
    n = prec + BigDecimal.double_fig
    y = d = i = one = BigDecimal('1')
    while d.nonzero? && (m = n - (y.exponent - d.exponent).abs) > 0
      m = BigDecimal.double_fig if m < BigDecimal.double_fig
      d = d.div(i, m)
      i += one
      y += d
    end
    y
  end

Now, let's put it to the test:

  (1..1000).all? {|n| BigMath.E(n).round(n) == E(n).round(n) }
  => true
  BigMath.E(10000).round(10000) == E(10000).round(10000)
  => true

What about the speed then?

  timer { E(1_000) }     #-> 0.003832 ~ 10 times faster
  timer { E(10_000) }    #-> 0.139862 ~ 100 times faster
  timer { E(100_000) }   #-> 8.787411 ~ dunno?
  timer { E(1_000_000) } #-> ~11 minutes

The same simple rule might be applied to BigDecimal.exp() which originally uses the same straightforward interpretation of power series.
Feynman's pure ruby version of BigMath.exp (the ext version seems now pointless anyway):

  def exp(x, prec)
    raise ArgumentError, "Zero or negative precision for exp" if prec <= 0
    x = case x
    when Float
      BigDecimal(x, prec && prec <= Float::DIG ? prec : Float::DIG + 1)
    else
      BigDecimal(x, prec)
    end
    one = BigDecimal('1', prec)
    case x.sign
    when BigDecimal::SIGN_NaN
      return BigDecimal::NaN
    when BigDecimal::SIGN_POSITIVE_ZERO, BigDecimal::SIGN_NEGATIVE_ZERO
      return one
    when BigDecimal::SIGN_NEGATIVE_FINITE
      x = -x
      inv = true
    when BigDecimal::SIGN_POSITIVE_INFINITE
      return BigDecimal::INFINITY
    when BigDecimal::SIGN_NEGATIVE_INFINITE
      return BigDecimal.new('0')
    end
    n = prec + BigDecimal.double_fig
    if x.round(prec) == one
      y = E(prec)
    else
      y = d = i = one
      while d.nonzero? && (m = n - (y.exponent - d.exponent).abs) > 0
        m = BigDecimal.double_fig if m < BigDecimal.double_fig
        d = d.mult(x, m).div(i, m)
        i += one
        y += d
      end
    end
    y = one.div(y, n) if inv
    y.round(prec - y.exponent)
  end

  (1..1000).all? {|n| exp(E(n),n) == BigMath.exp(BigMath.E(n),n) }
  # => true
  (1..1000).all? {|n| exp(-E(n),n) == BigMath.exp(-BigMath.E(n),n) }
  # => true
  (-10000..10000).all? {|n| exp(BigDecimal(n)/1000,100) == BigMath.exp(BigDecimal(n)/1000,100) }
  # => true
  (1..1000).all? {|n| exp(BigMath.PI(n),n) == BigMath.exp(BigMath.PI(n),n) }
  # => true

  timer { BigMath.exp(BigDecimal('1').div(3, 10), 100) }    #-> 0.000496
  timer { exp(BigDecimal('1').div(3, 10), 100) }            #-> 0.000406 faster but not that really

  timer { BigMath.exp(BigDecimal('1').div(3, 10), 1_000) }  #-> 0.029231
  timer { exp(BigDecimal('1').div(3, 10), 1_000) }          #-> 0.004554 here we go... 

  timer { BigMath.exp(BigDecimal('1').div(3, 10), 10_000) } #-> 12.554197
  timer { exp(BigDecimal('1').div(3, 10), 10_000) }         #->  0.189462 oops :)

  timer { exp(BigDecimal('1').div(3, 10), 100_000) }        #-> 11.914613 who has the patience to compare?

Arguments with large mantissa should slow down the results of course:

  timer { BigMath.exp(BigDecimal('1').div(3, 1_000), 1_000) }   #->  0.119048
  timer { exp(BigDecimal('1').div(3, 1_000), 1_000) }           #->  0.066177

  timer { BigMath.exp(BigDecimal('1').div(3, 10_000), 10_000) } #-> 68.083222
  timer { exp(BigDecimal('1').div(3, 10_000), 10_000) }         #-> 29.439336

Though still two times faster than the ext version.

It seems Dick Feynman was not such a joker after all. I think he was a master in treating lightly "serious" things and treating very seriously things that didn't matter to anybody else.

I'd write a patch for ext version if you are with me. Just let me know.

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