```Issue #6857 has been updated by mame (Yusuke Endoh).

Status changed from Open to Assigned
Target version set to next minor

----------------------------------------
Feature #6857: bigdecimal/math BigMath.E/BigMath.exp R. P. Feynman inspired optimization
https://bugs.ruby-lang.org/issues/6857#change-33330

Author: royaltm (Rafa?? Michalski)
Status: Assigned
Priority: Normal
Assignee: mrkn (Kenta Murata)
Category:
Target version: next minor

The algorythms to calculate E and exp programmed in BigMath module are the very straightforward interpretation of the series 1 + x + x^2/2! +
x^3/3! + ....
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 e^x, which is 1 + x + x^2/2! +
x^3/3! 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 x^4/4! 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

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.

--
http://bugs.ruby-lang.org/

```