```Issue #6857 has been updated by royaltm (Rafa?? Michalski).

Having fast exp() allows us to speed up BigMath.log(). Especially for calculations with large precision.

The area hyperbolic tangent power series performs better when the domain (x) of the function is closer to 1.
Additionally for x > 10 there is a significant linear performance degradation proportional to x.

So the first thing would be to narrow "no decimal shift" domain limitation to just 0.1 <= x <= 10.
The current implementation of BigMath.log uses range: 0.1 <= x < 100.

But this is just a prerequisite.

The real performance boost we gain from the following rule:

Let's suppose y ~ log(x) where y is calculated with much lesser precision than we actually need.
We may find then such an A:

A = x / exp(y)

which is very close to 1.

Now we can use it to calculate logarithm with the accurate precision from:

log(x) = y + log(a)

The implementation:

def log(x, prec)
raise ArgumentError, "Zero or negative precision for log" if prec <= 0
raise ArgumentError, "Zero or negative argument for log" if x.round(prec) <= 0
return BigDecimal('0') if x.round(prec) == BigDecimal('1')
return BigDecimal::INFINITY if x.infinite?

n = prec + BigDecimal.double_fig

shift = x.exponent
ten = BigDecimal('10')
if shift < 0 || x > 10
x = x.mult(BigDecimal("1E#{-shift}"), n)
else
shift = 0
end

if prec < 26 # 26 was chosen based on experiments
y = BigMath.log(x, prec)
else
y  = log(x, Math.exp(Math.log(prec)/2).round)

a = x.div(exp(y, n), n)
y += BigMath.log(a, prec)
end

y += log(ten, prec).mult(shift, n) unless shift.zero?
y
end

require 'benchmark'
require 'bigdecimal/util'

def testlog(p, range=100.0, iter=100, count=1000)
Benchmark.bm(20, 'ext', 'new') do |b|
count.times.map { rand*range }.inject([0,0]) do |(tt1,tt2), n|
nbig = n.to_d
a1 = a2 = nil
GC.disable
t1 = b.report("#{n} ext") { iter.times { a1 = BigMath.log(nbig, p) } }
t2 = b.report("#{n} new") { iter.times { a2 = log(nbig, p) } }
GC.enable
unless a1.round(p - a1.exponent) == a2.round(p - a2.exponent)
raise "bad #{a1.round(p - a1.exponent)} <> #{a2.round(p - a2.exponent)}"
end
[t1/count + tt1, t2/count + tt2]
end
end
nil
end

To get the idea of speed up factor I'll present some summaries:

testlog(9, 10.0)
ext                    0.026100   0.000000   0.026100 (  0.025777)
new                    0.025600   0.000000   0.025600 (  0.025944)

we didn't optimize anything within the domain range of 0 < x < 10.0 and precision (< 26) so the new implementation performs similarly
(it's slightly slower due to some overhead of wrapper code)

testlog(9, 100.0)
ext                    0.236000   0.000000   0.236000 (  0.235998)
new                    0.055900   0.000000   0.055900 (  0.055529)

just narrowing the domain range calculated wihtout decimal shift to 0.1 <= x <= 10 gives as a significant speed increase.

Now let's try some serious BigDecimal precision:

testlog(99, 10.0)
ext                    0.202900   0.000000   0.202900 (  0.201852)
new                    0.075600   0.000000   0.075600 (  0.076487)

we can now see the effect of approximation algorithm

let's increase the domain range:

testlog(99, 100.0)
ext                    2.387300   0.004000   2.391300 (  2.390849)
new                    0.158300   0.001700   0.160000 (  0.160178)

the combined effect of both approximation and domain decimal shift range limitation gives us more than 10 times performance boost (average)

testlog(999, 10.0, 2)
ext                    1.470000   0.000000   1.470000 (  1.469803)
new                    0.031300   0.000000   0.031300 (  0.031546)

Large mantissa tests:

e = E(10000)
l1 = timer{ BigMath.log(e, 10000) } # -> 318.629882
l2 = timer{ log(e, 10000) }         # -> 1.524671
l1.round(10000) == l2.round(10000)
=> true
l1.round(10000) == 1
=> true

pi = BigMath.PI(10000)
l1 = timer{ BigMath.log(pi, 10000) } # -> 371.913958
l2 = timer{ log(pi,10000) }         # -> 1.892104

l1.round(10000) == l2.round(10000)
=> true

----------------------------------------
Feature #6857: bigdecimal/math BigMath.E/BigMath.exp R. P. Feynman inspired optimization
https://bugs.ruby-lang.org/issues/6857#change-28836

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

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/

```