Related
I am trying to do Ruby codewars challenge and I am stuck since I pass sample tests but can't pass final one. I am getting error Expected: [8, 597], instead got: [8, 563].
Instructions :
A man has a rather old car being worth $2000. He saw a secondhand car
being worth $8000. He wants to keep his old car until he can buy the
secondhand one.
He thinks he can save $1000 each month but the prices of his old car
and of the new one decrease of 1.5 percent per month. Furthermore the
percent of loss increases by a fixed 0.5 percent at the end of every
two months.
Example of percents lost per month:
If, for example, at the end of first month the percent of loss is 1,
end of second month percent of loss is 1.5, end of third month still
1.5, end of 4th month 2 and so on ...
Can you help him? Our man finds it difficult to make all these
calculations.
How many months will it take him to save up enough money to buy the
car he wants, and how much money will he have left over?
def nbMonths(startPriceOld, startPriceNew, savingperMonth, percentLossByMonth)
months = 0
leftover = 0
currentSavings = 0
until (currentSavings + startPriceOld) >= (startPriceNew)
months += 1
months.even? ? percentLossByMonth = percentLossByMonth + 0.5 : percentLossByMonth
startPriceNew = startPriceNew * (1 - (percentLossByMonth/100))
startPriceOld = startPriceOld * (1 - (percentLossByMonth/100))
currentSavings = currentSavings + savingperMonth
end
leftover = currentSavings + startPriceOld - startPriceNew
return [months, leftover.abs.to_i]
end
I don't want to look at solutions and I don't need one here just a nudge in the right direction would be very helpful.
Also, I get that code is probably sub-optimal in a lot of ways but I have started coding 2 weeks ago so doing the best I can.
Tnx guys
Your algorithm is good. But you have two coding errors:
1) percentLossByMonth needs to be converted to float before dividing it by 100 ( 5 / 100 = 0 while (5.to_f) / 100 = 0.05 )
2) It's said in the instructions that you need to return the nearest integer of the leftover, which is leftover.round
def nbMonths(startPriceOld, startPriceNew, savingperMonth, percentLossByMonth)
months = 0
leftover = 0
currentSavings = 0
until (currentSavings + startPriceOld) >= (startPriceNew)
months += 1
percentLossByMonth += months.even? ? 0.5 : 0
startPriceNew = startPriceNew * (1 - (percentLossByMonth.to_f/100))
startPriceOld = startPriceOld * (1 - (percentLossByMonth.to_f/100))
currentSavings += savingperMonth
end
leftover = currentSavings + startPriceOld - startPriceNew
return [months, leftover.round]
end
The problem with your code has been identified, so I will just offer an alternative calculation.
r = 0.015
net_cost = 8000-2000
n = 1
months, left_over = loop do
r += 0.005 if n.even?
net_cost *= (1-r)
tot = n*1000 - net_cost
puts "n=#{n}, r=#{r}, net_cost=#{net_cost.to_i}, " +
"savings=#{(n*1000).to_i}, deficit=#{-tot.to_i}"
break [n, tot] if tot >= 0
n += 1
end
#=> [6, 766.15...]
months
#=> 6
left_over
#=> 766.15...
and prints
n=1, r=0.015, net_cost=5910, savings=1000, deficit=4910
n=2, r=0.020, net_cost=5791, savings=2000, deficit=3791
n=3, r=0.020, net_cost=5675, savings=3000, deficit=2675
n=4, r=0.025, net_cost=5534, savings=4000, deficit=1534
n=5, r=0.025, net_cost=5395, savings=5000, deficit=395
n=6, r=0.030, net_cost=5233, savings=6000, deficit=-766
I have the following program which should count the number of years it should take a population to grow to the desired size. Whenever I run this I get an infinite loop. Can someone help me identify my error?
def pop_growth(start, percent, desired)
year_count = 0
while start <= desired
year_count += 1
start = start + (start * (percent / 100))
end
return year_count
end
I'm sure that you are trying with Integers (instead floats), so you are losing precision try this
def pop_growth(start, percent, desired)
year_count = 0
while start <= desired
year_count += 1
start = start + (start * (percent.to_f / 100))
end
return year_count
end
and let me know if it works for you. if not can you send me your start, percent and desired values?
The proper answer is given by Horacio, let me rewrite this in idiomatic ruby:
def pop_growth start, percent, desired
(0..Float::INFINITY).inject(start) do |memo, years|
break years if memo > desired
memo *= (1.0 + percent / 100.0)
end
end
or, with infinite loop:
def pop_growth start, percent, desired
loop.each_with_object(years: 0, count: start) do |_, memo|
break memo[:years] if memo[:count] > desired
memo[:years] += 1
memo[:count] *= (1.0 + percent / 100.0)
end
end
Three ways.
#1 Solve equation
Solve desired = start * (1.0 + 0.01 * percent)**n for n:
def pop_growth(start, percent, desired)
Math.log(desired.to_f/start)/Math.log(1.0 + percent/100.0)
end
years = pop_growth(100, 10, 200)
#=> 7.272540897341713
years.ceil #=> 8 if desired.
#2 Compound until desire met
def pop_growth(start, percent, desired)
return 0 if start >= desired
alpha = 1.0 + 0.01 * percent
1.step.find { (start *= alpha) >= desired }
end
pop_growth 100, 10, 200
#=> 8
#3 Use recursion
def pop_growth(start, percent, desired, years=0)
return years if start >= desired
pop_growth(start*(1.0+0.01*percent), percent, desired, years+1)
end
pop_growth 100, 10, 200
#=> 8
Just add .to_f method to percent or divide by 100.0, which will convert the integer into float.
start + (start * (percent / 100))
When you are dividing, you need at least one float number in order to return the exact division answer, else Ruby will round it down to nearest whole number, which in this case percent / 100 will result in 0, assuming that the value in percent is less than 100. This will cause this statement start + (start * (percent / 100)) to become start = start + 0, which is why you are seeing the infinite loop.
This is another Codewars Ruby problem that's got me stumped:
Description:
In this kata you have to implement a base converter, which converts between arbitrary bases / alphabets. Here are some pre-defined alphabets:
bin='01'
oct='01234567'
dec='0123456789'
hex='0123456789abcdef'
allow='abcdefghijklmnopqrstuvwxyz'
allup='ABCDEFGHIJKLMNOPQRSTUVWXYZ'
alpha='abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ'
alphanum='0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ'
The function convert() should take an input (string), the source alphabet (string) and the target alphabet (string). You can assume that the input value always consists of characters from the source alphabet. You don't need to validate it.
Examples:
convert("15", dec, bin) #should return "1111"
convert("15", dec, oct) #should return "17"
convert("1010", bin, dec) #should return "10"
convert("1010", bin, hex) #should return "a"
convert("0", dec, alpha) #should return "a"
convert("27", dec, allow) #should return "bb"
convert("hello", allow, hex) #should return "320048"
Additional Notes:
The maximum input value can always be encoded in a number without loss of precision in JavaScript. In Haskell, intermediate results will probably be to large for Int.
The function must work for any arbitrary alphabets, not only the pre-defined ones.
You don't have to consider negative numbers.
I've been playing with this for a couple of days and managed to get the numeric-base-conversion portion working. It's the alphabetical part of it that I can't figure out how to approach, and my brain is tired from trying. Here's my code:
def convert(input, source, target)
bases = {
:bin => '01',
:oct => '01234567',
:dec => '0123456789',
:hex => '0123456789abcdef',
:allow => 'abcdefghijklmnopqrstuvwxyz',
:allup => 'ABCDEFGHIJKLMNOPQRSTUVWXYZ',
:alpha => 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ',
:alphanum => '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ'
}
base_from , base_to = 0
src_num_switch = 1 if source == bases[:bin] || [:oct] || [:dec] || [:hex]
tgt_num_switch = 1 if target == bases[:bin] || [:oct] || [:dec] || [:hex]
src_num_switch = 0 if source == bases[:allow] || [:allup] || [:alpha] || [:alphanum]
tgt_num_switch = 0 if target == bases[:allow] || [:allup] || [:alpha] || [:alphanum]
if source == bases[:bin] then base_from = 2
elsif source == bases[:oct] then base_from = 8
elsif source == bases[:dec] then base_from = 10
elsif source == bases[:hex] then base_from = 16
elsif source == bases[:allow] then base_from = 13
elsif source == bases[:allup] then base_from = 13
elsif source == bases[:alpha] then base_from = 13
elsif source == bases[:alphanum] then base_from = 13
else puts ":( no source match found :("
end
if target == bases[:bin] then puts base_to = 2
elsif target == bases[:oct] then base_to = 8
elsif target == bases[:dec] then base_to = 10
elsif target == bases[:hex] then base_to = 16
elsif target == bases[:allow] then base_to = 13
elsif target == bases[:allup] then base_to = 13
elsif target == bases[:alpha] then base_to = 13
elsif target == bases[:alphanum] then base_to = 13
else puts ":( no target match found :("
end
if base_from == base_to then
return input
elsif src_num_switch == 1 && tgt_num_switch == 1 then
return Integer(input, base_from).to_s(base_to)
elsif src_num_switch == 0 && tgt_num_switch == 0 then
return Integer(input, base_from).to_s(base_to)
# ### # :::::::::::::::::::::::::::::::::::::::::::::
else
puts "ouch, something broke"
end
end
I've got everything down to the "# ### #" portion working for me. Can anyone give me an idea of how to do the alpha-base portion? I've tried the following but had no luck:
if base_from == base_to then return input
elsif src_num_switch == 1 && tgt_num_switch == 1 then
return Integer(input, base_from).to_s(base_to)
elsif src_num_switch == 1 && tgt_num_switch == 0 then
if target == bases[:allup] then return bases[input.index].to_s.upcase
elsif target == bases[:allow] then return bases[input.index].to_s.downcase
end
end
elsif src_num_switch == 0 && tgt_num_switch == 1 then
return input.index.to_s(base_to)
elsif src_num_switch == 0 && tgt_num_switch == 0 then
return Integer(input, base_from).to_s(base_to)
else
puts "ouch, something broke"
end
This one too:
elsif src_num_switch == 1 && tgt_num_switch == 0 then # number-base to alphanumeric-base
if target == bases[:allup] then
return bases[input.index].to_s.upcase
elsif target == bases[:allow] then
return bases[input.index].to_s.downcase
end
elsif src_num_switch == 0 && tgt_num_switch == 1 then # alpha-base to number-base
return input.index.to_s(base_to)
There may be a very clever built-in Ruby solution, but I would guess based on the custom alphabets describing the number systems that there is not. So, I don't have a direct answer to how to complete your code, but I would suggest a slightly different strategy.
Converting from a decimal
Any number system can be converted from the decimal system like so:
vals_in_system = system.length
output_in_system = []
while (decimal_num != 0)
index_of_next_val = decimal_num % system.length
output_in_system.unshift(system[index_of_next_val])
decimal_num = decimal_num / vals_in_system # truncating is desired here
end
It's a bit tricky. This algorithm first tries to determine what value it has to put in the last position (which has the most granularity in whatever number system you're using). E.g. if you were to represent 12 in decimal (yes, it already is, but using this algorithm), a 2 has to go in the last position - no number you put in the tens place or higher will otherwise help you represent 12. If you were to represent 3 in binary, a 1 has to go in the last position of the binary - nothing you put in the next position will get you to a 3. Once it determines this, it can divide by the base, which will leave you with the number you would use to calculate the remaining positions. For example, if you were to represent 123 in decimal, dividing by 10 (the decimal base) and truncating would give you 12. 12 is the representation of the original number except for the final position (which was chopped off by dividing by the base). (I realize this isn't the clearest explanation so let me know if you have questions.) Some examples:
E.g. the decimal number 15 can be converted to binary:
15 % 2 = 1 # last position
15 / 2 = 7
7 % 2 = 1 # next to last position
7 / 2 = 3
3 % 2 = 1 # 3rd to last position
3 / 2 = 1
1 % 2 = 1 # 4th to last position
1 / 2 = 0 # stop
That's kinda boring, you just get 1111. Try something a little more interesting, like 10:
10 % 2 = 0 # last position
10 / 2 = 5
5 % 2 = 1 # next to last position
5 / 2 = 2
2 % 2 = 0 # 3rd to last position
2 / 2 = 1
1 % 2 = 1 # 4th to last position
1 / 2 = 0 # stop
And you get 1010, which is indeed 10 in binary. You can do this with any of those alphabets.
Converting to a decimal
Similarly, any number system can be converted to a decimal by doing the opposite:
vals_in_system = from.length
output_in_decimal = 0
val.each_char do |next_val|
output_in_decimal *= vals_in_system
output_in_decimal += from.index(next_val)
end
This is easier to understand than the "from decimal" algorithm. Consider if you were to apply this to the decimal number 123. This algorithm is essentially doing this equation
((1 * 10) + 2) * 10) + 3
or, much easier to read:
1 * (10 * 10) + 2 * (10) + 3
Just iteratively. It works for other number systems, by replacing the 10 with the base of the number system (i.e. the number of values the number system contains). The only other magic it does it converts a value in the number system into a decimal number using .index.
E.g. converting "bcdl" to decimal from their "allow" system. Using a 0-index, b = the 1st position, c = 2nd, d = 3rd, l = 11th
Start with 0
Multiply by the number system base, which is 26 (26 letters in the lowercase alphabet) = 0
Add the decimal value of b (1) => 1
1 * 26 = 26
Add decimal value of c (2) => 28
28 * 26 => 728
Add decimal value of d (3) => 731
731 * 26 => 19006
Add decimal value of l (11) => 19017 That's the decimal notation for "bcdl".
Putting it together
Once you have converters to and from decimal, you can write a pretty straightforward wrapper to handle every situation (I put DEC in a constant to make it visible in this method, it's the same as dec):
def convert(val, from, to)
case
when from == to then val
when from == DEC then convert_from_dec(val, to)
when to == DEC then convert_to_dec(val, from)
else
convert_from_dec(convert_to_dec(val, from), to)
end
end
After that, you mostly have to deal with edge cases.
As I said, not a direct answer to your question, but it seems like you'll have to use this general approach for the alpha number systems, at which point you may as well use it for everything :)
I honestly tried not to look at alexcavalli's solution, but in the end came to exact same algorithm with a different code. So for explanation why it works look at his much more explained answer. Here it's only code, written in a way if you save it under base_converter.rb name you can run it as:
$ ruby ../base_converer.rb 123 hex dec #=> 291
bases = {
bin: '01',
oct: '01234567',
dec: '0123456789',
hex: '0123456789abcdef',
allow: 'abcdefghijklmnopqrstuvwxyz',
allup: 'ABCDEFGHIJKLMNOPQRSTUVWXYZ',
alpha: 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ',
alphanum: '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ',
}
def to_int(num, src)
src_map = src.split('').map.with_index.to_h
num.reverse.each_char.with_index.sum{ |c, i| src_map[c] * (src.size ** i) }
end
def from_int(num, dst)
res = []
while num > 0
res << dst[num % dst.size]
num /= dst.size
end
res.join.reverse
end
def convert(num, src, dst)
from_int(to_int(num, src), dst)
end
if ARGV.size > 2
puts convert(ARGV[0], bases[ARGV[1].to_sym], bases[ARGV[2].to_sym])
end
I recently had to merge images like these:
These images represents different sorts of events happening at different locations. The idea is to merge these images in a way to keep the "hot" zones of each images (red-yellow-green) to get a global picture of what happens globally.
In my current approach, I take take the second image and extracts the red/green channel, in order to form a mask of the relevant parts, like this:
Then I merge it with the first image by using this mask so only the relevant parts gets copied over.
Here is the script used for this:
#!/bin/bash
# Extract RGB
convert b.png -colorspace RGB -separate b-sep-%d.png
# Keep red & green only
convert b-sep-2.png b-sep-0.png -compose minus -composite b-tmp-br.png
convert b-sep-2.png b-sep-1.png -compose minus -composite b-tmp-bg.png
convert b-tmp-br.png b-tmp-bg.png -compose plus -composite -level 10%,100% b-mask.png
# Composite!
composite b.png a.png b-mask.png final.png
Here is my current result so far:
As you can see, it works well for the red-yellow-green part, but the blue part is missing. The problem is that if I enlarge the mask to include the blue part, then it'll overwrite red-yellow-green parts from the first image with blue parts from the second one! This is already visible in the final result, the top left first image red part is overwritten by the green part of the second image.
Getting the blue part correctly is trickier, but I think the following algorithm should work (pseudo code):
function merge_pixel(pixel a, pixel b)
{
points = { :red => 4, :yellow => 3, :green => 2, :blue => 1, :default => 0 }
a_points = points[a.color()]
b_points = points[b.color()]
return a_points > b_points ? a : b
}
That is, when merging images, copy the pixel from image a or b depending on which color is the most important for the final image. Maybe this algorithm isn't sound (e.g how to handle the gradient part, maybe with a threshold), feel free to debunk it.
REAL QUESTION:
using imagemagick, how to:
get the desired result using any technique/whatever?
implement the algorithm from above?
You don't need to answer both questions, just finding an imagemagick way of getting the desired result is fine.
[EDIT]
Hint: I just had an idea, I think you can generate the masks (including blue parts) for both images and do some "set intersection/union/difference/whatever" of the masks to generate the appropriate "final" mask so only the real relevant parts of image b is copied over.
Ok, I did the "merge_pixel" strategy and it worked!
require 'RMagick'
include Magick
def pixel_score(p)
r, g, b = [p.red, p.green, p.blue].map{ |i| i / 256 }
is_flat = (r-g).abs < 20 && (r-b).abs < 20 && (g-b).abs < 20
is_grey = is_flat && r < 200
is_red = r >= 240 && g <= 100 # && b < 10
is_yellow = r >= 150 && g >= 100 && b <= 10
is_green = r <= 200 && g >= 200 && b <= 100
is_cyan = r <= 10 && g >= 100 && b >= 30
is_blue = r <= 10 && g <= 100 && b >= 200
if is_red
(999**8) + (r - g)
elsif is_yellow
(999**7) + (r + g)
elsif is_green
(999**6) + (g - b)
elsif is_cyan
(999**5) + (g + b)
elsif is_blue
(999**4) + (b - g)
else
(999**1) + r ** 3 + g ** 2 + b
end
end
def rmagick_merge(file_a, file_b, file_merged)
img_a = ImageList.new(file_a)
img_b = ImageList.new(file_b)
result = Image.new(img_a.columns, img_a.rows)
img_a.columns.times do |col|
img_a.rows.times do |row|
pixel_a = img_a.pixel_color(col, row)
pixel_b = img_b.pixel_color(col, row)
pixel = [pixel_a, pixel_b].sort_by{ |p| pixel_score(p) }.last
#pixel = [pixel_a, pixel_b].sort_by{ |p| [p.red - p.green, p.green, p.blue] }.first
#pixel = [pixel_a, pixel_b].sort_by{ |p| [p.red - p.green - p.blue * 100, p.green, p.blue] }.last
result.pixel_color(col, row, pixel)
end
end
result.format = "PNG"
result.write(file_merged)
end
if __FILE__ == $0
if ARGV.size < 3
puts "usage #{__FILE__} a.png b.png merged.png"
exit 1
end
rmagick_merge(ARGV[0], ARGV[1], ARGV[3])
end
Here is the result (not perfect but fine tuned for my needs on the real pictures):
Recently I found this in some code I wrote a few years ago. It was used to rationalize a real value (within a tolerance) by determining a suitable denominator and then checking if the difference between the original real and the rational was small enough.
Edit to clarify : I actually don't want to convert all real values. For instance I could choose a max denominator of 14, and a real value that equals 7/15 would stay as-is. It's not as clear that as it's an outside variable in the algorithms I wrote here.
The algorithm to get the denominator was this (pseudocode):
denominator(x)
frac = fractional part of x
recip = 1/frac
if (frac < tol)
return 1
else
return recip * denominator(recip)
end
end
Seems to be based on continued fractions although it became clear on looking at it again that it was wrong. (It worked for me because it would eventually just spit out infinity, which I handled outside, but it would be often really slow.) The value for tol doesn't really do anything except in the case of termination or for numbers that end up close. I don't think it's relatable to the tolerance for the real - rational conversion.
I've replaced it with an iterative version that is not only faster but I'm pretty sure it won't fail theoretically (d = 1 to start with and fractional part returns a positive, so recip is always >= 1) :
denom_iter(x d)
return d if d > maxd
frac = fractional part of x
recip = 1/frac
if (frac = 0)
return d
else
return denom_iter(recip d*recip)
end
end
What I'm curious to know if there's a way to pick the maxd that will ensure that it converts all values that are possible for a given tolerance. I'm assuming 1/tol but don't want to miss something. I'm also wondering if there's an way in this approach to actually limit the denominator size - this allows some denominators larger than maxd.
This can be considered a 2D minimization problem on error:
ArgMin ( r - q / p ), where r is real, q and p are integers
I suggest the use of Gradient Descent algorithm . The gradient in this objective function is:
f'(q, p) = (-1/p, q/p^2)
The initial guess r_o can be q being the closest integer to r, and p being 1.
The stopping condition can be thresholding of the error.
The pseudo-code of GD can be found in wiki: http://en.wikipedia.org/wiki/Gradient_descent
If the initial guess is close enough, the objective function should be convex.
As Jacob suggested, this problem can be better solved by minimizing the following error function:
ArgMin ( p * r - q ), where r is real, q and p are integers
This is linear programming, which can be efficiently solved by any ILP (Integer Linear Programming) solvers. GD works on non-linear cases, but lack efficiency in linear problems.
Initial guesses and stopping condition can be similar to stated above. Better choice can be obtained for individual choice of solver.
I suggest you should still assume convexity near the local minimum, which can greatly reduce cost. You can also try Simplex method, which is great on linear programming problem.
I give credit to Jacob on this.
A problem similar to this is solved in the Approximations section beginning ca. page 28 of Bill Gosper's Continued Fraction Arithmetic document. (Ref: postscript file; also see text version, from line 1984.) The general idea is to compute continued-fraction approximations of the low-end and high-end range limiting numbers, until the two fractions differ, and then choose a value in the range of those two approximations. This is guaranteed to give a simplest fraction, using Gosper's terminology.
The python code below (program "simpleden") implements a similar process. (It probably is not as good as Gosper's suggested implementation, but is good enough that you can see what kind of results the method produces.) The amount of work done is similar to that for Euclid's algorithm, ie O(n) for numbers with n bits, so the program is reasonably fast. Some example test cases (ie the program's output) are shown after the code itself. Note, function simpleratio(vlo, vhi) as shown here returns -1 if vhi is smaller than vlo.
#!/usr/bin/env python
def simpleratio(vlo, vhi):
rlo, rhi, eps = vlo, vhi, 0.0000001
if vhi < vlo: return -1
num = denp = 1
nump = den = 0
while 1:
klo, khi = int(rlo), int(rhi)
if klo != khi or rlo-klo < eps or rhi-khi < eps:
tlo = denp + klo * den
thi = denp + khi * den
if tlo < thi:
return tlo + (rlo-klo > eps)*den
elif thi < tlo:
return thi + (rhi-khi > eps)*den
else:
return tlo
nump, num = num, nump + klo * num
denp, den = den, denp + klo * den
rlo, rhi = 1/(rlo-klo), 1/(rhi-khi)
def test(vlo, vhi):
den = simpleratio(vlo, vhi);
fden = float(den)
ilo, ihi = int(vlo*den), int(vhi*den)
rlo, rhi = ilo/fden, ihi/fden;
izok = 'ok' if rlo <= vlo <= rhi <= vhi else 'wrong'
print '{:4d}/{:4d} = {:0.8f} vlo:{:0.8f} {:4d}/{:4d} = {:0.8f} vhi:{:0.8f} {}'.format(ilo,den,rlo,vlo, ihi,den,rhi,vhi, izok)
test (0.685, 0.695)
test (0.685, 0.7)
test (0.685, 0.71)
test (0.685, 0.75)
test (0.685, 0.76)
test (0.75, 0.76)
test (2.173, 2.177)
test (2.373, 2.377)
test (3.484, 3.487)
test (4.0, 4.87)
test (4.0, 8.0)
test (5.5, 5.6)
test (5.5, 6.5)
test (7.5, 7.3)
test (7.5, 7.5)
test (8.534537, 8.534538)
test (9.343221, 9.343222)
Output from program:
> ./simpleden
8/ 13 = 0.61538462 vlo:0.68500000 9/ 13 = 0.69230769 vhi:0.69500000 ok
6/ 10 = 0.60000000 vlo:0.68500000 7/ 10 = 0.70000000 vhi:0.70000000 ok
6/ 10 = 0.60000000 vlo:0.68500000 7/ 10 = 0.70000000 vhi:0.71000000 ok
2/ 4 = 0.50000000 vlo:0.68500000 3/ 4 = 0.75000000 vhi:0.75000000 ok
2/ 4 = 0.50000000 vlo:0.68500000 3/ 4 = 0.75000000 vhi:0.76000000 ok
3/ 4 = 0.75000000 vlo:0.75000000 3/ 4 = 0.75000000 vhi:0.76000000 ok
36/ 17 = 2.11764706 vlo:2.17300000 37/ 17 = 2.17647059 vhi:2.17700000 ok
18/ 8 = 2.25000000 vlo:2.37300000 19/ 8 = 2.37500000 vhi:2.37700000 ok
114/ 33 = 3.45454545 vlo:3.48400000 115/ 33 = 3.48484848 vhi:3.48700000 ok
4/ 1 = 4.00000000 vlo:4.00000000 4/ 1 = 4.00000000 vhi:4.87000000 ok
4/ 1 = 4.00000000 vlo:4.00000000 8/ 1 = 8.00000000 vhi:8.00000000 ok
11/ 2 = 5.50000000 vlo:5.50000000 11/ 2 = 5.50000000 vhi:5.60000000 ok
5/ 1 = 5.00000000 vlo:5.50000000 6/ 1 = 6.00000000 vhi:6.50000000 ok
-7/ -1 = 7.00000000 vlo:7.50000000 -7/ -1 = 7.00000000 vhi:7.30000000 wrong
15/ 2 = 7.50000000 vlo:7.50000000 15/ 2 = 7.50000000 vhi:7.50000000 ok
8030/ 941 = 8.53347503 vlo:8.53453700 8031/ 941 = 8.53453773 vhi:8.53453800 ok
24880/2663 = 9.34284641 vlo:9.34322100 24881/2663 = 9.34322193 vhi:9.34322200 ok
If, rather than the simplest fraction in a range, you seek the best approximation given some upper limit on denominator size, consider code like the following, which replaces all the code from def test(vlo, vhi) forward.
def smallden(target, maxden):
global pas
pas = 0
tol = 1/float(maxden)**2
while 1:
den = simpleratio(target-tol, target+tol);
if den <= maxden: return den
tol *= 2
pas += 1
# Test driver for smallden(target, maxden) routine
import random
totalpass, trials, passes = 0, 20, [0 for i in range(20)]
print 'Maxden Num Den Num/Den Target Error Passes'
for i in range(trials):
target = random.random()
maxden = 10 + round(10000*random.random())
den = smallden(target, maxden)
num = int(round(target*den))
got = float(num)/den
print '{:4d} {:4d}/{:4d} = {:10.8f} = {:10.8f} + {:12.9f} {:2}'.format(
int(maxden), num, den, got, target, got - target, pas)
totalpass += pas
passes[pas-1] += 1
print 'Average pass count: {:0.3}\nPass histo: {}'.format(
float(totalpass)/trials, passes)
In production code, drop out all the references to pas (etc.), ie, drop out pass-counting code.
The routine smallden is given a target value and a maximum value for allowed denominators. Given maxden possible choices of denominators, it's reasonable to suppose that a tolerance on the order of 1/maxden² can be achieved. The pass-counts shown in the following typical output (where target and maxden were set via random numbers) illustrate that such a tolerance was reached immediately more than half the time, but in other cases tolerances 2 or 4 or 8 times as large were used, requiring extra calls to simpleratio. Note, the last two lines of output from a 10000-number test run are shown following the complete output of a 20-number test run.
Maxden Num Den Num/Den Target Error Passes
1198 32/ 509 = 0.06286837 = 0.06286798 + 0.000000392 1
2136 115/ 427 = 0.26932084 = 0.26932103 + -0.000000185 1
4257 839/2670 = 0.31423221 = 0.31423223 + -0.000000025 1
2680 449/ 509 = 0.88212181 = 0.88212132 + 0.000000486 3
2935 440/1853 = 0.23745278 = 0.23745287 + -0.000000095 1
6128 347/1285 = 0.27003891 = 0.27003899 + -0.000000077 3
8041 1780/4243 = 0.41951449 = 0.41951447 + 0.000000020 2
7637 3926/7127 = 0.55086292 = 0.55086293 + -0.000000010 1
3422 27/ 469 = 0.05756930 = 0.05756918 + 0.000000113 2
1616 168/1507 = 0.11147976 = 0.11147982 + -0.000000061 1
260 62/ 123 = 0.50406504 = 0.50406378 + 0.000001264 1
3775 52/3327 = 0.01562970 = 0.01562750 + 0.000002195 6
233 6/ 13 = 0.46153846 = 0.46172772 + -0.000189254 5
3650 3151/3514 = 0.89669892 = 0.89669890 + 0.000000020 1
9307 2943/7528 = 0.39094049 = 0.39094048 + 0.000000013 2
962 206/ 225 = 0.91555556 = 0.91555496 + 0.000000594 1
2080 564/1975 = 0.28556962 = 0.28556943 + 0.000000190 1
6505 1971/2347 = 0.83979548 = 0.83979551 + -0.000000022 1
1944 472/ 833 = 0.56662665 = 0.56662696 + -0.000000305 2
3244 291/1447 = 0.20110574 = 0.20110579 + -0.000000051 1
Average pass count: 1.85
Pass histo: [12, 4, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
The last two lines of output from a 10000-number test run:
Average pass count: 1.77
Pass histo: [56659, 25227, 10020, 4146, 2072, 931, 497, 233, 125, 39, 33, 17, 1, 0, 0, 0, 0, 0, 0, 0]