# Khinchin's constant bad estimate [closed]

Inspired by How to write down numbers having an infinity of decimals? Link 🇫🇷

## Background

From Wikipedia: for almost all real numbers $$\x\$$, coefficients $$\a_i\$$ of the continued fraction expansion of $$\x\$$ have a finite geometric mean that is independent of the value of $$\x\$$ and is known as Khinchin's Constant.

Khinchin's Constant can be calculated using the following method:

1. Using the $$\n\$$ first terms of the simple continued fraction of a real number (For example Pi).
2. Compute their product
3. Apply the $$\n^{\text{th}}\$$ root on the absolute value of product calculated above

## What is a simple continued fraction

It can be expressed as: $$x=[a_0; a_1, a_2, \dots, a_n]$$ Or $$x=a_0+ \cfrac{1}{a_1+ \cfrac{1}{a_2+ \cfrac{1}{\ddots{+ \cfrac{1}{a_n}}}}}$$

And can be calculated using the following:

1. Separate $$\x\$$ into its integer part $$\a_n\$$ and decimal part $$\d\$$
• $$\a_n = \lfloor x \rfloor\$$
• $$\d = x - \lfloor x \rfloor\$$
2. Repeat using the inverse of $$\d\$$ in place of $$\x\$$ while $$\d\$$ is not 0

It is a simple continued fraction because the numerator is always 1.

For negative numbers, the same rule applies, the first term of the sequence will be negative.

For example for $$\π\$$: $$π = [a_0; a_1, a_2, a_3, \dots] = [3; 7, 15, 1, 292, \dots]$$

$$π=3+\cfrac{1}{7+ \cfrac{1}{15+ \cfrac{1}{1+ \cfrac{1}{292+ \cdots}}}}$$

And $$\-π = [-4; 1, 6, 15, 1, 292, \dots]\$$

## Challenge

Using the method used to approximate Khinchin's Constant described above:
Given a number of terms $$\n\$$ and a real number $$\x\$$, compute the geometric mean of the coefficients of the continued fraction expansion of $$\x\$$.

Standard rules apply.

## Test cases:

Terms to use Real number Expected Result Note
1 π 3
2 π 4.5825...
3 π 6.8040...
7 π 5.1179...
7 -π 5.2165...
15 1.8156...
15 - 1.9164...
20 -φ 1.0717...
20 φ 1
100 φ 1 Optional, might not work because of consecutive floating point errors

# Scoring

This is , so the answer with the least amount of bytes wins.

• Can we work with floating-point numbers and accept some small inaccuracy derived from that? Sep 29, 2021 at 22:05
• Which invalidates my answer. \o/ Sep 29, 2021 at 22:08
• I think the problem with phi using floating-point numbers may be in defining the input, rather than in the computations. phi cannot be defined exactly as a floating-point value Sep 29, 2021 at 22:39
• Being pedantic, the challenge isn’t to calculate Khinchin’s constant but rather the (related) geometric mean of the coefficients of the continued fraction expansion of $x$. According to Wikipedia, $\phi$ and $e$ are among the exceptions to the ‘almost all’ clause: the geometric means for these numbers do not converge to Khinchin’s constant. None of this changes the task as specified, but perhaps the wording could be clarified. Sep 29, 2021 at 23:39
• To add to Luis Mendo's comment, I believe that -pi has a very different continued fraction expansion from pi if we adhere to wikipedia's definition, so (7,-pi) should give a very different answer from (7,pi). Sep 30, 2021 at 12:50

# Husk, 14 bytes

^\¹Π↑¡§,o\%1⌊²


Try it online! for the last test case (or try it here for n=7, number=pi).

     ¡          # Apply function repeatedly to first results,
# collecting second results into infinite list:
§,     ²  #  combine pair of results of functions applied to arg 2:
⌊   #   floor
o\%1    #   reciprocal of fractional part
↑           # Now take arg1 elements from list,
Π            # calculate the product,
^               # and raise to the power of
\¹             # reciprocal of arg1


# Factor + math.continued-fractions math.unicode, 66 bytes

[ 1vector over [ dup next-approx ] times 1 head* Π abs nth-root ]


Try it online!

## Explanation

It's a quotation (anonymous function) that takes an integer signifying the number of terms to use and a real number and returns a real number. Assuming 3 3.141592653589793 is on the data stack when this quotation is called...

Snippet Comment Data stack (top on right)
1vector Make a vector out of the object on top of the data stack. This is how next-approx expects to take its input. 3 V{ 3.141592653589793 }
over Put a copy of the object second from the top on top of the data stack. 3 V{ 3.141592653589793 } 3
[ dup next-approx ] Push a quotation to the data stack for times to use later. 3 V{ 3.141592653589793 } 3 [ dup next-approx ]
times Take an integer and a quotation and call the quotation that many times. In this case, equivalent to dup next-approx dup next-approx dup next-approx
Inside the quotation now... 3 V{ 3.141592653589793 }
dup Copy the top data stack object 3 V{ 3.141592653589793 } V{ 3.141592653589793 }
next-approx Add the next term in the continued fraction to our vector. next-approx has stack effect ( seq -- ) so we made a copy so we don't lose it 3 V{ 3 7.062513305931052 }
dup next-approx Iteration 2 3 V{ 3 7 15.9965944066841 }
dup next-approx Iteration 3 3 V{ 3 7 15 1.003417231015 }
1 head* Remove last element 3 V{ 3 7 15 }
Π Take the product 3 315
abs Take the absolute value 3 315
nth-root Take the nth root of a number. In this case, take the cube root of 315 6.804092115953367

# Mathematica, 42 38 bytes

N@1##&@@ContinuedFraction[#2,#]^(1/#)&


Corrected formatting and size reduced by @theorist.

Inputting n=100 and variable as Pi we get the output as

2.69405


You can save 2 bytes by removing N@ but this will give an exact expression and not numeric.

The code also passes all the tests giving the exact value for each number except for -Pi for which it returns a complex number. However the magnitude of this complex number is exactly the same as that given in the expected value.

• For this site, you'll want to format your code as a program that can take the arguments as input, e.g.: N@1##&@@ContinuedFraction[#2,#]^(1/#)& (38 bytes). This would be implemented as: N@1##&@@ContinuedFraction[#2,#]^(1/#)&@@{n,x}, e.g., N@1##&@@ContinuedFraction[#2,#]^(1/#)&@@{100,Pi} Sep 30, 2021 at 6:55
• You can remove the whitespace Sep 30, 2021 at 6:57
• Thanks. I'll fix the formatting. Sep 30, 2021 at 6:58

# JavaScript (ES7), 47 bytes

Expects (n)(real).

n=>v=>(g=n=>n?~~v*g(n-1,v=1/(v%1)):1)(n)**(1/n)


Try it online!

Note: With 40+ terms, the last test case will diverge from 1 because of cumulated floating point errors.

# Ruby, 57 bytes

->n,l,r=1{n=n.abs;l.times{r*=n.to_i**(1.0/l);n=1/n%=1};r}


Try it online!