# High Precision Metallic Means

## Background

The metallic means, starting with the famous golden mean, are defined for every natural number (positive integer), and each one is an irrational constant (it has an infinite non-recurring decimal expansion).

For a natural number $n$, the $n^{th}$ metallic mean is the root of a quadratic equation $x^2-nx=1$

The roots are always

$\frac{n\pm\sqrt{4+n^2}}{2}$

but the metallic mean is usually given as the positive root. So for this question it will be defined by:

$\frac{n+\sqrt{4+n^2}}{2}$

For $n=1$ the result is the famous golden ratio:

$\frac{1+\sqrt{5}}{2}$

## Challenge

Your code should take 2 inputs: n and p (the order is not important as long as it is consistent)

• n is a natural number indicating which metallic mean
• p is a natural number indicating how many decimal places of precision

Your code should output the nth metallic mean to p decimal places precision.

### Validity

Your code is valid if it works for values of n and p from 1 to 65,535.

You must output a decimal in the form

digit(s).digit(s) (without spaces)

For example, the golden mean to 9 decimal places is

1.618033988

Display the last digit without rounding, as it would appear in a longer decimal expansion. The next digit in the golden mean is a 7, but the final 8 in the example should not be rounded up to a 9.

The number of decimal digits must be p, which means any trailing zeroes must also be included.

$\frac{1+\sqrt{5}}{2}$

are not valid - you must use a decimal expansion.

You may output up to 1 leading newline and up to 1 trailing newline. You may not output any spaces, or any other characters besides digits and the single point/full stop/period.

## Score

This is standard code golf: your score is the number of bytes in your code.

1.618033988
$ • Right tool for the job. Jul 2, 2015 at 3:06 • @Dennis its got to be the first time CJam is nearly 3 times as long as something else ;-) Jul 2, 2015 at 3:20 # R, 116 bytes library(Rmpfr);s=scan();n=mpfr(s[1],1e6);r=(n+(4+n^2)^.5)/2;t=toString(format(r,s[2]+2));cat(substr(t,1,nchar(t)-1))  This reads two integers from STDIN and prints the result to STDOUT. You can try it online. Ungolfed + explanation: # Import the Rmpfr library for arbitrary precision floating point arithmetic library(Rmpfr) # Read two integers from STDIN s <- scan() # Set n equal to the first input as an mpfr object with 1e6 bits of precision n <- mpfr(s[1], 1e6) # Compute the result using the basic formula r <- (n + sqrt(4 + n^2)) / 2 # Get the rounded string representation of r with 1 more digit than necessary t <- toString(format(r, s[2] + 2)) # Print the result with p unrounded digits cat(substr(t, 1, nchar(t) - 1))  If you don't have the Rmpfr library installed, you can install.packages("Rmpfr") and all of your dreams will come true. # Mathematica, 50 bytes SetAccuracy[Floor[(#+Sqrt[4+#^2])/2,10^-#2],#2+1]&  Defines an anonymous function that takes n and p in order. I use Floor to prevent rounding with SetAccuracy, which I need in order to get decimal output. • @Arcinde I can't use machine precision numbers unfortunately, since they wouldn't be able to handle p>15. Jul 2, 2015 at 13:43 # CJam, 35 bytes 1'el+~1$*_2#2$2#4*+mQ+2/1$md@+s0'.t


Try it online in the CJam interpreter.

### How it works

We simply compute the formula from the question for n × 10p, get the integer and fractional part of the result divided by 10p, pad the fractional part with leading zeroes to obtain p digits and print the parts separated by a dot.

1'e  e# Push 1 and 'e'.
l+   e# Read a line from STDIN and prepend the 'e'.
~    e# Evaluate. This pushes 10**p (e.g., 1e3 -> 1000) and n.
1$* e# Copy 10**p and multiply it with n. _2# e# Copy n * 10**p and square it. 2$   e# Copy 10**p.
2#4* e# Square and multiply by 4.
+    e# Add (n * 10**p)**2 and 4 * 10**2p.
mQ   e# Push the integer part of the square root.
+2/  e# Add to n * 10**p and divide by 2.
1$md e# Perform modular division by 10**p. @+s e# Add 10**p to the fractional part and convert to string. 0'.t e# Replace the first character ('1') by a dot.  # Python 2, 92 Bytes As I am now looking at the answers, it looks like the CJam answer uses the same basic method as this. It calculates the answer for n*10**p and then adds in the decimal point. It is incredibly inefficient due to the way it calculates the integer part of the square root (just adding 1 until it gets there). n,p=input() e=10**p;r=0 while(n*n+4)*e*e>r*r:r+=1 s=str((n*e+r-1)/2);print s[:-p]+'.'+s[-p:]  # PHP, 85 78 bytes echo bcdiv(bcadd($n=$argv[bcscale($argv[2])],bcsqrt(bcadd(4,bcpow($n,2)))),2);  It uses the BC Math mathematical extension which, on some systems, could not be available. It needs to be included on the compilation time by specifying the --enable-bcmath command line option. It is always available on Windows and it seems it is included in the PHP version bundled with OSX too. Update: I applied all the hacks suggested by @blackhole in their comments (thank you!) then I squeezed the initialization of $n into its first use (3 more bytes saved) and now the code fits in a single line in the code box above.

• @Blackhole. 85, indeed. I have probably read 86 (did a slightly larger selection) and wrote 68 by mistake. Fixed now. Aug 9, 2015 at 22:41
• No problem :). You can have 1 byte less by the way: remove the parenthesis around the echo, just leave a space after it. Aug 9, 2015 at 22:43
• And since you expect bcscale to return true, you can use $n=$argv[bcscale($argv[2])]; and save 2 more bytes. Aug 9, 2015 at 22:45 • That's a nice hack. Aug 9, 2015 at 22:46 • Code dirtiness is an art :P. Oh, the last one: bcpow($n,2) instead of bcmul($n,$n) saves you 1 byte. Aug 9, 2015 at 22:48

# J, 27 Bytes

4 :'}:":!.(2+x)-:y+%:4+*:y'


### Explanation:

4 :'                      '   | Define an explicit dyad
*:y    | Square y
%:         | Square root
-:             | Half
":!.(2+x)               | Set print precision to 2+x
}:                        | Remove last digit, to fix rounding


Call it like this:

    9 (4 :'}:":!.(2+x)-:y+%:4+*:y') 1
1.618033988


Another, slightly cooler solution:

4 :'}:":!.(2+x){.>{:p._1,1,~-y'


Which calculates the roots of the polynomial x^2 - nx - 1. Unfortunately, the way J formats the result makes retreving the desired root slightly longer.