Sum the means of the two integers

Question

There are quite a few means in mathematics, such as the arithmetic mean, the geometric mean, and many others...

Definitions and Task

Note that these are the definitions for two positive integers*:

• The root mean square is the square root of the sum of their squares halved ().

• The arithmetic mean is their sum, halved ().

• The geometric mean is the square root of their product ().

• The harmonic mean is 2 divided by the sum of their inverses ( = ).

Given two integers a and b such that a, b ∈ [1, +∞), sum the means mentioned above of a and b. Your answers must be accurate to at least 3 decimal places, but you do not have to worry about rounding or floating-point precision errors.

Test Cases

a, b -> Output

7, 6 -> 25.961481565148972
10, 10 -> 40
23, 1 -> 34.99131878607909
2, 4 -> 11.657371451581236
345, 192 -> 1051.7606599443843

You can see the correct results for more test cases using this program. This is code-golf, so the shortest valid submissions that follows the standard rules wins.

_{* There are many other means, but for the purposes of this challenge we'll use the ones mentioned in the "Definitions" section.}

Answer 1

Haskell, 48 bytes

a%b=sum[((a**p+b**p)/2)**(1/p)|p<-[2,1,-1,1e-9]]

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This uses the fact that the root-square, arithmetic, harmonic, and geometric means are all special cases of the generalized mean ((a**p+b**p)/2)**(1/p) for p=2,1,-1,0. The geometric mean uses the limit p->0+, approximated as p=1e-9 which suffices for precision.

Answer 2

Mathematica, 37 bytes

-2 bytes thanks to Martin Ender. -6 bytes thanks to Jenny_mathy and function reusability thanks to JungHwan Min.

(t=1##)^.5+(2(s=+##/2)^2-t)^.5+s+t/s&

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Mathematica, 55 bytes

RootMeanSquare@#+Mean@#+GeometricMean@#+HarmonicMean@#&

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¯\_(ツ)_/¯

Answer 3

Python 3, 57 bytes

lambda a,b:(a+b+(a*a+b*b<<1)**.5)/2+(a*b)**.5+2*a*b/(a+b)

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Answer 4

R, 52 bytes

function(a,b,m=(a+b)/2,p=a*b)m+p^.5+(m^2*2-p)^.5+p/m

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Answer 5

Octave, 44 42 41 bytes

@(n)(q=mean(n))+rms(n)+(z=prod(n))^.5+z/q

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Note that TIO does not have the signal package installed, so I defined rms() in the header. On Octave Online, you can try it if you pkg load nan. I'm not sure if there are any online interpreters that load it by default, but most systems would have this package loaded by default.

Thanks to Tom Carpenter for spotting a small mistake of 2 bytes.

This defines an anonymous function, taking the input as a vector n=[a,b]. We then use inline assignment to reduce the calculation of the HM to just z/q.

Answer 6

Haskell, 48 bytes

a?b|s<-a+b,p<-a*b=s/2+sqrt(s^2/2-p)+sqrt p+2*p/s

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Explanation:

s/2 = (a+b)/2: The arithmetic mean.

sqrt(s^2/2-p) = sqrt((a^2+2*a*b+b^2)/2-a*b) = sqrt((a^2+b^2)/2): The root mean square.

sqrt p = sqrt(a*b). The geometric mean.

2*p/s = 2*a*b/(a+b). The harmonic mean.

Answer 7

Jelly, 17 bytes

²Æm,P½S
PḤ÷S+Ç+Æm

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Answer 8

MATL, 21 18 17 bytes

UYmGphX^GYmGpy/vs

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-3 bytes thanks to Luis Mendo.

Explanation

UYm               % Mean of squares, 
                  % Stack: { (a^2+b^2)/2 }
   Gp             % Product of input, a*b
                  % Stack: { (a^2+b^2)/2, a*b }
     hX^          % Concatenate into array, take square root of each element.
                  % Stack: { [RMS, HM] } 
        GYm       % Arithmetic mean of input.
                  % Stack: { [RMS,GM], AM }
           Gpy    % Product of input, duplicate AM from below.
                  % Stack: { [RMS,GM], AM, a*b, AM
              /   % Divide to get HM
                  % Stack { [RMS,GM], AM, HM}
               vs % Concatenate all to get [RMS,GM,AM,HM], sum.

Answer 9

05AB1E, 18 16 bytes

-2 bytes thanks to Erik the Outgolfer

nO;t¹O;¹Pt2¹zO/O

Explanation:

nO;t                Root mean square
n                    Raise [a, b] to [a ** 2, b ** 2]
 O                   Sum
  ;                  Half
   t                 Square root
    ¹O;             Arithmetic mean
    ¹                Retrieve stored [a, b]
     O               Sum
      ;              Half
       ¹Pt          Geometric mean
       ¹             Retrieve stored [a, b]
        P            Product
         t           Square root
          2¹zO/     Harmonic mean
           ¹         Retrieved stored [a, b]
            z        Vectorised inverse to [1 / a, 1 / b]
             O       Sum
          2   /      Get 2 divided by the sum
               O    Sum of all elements in stack

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Answer 10

Husk, 19 bytes

ṁëȯ√½ṁ□o½Σo√Π§/ΣoDΠ

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-1 thanks to H.PWiz.

Answer 11

Ohm v2, 16 bytes

²Σ½¬³Π¬³Σ½D³Πs/Σ

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Explanation

square sum halve sqrt input product sqrt input sum halve dupe input product swap div sum

...if Ohm had a verbose mode of sorts. :P

²Σ½¬³Π¬³Σ½D³Πs/Σ

                  implicit input       [[7, 6]]
²Σ½¬              root mean square
²                  square              [[49, 36]]
 Σ                 sum                 [85]
  ½                halve               [42.5]
   ¬               square root         [6.519]
    ³Π¬           geometric mean
    ³              push first input    [6.519, [7, 6]]
     Π             product             [6.519, 42]
      ¬            square root         [6.519, 6.481]
       ³Σ½        arithmetic mean
       ³           push first input    [6.519, 6.481, [7, 6]]
        Σ          sum                 [6.519, 6.481, 13]
         ½         halve               [6.519, 6.481, 6.500]
          D³Πs/   harmonic mean
          D        duplicate           [6.519, 6.481, 6.500, 6.500]
           ³       push first input    [6.519, 6.481, 6.500, 6.500, [7, 6]]
            Π      product             [6.519, 6.481, 6.500, 6.500, 42]
             s     swap                [6.519, 6.481, 6.500, 42, 6.500]
              /    divide              [6.519, 6.481, 6.500, 6.461]
               Σ  sum                  [25.961]
                  implicit output      [25.961]

Answer 12

TI-Basic (TI-84 Plus CE), 27 25 bytes

√(sum(Ans2)/2)+mean(Ans)+2prod(Ans)/sum(Ans)+√(prod(Ans

-2 bytes from Scrooble

Takes a list of two numbers in Ans, and implicitly returns the sum of the four means; e.g. run with {7,6}:prgmNAME to get 25.96148157.

Explanation:

√(sum(Ans2)/2): 8 bytes: root mean square

mean(Ans): 5 3 bytes: arithmetic mean (old: sum(Ans)/2)

2prod(Ans)/sum(Ans): 8 bytes: harmonic mean

√(prod(Ans: 3 bytes: geometric mean

+3 bytes for 3 +es

Answer 13

SOGL V0.12, 22 bytes

+:A½.².²+½√..*:√;«a/¹∑

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Answer 14

Dyalog APL, 44 bytes

{+/(2×o÷k),(.5×k←⍺+⍵),.5*⍨(o←⍺×⍵),.5×+/⍺⍵*2}

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Dyadic dfns with a on the left and b on the right.

Answer 15

C# (.NET Core), 76 bytes

+13 bytes for using System;

a=>b=>Math.Sqrt((a*a+b*b)/2)+(a+b)/2+Math.Sqrt(a*b)+2/(1/a+1/b)

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Answer 16

Julia, 49 47 bytes

a$b=(x=a+b)/2+((a^2+b^2)/2)^.5+(y=a*b)^.5+2*y/x

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Answer 17

JavaScript, 47 bytes

a=>b=>(c=a+b)/2+(c*c/2-(d=a*b))**.5+d**.5+2*d/c

quite trivial

f =

a=>b=>(c=a+b)/2+(c*c/2-(d=a*b))**.5+d**.5+2*d/c

<div oninput="r.value = f(+a.value)(+b.value)">
<p><label>a = <input id="a" type="number" step="any" value="1" /></label></p>
<p><label>b = <input id="b" type="number" step="any" value="1" /></label></p>
</div>
<p>result = <output id="r">4</output></label></p>

Answer 18

ARBLE, 49 45 bytes

-4 bytes thanks to Mr. Xcoder

((a^2+b^2)/2)^.5+(a+b)/2+(a*b)^.5+2*a*b/(a+b)

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Answer 19

Java 8, 63 bytes

a->b->Math.sqrt((a*a+b*b)/2)+(a+b)/2+Math.sqrt(a*b)+2/(1/a+1/b)

Takes both parameters as Double and outputs as Double.
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Or (also 63 bytes):

a->b->(a+b+Math.sqrt(a*a+b*b<<1))/2+Math.sqrt(a*b)+2d*a*b/(a+b)

Takes both parameters as Integer and outputs as Double.
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Answer 20

Python 2, 58 bytes

lambda a,b:((a*a+b*b)/2)**.5+(a+b)/2+(a*b)**.5+2*a*b/(a+b)

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Takes input as floats

Answer 21

Actually, 15 bytes

æßπ√+ßΣßπτ/+ßµ+

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Yay Actually has a built-in for Root Square Mean!

æßπ√+ßΣßπτ/+ßµ+  ~ Full program.

æ                ~ Arithmetic mean.
 ßπ√             ~ Product, Square root (computes geometric mean).
    +            ~ Addition.
     ßΣ          ~ Push the sum of the input.
       ßπτ       ~ Push the product of the input doubled.
          /      ~ Divide.
           +     ~ Addition.
            ßµ   ~ Push Root Square Mean.
              +  ~ Addition.

Answer 22

Groovy, 54 bytes

{a,b->c=a+b;((a*a+b*b)/2)**0.5+c/2+(a*b)**0.5+2*a*b/c}

-2 thanks to Mr. Xcoder for an edit that made me feel dumb.

Answer 23

Jq 1.5, 76 bytes

[pow((map(pow(.;2))|add)/2;.5),add/2,pow(.[0]*.[1];.5),2/(map(1/.)|add)]|add

Expanded

[
  pow((map(pow(.;2))|add)/2;.5)  # root mean square
, add/2                          # arithmetic mean
, pow(.[0]*.[1];.5)              # geometric mean
, 2/(map(1/.)|add)               # harmonic mean
]
| add                            # that just about sums it up for mean

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