# Calculate the Mean mean of two numbers

disclaimer: the Mean mean is made up by me

Define the arithmetic mean of $$\n\$$ numbers as $$M_1(x_1,...,x_n)=\frac{x_1+x_2+...+x_n}{n}$$ Define the geometric mean of $$\n\$$ numbers as $$M_0(x_1,...,x_n)=\root{n}\of{x_1x_2...x_n}$$ Define the harmonic mean of $$\n\$$ numbers as $$M_{-1}(x_1,...,x_n)=\frac{n}{\frac{1}{x_2} + \frac{1}{x_2} + ... + \frac{1}{x_n}}$$ Define the quadratic mean of $$\n\$$ numbers as $$M_2(x_1,...,x_n)=\root\of{\frac{x_1^2+x_2^2+...+x_n^2}{n}}$$ The Mean mean ($$\M_M\$$) is defined as follows: Define four sequences ($$\a_k, b_k, c_k, d_k\$$) as $$a_0=M_1(x_1,...,x_n),\\b_0=M_0(x_1,...,x_n),\\c_0=M_{-1}(x_1,...,x_n),\\d_0=M_2(x_1,...,x_n),\\ a_{k+1}=M_1(a_k,b_k,c_k,d_k),\\b_{k+1}=M_0(a_k,b_k,c_k,d_k),\\c_{k+1}=M_{-1}(a_k,b_k,c_k,d_k),\\d_{k+1}=M_2(a_k,b_k,c_k,d_k)$$ All four sequences converge to the same number, $$\M_M(x_1,x_2,...,x_n)\$$.

## Example

The Mean mean of 1 and 2 is calculated as follows: start with $$a_0 = (1+2)/2 = 1.5, b_0 = \root\of{1 * 2} = \root\of2 \approx 1.4142,\\ c_0 = \frac2{\frac1{1}+\frac1{2}} = \frac4{3} \approx 1.3333, d_0 = \root\of{\frac{1^2+2^2}2} = \root\of{\frac5{2}} \approx 1.5811.$$ Then $$a_1 = \frac{1.5+1.4142+1.3333+1.5811}4 \approx 1.4571,\\ b_1 = \root^4\of{1.5*1.4142*1.3333*1.5811} \approx 1.4542,\\ c_1 = \frac4{\frac1{1.5}+\frac1{1.4142}+\frac1{1.3333}+\frac1{1.5811}} \approx 1.4512,\\ d_1 = \root\of{\frac{1.5^2+1.4142^2+1.3333^2+1.5811^2}4} \approx 1.4601.$$ The further calculation of the sequences should be clear. It can be seen that they converge to the same number, approximately $$\1.45568889\$$.

## Challenge

Given two positive real numbers, $$\a\$$ and $$\b\$$ ($$\a), calculate their Mean mean $$\M_M(a,b)\$$.

## Test cases

1 1 => 1
1 2 => 1.45568889
100 200 => 145.568889
2.71 3.14 => 2.92103713
0.57 1.78 => 1.0848205
1.61 2.41 => 1.98965438
0.01 100 => 6.7483058


# Notes

• Your program is valid if the difference between its output and the correct output is not greater than 1/100000 of the absolute value of the difference between input numbers.
• The output should be a single number.

This is , so the shortest code wins!

# Wolfram Language (Mathematica), 52 bytes

#//.x_:>N@{M@x,E^M@Log@x,1/M[1/x],M[x^2]^.5}&
M=Mean


Try it online!

In my first approach I used these builtins
Mean GeometricMean HarmonicMean and RootMeanSquare

Here are some substitutions for saving bytes

HarmonicMean --> 1/Mean[1/x] by @Robin Ryder (3 bytes saved)
GeometricMean --> E^Mean@Log@x by @A. Rex (2 bytes saved)
RootMeanSquare --> Mean[x^2]^.5 by @A. Rex (4 bytes saved)

finally we can assign Mean to M (as proposed by @ovs) and save 5 more bytes

• Save 2 bytes by recoding GeometricMean – Robin Ryder Apr 1 '19 at 9:58
• @RobinRyder I believe you mean Harmonic.. nice! – J42161217 Apr 1 '19 at 10:04
• Save 8 more bytes: #//.x_:>N@{Mean@x,E^Mean@Log@x,1/Mean[1/x],Mean[x^2]^.5}& – A. Rex Apr 1 '19 at 10:41
• @ovs edited..... – J42161217 Apr 1 '19 at 11:27

# R, 7069 67 bytes

x=scan();?=mean;while(x-?x)x=c((?x^2)^.5,?x,2^?log2(x),1/?1/x);?x


Try it online!

-1 byte with better conditioning.
-2 bytes by switching to base 2.

Like some other answers, this uses the expression of the geometric mean as an arithmetic mean on the log scale (here in base 2): $$M_0(x_1,\ldots, x_n) = 2^{M_1(\log_2 x_1,\ldots,\log_2 x_n)}.$$

It also uses the fact that $$\\forall k, d_k\geq a_k \geq b_k\geq c_k\$$, i.e. $$\d_k=\max(a_k, b_k, c_k, d_k)\$$. The condition $$\a_k=b_k=c_k=d_k\$$ is therefore equivalent to $$\d_k = M_1(a_k, b_k, c_k, d_k)\$$, which is what I use in the while loop; this is achieved by abusing the syntax of while which only considers the first element when the condition is a vector, hence the order in which the means are stored. (Note that we could also use $$\c_k\$$ instead since it is the minimum of the four, but we could not use $$\a_k\$$ or $$\b_k\$$ in the condition.)

When we exit the while loop, x is a constant vector. The final ?x computes its mean to reduce it to a scalar.

• Shouldn't it be $ln x_n$ instead of $log x_n$ ? – Tau Mar 31 '19 at 23:40
• @Tau Yes, I was denoting natural logarithm by $log$, which is the default in R. Anyway, I have now changed it to base 2 logarithm for -2 bytes. – Robin Ryder Apr 1 '19 at 7:14

# J, 34 bytes

(31 as an expression without the assignment to variable f)

f=:1{(^.z,%z,*:z,[z=:(+/%#)&.:)^:_


Try it online!

For functions a and b, a &.: b ("a under b" (related challenge)) is equivalent to (b inv) a b -- apply b, then a, then inverse of b. In this case, geometric/harmonic/quadratic mean is the arithmetic mean "under" logarithm, inversion, and square respectively.

# TI-BASIC, 4235 34 bytes

-1 byte thanks to @SolomonUcko

While max(ΔList(Ans:{mean(Ans),√(mean(Ans²)),mean(Ans^-1)^-1,e^(mean(ln(Ans:End:Ans(1


Input is a list of two integers in Ans.
Output is stored in Ans and is automatically printed out when the program completes.

Formulas used for geometric, harmonic, and quadratic means are based off of user202729's explanation.

Example:

{1,2
{1 2}
prgmCDGFB
1.455688891
{100,200
{100 200}
prgmCDGFB
145.5688891


Explanation:
(Newlines have been added for clarification. They do NOT appear in the code.)

While max(ΔList(Ans           ;loop until all elements of the current list are equal
; the maximum of the change in each element will be 0
{                             ;create a list containing...
mean(Ans),                   ; the arithmetic mean
mean(Ans^-1)^-1,             ; the harmonic mean
e^(mean(ln(Ans               ; and the geometric mean
End
Ans(1                         ;keep the first element in "Ans" and implicitly print it


Notes:

TI-BASIC is a tokenized language. Character count does not equal byte count.

e^( is this one-byte token.

^-1 is used for this one-byte token.
I opted for writing ^-1 instead because the token looks like ֿ¹ when in a code block.

√( is this one-byte token.

ΔList( is this two-byte token.

• I think you can save a parenthesis by putting the geometric mean last. – Solomon Ucko Apr 1 '19 at 11:06
• @SolomonUcko ah, thanks for noticing! Didn't consider that before. – Tau Apr 1 '19 at 13:01
• max(DeltaList(Ans -> variance(Ans. – lirtosiast Jun 12 '19 at 2:29

# Java 10, 234229214211215206203196180 177 bytes

a->{for(;a-a>4e-9;){double l=a.length,A[]={0,0,0,1};for(var d:a){A+=d/l;A*=Math.pow(d,1/l);A+=1/d;A+=d*d;}A=l/A;A=Math.sqrt(A/l);a=A;}return a;}


-5 bytes thanks to @PeterCordes.
-15 more bytes thanks to @PeterCordes, inspired by @RobinRyder's R answer.
+4 bytes because I assumed the inputs are pre-ordered.
-27 bytes thanks to @OlivierGrégoire.

Try it online.

Explanation:

a->{                        // Method with double-array parameter and double return-type
for(;a-a            //  Loop as long as the difference between the 2nd and 1st items
>4e-9;){    //  is larger than 0.000000004:
double l=a.length,      //   Set l to the amount of values in the array a
A[]={0,0,0,1};   //   Create an array A, filled with the values [0,0,0,1]
for(var d:a){           //   Inner loop over the values of a:
A+=d/l;            //    Calculate the sum divided by the length in the third spot
A*=Math.pow(d,1/l);//    The product of the power of 1/length in the fourth spot
A+=1/d;            //    The sum of 1/value in the first spot
A+=d*d;            //    And the sum of squares in the second spot
}                       //   After the inner loop:
//   (the third spot of the array now holds the Arithmetic Mean)
//   (the fourth spot of the array now holds the Geometric Mean)
A=l/A;            //   Divide the length by the first spot
//   (this first spot of the array now holds the Harmonic Mean)
A=Math.sqrt(A/l); //   Take the square of the second spot divided by the length
//   (this second spot of the array now holds the Quadratic Mean)
a=A;                    //   And then replace input a with array A
}                         //  After the outer loop when all values are approximately equal:
return a;}             //  Return the value in the first spot as result

• In C you could f+=Math.abs(d-D)<1e-9; and get implicit conversion from a boolean compare result to a 0 / 1 integer and then double. Does Java have any compact syntax for that? Or is it possible to do f+=Math.abs(d-D) and then check that the sum of absolute differences is small enough? – Peter Cordes Apr 2 '19 at 2:28
• Yup, for your test cases, f>1e-8 works as a loop condition: 229 bytes. a->{for(double f=1,D,A[],l;f>1e-8;a=A){D=a;A=new double[]{f=0,1,0,0};for(var d:a){f+=Math.abs(d-D);A+=d;A*=d;A+=1/d;A+=d*d;}A/=l=a.length;A=Math.pow(A,1/l);A=l/A;A=Math.sqrt(A/l);}return a;}. With 1e-9, it runs slower (about twice the CPU time), having to do more iterations to get essentially 4 * d-D down that small. With 1e-7, it's about the same speed as 1e-8. With 1e-6, some of the trailing digits differ for one case. – Peter Cordes Apr 2 '19 at 2:36
• @RobinRyder's answer points out that the quadratic mean is always the largest, and harmonic always the smallest, so perhaps you can ditch f entirely and only check a-a<4e-9. – Peter Cordes Apr 2 '19 at 7:32
• @PeterCordes l==2|| you mean (golfed to l<3|). But yes, good point; I've added it. :) – Kevin Cruijssen Apr 2 '19 at 8:03
• 180 bytes by aggregating aggregable reducers. – Olivier Grégoire Apr 2 '19 at 8:45

# Charcoal, 40 bytes

Ｗ‹⌊θ⌈θ≔⟦∕ΣθＬθＸΠθ∕¹Ｌθ∕ＬθΣ∕¹θ₂∕ΣＸθ²Ｌθ⟧θＩ⊟θ


Try it online! Link is to verbose version of code. Takes input as an array of numbers. Explanation:

Ｗ‹⌊θ⌈θ


Repeat while the array contains different values...

≔⟦....⟧θ


... replace the array with a list of values:

∕ΣθＬθ


... the mean...

ＸΠθ∕¹Ｌθ


... the geometric mean...

∕ＬθΣ∕¹θ


... the harmonic mean...

₂∕ΣＸθ²Ｌθ


... and the root mean square.

Ｉ⊟θ


Cast an element of the array to string and implicitly print it.

# Jelly, 24 bytes

;µ*Ɱ-;Ø½¤©Æm*İɼ;P½½ƊµÐLḢ


Try it online!

# PowerShell, 182180 183 bytes

$f={$a=$c=$d=$n=0$b=1
$m=[math]$args|%{$n++$a+=$_$b*=$_$c+=1/$_$d+=+$_*$_}
$p=($a/$n),$m::pow($b,1/$n),($n/$c),$m::sqrt($d/$n)|%{$m::round($_,9)} if($p-ne$p){$p=&$f @p}$p}


Try it online!

• 171 bytes – mazzy Apr 1 '19 at 12:10
• @mazzy, impressively :) – Andrei Odegov Apr 1 '19 at 14:14

# 05AB1E, 2624 23 bytes

Δ©ÅA®.²ÅAo®zÅAz®nÅAt)}н


-1 byte thanks to @Grimy.

23 byter alternative for Geometric mean:

Δ©P®gzm®ÅA®zÅAz®nÅAt)}н


Explanation:

Δ         # Loop until the list no longer changes:
©        #  Store the current list in variable ® (without popping)
#  (which is the implicit input-list in the first iteration)
#  Arithmetic mean:
ÅA      #   Builtin to calculate the arithmetic mean of the list
#  Geometric mean:
®.²     #   Take the base-2 logarithm of each value in the list ®
ÅA   #   Get the arithmetic mean of that list
o  #   And take 2 to the power of this mean
#  Harmonic mean:
®z      #   Get 1/x for each value x in the list ®
ÅA    #   Get the arithmetic mean of that list
z   #   And calculate 1/y for this mean y
®n      #   Take the square of each number x in the list from the register
ÅA    #   Calculate the arithmetic mean of this list
t   #   And take the square-root of that mean
)       #  Wrap all four results into a list
}н        # After the list no longer changes: pop and push its first value
# (which is output implicitly as result)

• 23: Δ©P®gzm®ÅA®zÅAz®nÅAt)}н – Grimmy Sep 11 '19 at 16:44
• @Grimy Thanks! Can't believe I hadn't thought about using the length instead of Y for 2/4. :) – Kevin Cruijssen Sep 12 '19 at 5:20
• Another 23 that betters shows the similarity of geometric mean to the other ones: Δ©ÅA®.²ÅAo®zÅAz®nÅAt)}н. Unfortunately, it doesn't look like we can refactor all those ÅAs. – Grimmy Sep 12 '19 at 11:27
• @Grimy Oh, I like this second version. :) EDIT: Oops.. thanks for noticing my mistake in the explanation.. >.> – Kevin Cruijssen Sep 12 '19 at 12:23
• I don't program in 05ab1e very well, but can you compute sums and then divide them all by the length later? – my pronoun is monicareinstate Sep 12 '19 at 12:28

# Jelly, 25 24 bytes

Wẋ4¹ÆlÆeƭ²½ƭİ4ƭÆm$€⁺µÐLḢ  Try it online! ### Explanation  µÐL | Repeat until unchanged: W | Wrap as a list ẋ4 | Copy list 4 times ⁺ | Do twice:$€     |     For each copy of the list:
4ƭ         |     One of these functions, cycling between them:
¹                    |       Identity
ÆlÆeƭ               |       Alternate between log and exp
²½ƭ            |       Alternate between square and square root
İ           |       Reciprocal
Æm       |    Then take the mean
Ḣ| Finally take the first item

• I am fairly bad at Jelly, but could something similar to P*İL work for the geometric mean? – my pronoun is monicareinstate Mar 31 '19 at 15:08
• @someone it would need to be P*Lİ so wouldn’t save bytes. It would mean I could bring Æm back down a line without costing bytes, but I quite like the fact each one currently has an arithmetic mean at its core. – Nick Kennedy Mar 31 '19 at 15:19 # Python 3, 152 bytes from math import* s=sum def f(*a):l=len(a);return 2>len({*a})and{*a}or f(s(a)/l,l/s(map(pow,a,l*[-1])),exp(s(map(log,a))/l),(s(map(pow,a,l*))/l)**.5)  Try it online! Recursive function f, will converge to floating point precision. Works in principle for all lists of positive numbers of any size, but is limited by Python's recursion limit a rounding error for some test cases. Alternatively, settling for 9 decimals precision: # Python 3, 169 bytes from math import* s=sum def f(*a):l=len(a);return(2>len({round(i,9)for i in a}))*aor f(s(a)/l,l/s(map(pow,a,l*[-1])),exp(s(map(log,a))/l),(s(map(pow,a,l*))/l)**.5)  Try it online! # C#, 173 bytes double m(int n,params double[]a)=>(n<1?a:m(n-1,a.Sum()/a.Length,Math.Pow(a.Aggregate((t,x)=>t*x),1.0/a.Length),a.Length/a.Sum(x=>1/x),Math.Sqrt(a.Sum(x=>x*x)/a.Length)));  Try it online! • This seems to really on a variable that must be passed. Also, you have to include using System and using System.Linq in your byte count, since they are required for the program to run. You can change your compiler to the C# Visual Interactive Compiler, which doesn't need those imports. Also, 1.0 -> 1d – Embodiment of Ignorance Mar 31 '19 at 20:59 # Clean, 124 bytes import StdEnv f=avg o limit o iterate\l=let n=toReal(length l)in[avg l,prod l^(1.0/n),n/sum[1.0/x\\x<-l],avg[x*x\\x<-l]^0.5]  Try it online! Performs the operation until the result stops changing. Hurray for limited precision floating point! # Pyth, 32 bytes h.Wt{H[.OZ@*FZJlZcJscL1Z@.O^R2Z2  Try it online here, or verify all the test cases (bar two, see note below) at once here. Accepts input as a list. There seem to be some issues with rounding, as certain inputs don't converge correctly when they otherwise should. In particular, test case 0.01 100 gets stuck at values [6.748305820749738, 6.748305820749738, 6.748305820749739, 6.748305820749738], and test case 1.61 2.41 gets stuck at [1.9896543776640825, 1.9896543776640825, 1.9896543776640827, 1.9896543776640825] - note in both cases that the 3rd mean (harmonic mean) differs from the others. I'm not sure if this problem invalidates my entry, but I'm posting it anyway as it should work. If this isn't acceptable, it can be fixed by instering .RRT before the [, to round each of the means to 10 decimal places, as seen in this test suite. h.Wt{H[.OZ@*FZJlZcJscL1Z@.O^R2Z2)Q Implicit: Q=eval(input()) Trailing )Q inferred .W Q Funcitonal while: While condition is true, call inner. Starting value Q t{H Condition function: current input H {H Deduplicate H t Discard first value Empty list is falsey, so while is terminated when means converge [.OZ@*FZJlZcJscL1Z@.O^R2Z2) Inner function: current input Z JlZ Take length of Z, store in J .OZ (1) Arithmetic mean of Z *FZ Product of Z @ J (2) Jth root of the above L Z Map each element of Z... c 1 ... to its reciprocal s Sum the above cJ (3) J / the above R Z Map each element of Z... ^ 2 ... to its square .O Arithmetic mean of the above @ 2 (4) Square root of the above [ ) Wrap results (1), (2), (3), and (4) in a list This is used as the input for the next iteration of the loop h Take the first element of the result, implicit print  • Since I'm pretty sure the repeated calculation won't jump around to previous values, you could replace .Wt{H with u for -4 bytes (and change Z to G) – ar4093 Sep 12 '19 at 8:14 # Japt v2.0a0 -g, 42 38 bytes â ÊÉ?ß[Ux²÷(V=UÊ)¬Ux÷V U×qV V÷Ux!÷1]:U  There's got to be a shorter way... This is a monstrosity! Saved 4 bytes thanks to @Shaggy! Try it • 38 bytes. But I agree, there must be a shorter way! – Shaggy Apr 1 '19 at 17:12 # C# (Visual C# Interactive Compiler), 177 bytes double f(double[]g)=>g.All(c=>Math.Abs(c-g)<1e-9)?g:f(new[]{g.Sum()/(z=g.Length),Math.Pow(g.Aggregate((a,b)=>a*b),1d/z),z/g.Sum(x=>1/x),Math.Sqrt(g.Sum(x=>x*x)/z)});int z;  Thanks to @KevinCruijjsen for pointing out that using floating point precision was causing problems! Would be 163 bytes if doubles were perfectly precise. Try it online! • The last two test cases give a StackOverflowException due to floating point precision. Instead of c==g you could do something like Math.Abs(c-g)<1e-9. Try it online. – Kevin Cruijssen Apr 1 '19 at 9:01 • @KevinCruijssen Thanks, it's such a pain dealing with floating point numbers – Embodiment of Ignorance Apr 2 '19 at 15:32 # x86 machine code (SIMD 4x float using 128-bit SSE1&AVX) 94 bytes # x86 machine code (SIMD 4x double using 256-bit AVX) 123 bytes float passes the test cases in the question, but with a loop-exit threshold small enough to make that happen, it's easy for it to get stuck in an infinite loop with random inputs. SSE1 packed-single-precision instructions are 3 bytes long, but SSE2 and simple AVX instructions are 4 bytes long. (Scalar-single instructions like sqrtss are also 4 bytes long, which is why I use sqrtps even though I only care about the low element. It's not even slower than sqrtss on modern hardware). I used AVX for non-destructive destination to save 2 bytes vs. movaps+op. In the double version we can still do a couple movlhps to copy 64-bit chunks (because often we only care about the low element of a horizontal sum). Horizontal sum of a 256-bit SIMD vector also requires an extra vextractf128 to get the high half, vs. the slow but small 2x haddps strategy for float. The double version also needs 2x 8-byte constants, instead of 2x 4-byte. Overall it comes out at close to 4/3 the size of the float version. mean(a,b) = mean(a,a,b,b) for all 4 of these means, so we can simply duplicate the input up to 4 elements and never have to implement length=2. Thus we can hardcode geometric mean as 4th-root = sqrt(sqrt), for example. And we only need one FP constant, 4.0. We have a single SIMD vector of all 4 [a_i, b_i, c_i, d_i]. From that, we calculate the 4 means as scalars in separate registers, and shuffle them back together for the next iteration. (Horizontal operations on SIMD vectors are inconvenient, but we need to do the same thing for all 4 elements in enough cases that it balances out. I started on an x87 version of this, but it was getting very long and not fun.) The loop-exit condition of }while(quadratic - harmonic > 4e-5) (or a smaller constant for double) is based on @RobinRyder's R answer, and Kevin Cruijssen's Java answer: quadratic mean is always the largest magnitude, and harmonic mean is always the smallest (ignoring rounding errors). So we can check the delta between those two to detect convergence. We return the arithmetic mean as the scalar result. It's usually between those two and is probably the least susceptible to rounding errors. Float version: callable as float meanmean_float_avx(__m128); with the arg and return value in xmm0. (So x86-64 System V, or Windows x64 vectorcall, but not x64 fastcall.) Or declare the return-type as __m128 so you can get at the quadratic and harmonic mean for testing. Letting this take 2 separate float args in xmm0 and xmm1 would cost 1 extra byte: we'd need a shufps with an imm8 (instead of just unpcklps xmm0,xmm0) to shuffle together and duplicate 2 inputs.  40 address align 32 41 code bytes global meanmean_float_avx 42 meanmean_float_avx: 43 00000000 B9 mov ecx, .arith_mean ; allows 2-byte call reg, and a base for loading constants 44 00000005 C4E2791861FC vbroadcastss xmm4, [rcx-4] ; float 4.0 45 46 ;; mean(a,b) = mean(a,b,a,b) for all 4 types of mean 47 ;; so we only ever have to do the length=4 case 48 0000000B 0F14C0 unpcklps xmm0,xmm0 ; [b,a] => [b,b,a,a] 49 50 ; do{ ... } while(quadratic - harmonic > threshold); 51 .loop: 52 ;;; XMM3 = geometric mean: not based on addition. (Transform to log would be hard. AVX512ER has exp with 23-bit accuracy, but not log. vgetexp = floor(lofg2(x)), so that's no good.) 53 ;; sqrt once *first*, making magnitudes closer to 1.0 to reduce rounding error. Numbers are all positive so this is safe. 54 ;; both sqrts first was better behaved, I think. 55 0000000E 0F51D8 sqrtps xmm3, xmm0 ; xmm3 = 4th root(x) 56 00000011 F30F16EB movshdup xmm5, xmm3 ; bring odd elements down to even 57 00000015 0F59EB mulps xmm5, xmm3 58 00000018 0F12DD movhlps xmm3, xmm5 ; high half -> low 59 0000001B 0F59DD mulps xmm3, xmm5 ; xmm3 = hproduct(sqrt(xmm)) 60 ; sqrtps xmm3, xmm3 ; sqrt(hprod(sqrt)) = 4th root(hprod) 61 ; common final step done after interleaving with quadratic mean 62 63 ;;; XMM2 = quadratic mean = max of the means 64 0000001E C5F859E8 vmulps xmm5, xmm0,xmm0 65 00000022 FFD1 call rcx ; arith mean of squares 66 00000024 0F14EB unpcklps xmm5, xmm3 ; [quad^2, geo^2, ?, ?] 67 00000027 0F51D5 sqrtps xmm2, xmm5 ; [quad, geo, ?, ?] 68 69 ;;; XMM1 = harmonic mean = min of the means 70 0000002A C5D85EE8 vdivps xmm5, xmm4, xmm0 ; 4/x 71 0000002E FFD1 call rcx ; arithmetic mean (under inversion) 72 00000030 C5D85ECD vdivps xmm1, xmm4, xmm5 ; 4/. (the factor of 4 cancels out) 73 74 ;;; XMM5 = arithmetic mean 75 00000034 0F28E8 movaps xmm5, xmm0 76 00000037 FFD1 call rcx 77 78 00000039 0F14E9 unpcklps xmm5, xmm1 ; [arith, harm, ?,?] 79 0000003C C5D014C2 vunpcklps xmm0, xmm5,xmm2 ; x = [arith, harm, quad, geo] 80 81 00000040 0F5CD1 subps xmm2, xmm1 ; largest - smallest mean: guaranteed non-negative 82 00000043 0F2E51F8 ucomiss xmm2, [rcx-8] ; quad-harm > convergence_threshold 83 00000047 73C5 jae .loop 84 85 ; return with the arithmetic mean in the low element of xmm0 = scalar return value 86 00000049 C3 ret 87 88 ;;; "constant pool" between the main function and the helper, like ARM literal pools 89 0000004A ACC52738 .fpconst_threshold: dd 4e-5 ; 4.3e-5 is the highest we can go and still pass the main test cases 90 0000004E 00008040 .fpconst_4: dd 4.0 91 .arith_mean: ; returns XMM5 = hsum(xmm5)/4. 92 00000052 C5D37CED vhaddps xmm5, xmm5 ; slow but small 93 00000056 C5D37CED vhaddps xmm5, xmm5 94 0000005A 0F5EEC divps xmm5, xmm4 ; divide before/after summing doesn't matter mathematically or numerically; divisor is a power of 2 95 0000005D C3 ret 96 0000005E 5E000000 .size: dd - meanmean_float_avx
0x5e = 94 bytes


(NASM listing created with nasm -felf64 mean-mean.asm -l/dev/stdout | cut -b -34,$((34+6))-. Strip the listing part and recover the source with cut -b 34- > mean-mean.asm) SIMD horizontal sum and divide by 4 (i.e. arithmetic mean) is implemented in a separate function that we call (with a function pointer to amortize the cost of the address). With 4/x before/after, or x^2 before and sqrt after, we get the harmonic mean and quadratic mean. (It was painful to write these div instructions instead of multiplying by an exactly-representable 0.25.) Geometric mean is implemented separately with multiply and chained sqrt. Or with one sqrt first to reduce exponent magnitude and maybe help numerical precision. log is not available, only floor(log2(x)) via AVX512 vgetexpps/pd. Exp is sort of available via AVX512ER (Xeon Phi only), but with only 2^-23 precision. Mixing 128-bit AVX instructions and legacy SSE is not a performance problem. Mixing 256-bit AVX with legacy SSE can be on Haswell, but on Skylake it just potentially creates a potential false dependency for SSE instructions. I think my double version avoids any unnecessary loop-carried dep chains, and bottlenecks on div/sqrt latency/throughput. Double version:  108 global meanmean_double_avx 109 meanmean_double_avx: 110 00000080 B9[E8000000] mov ecx, .arith_mean 111 00000085 C4E27D1961F8 vbroadcastsd ymm4, [rcx-8] ; float 4.0 112 113 ;; mean(a,b) = mean(a,b,a,b) for all 4 types of mean 114 ;; so we only ever have to do the length=4 case 115 0000008B C4E37D18C001 vinsertf128 ymm0, ymm0, xmm0, 1 ; [b,a] => [b,a,b,a] 116 117 .loop: 118 ;;; XMM3 = geometric mean: not based on addition. 119 00000091 C5FD51D8 vsqrtpd ymm3, ymm0 ; sqrt first to get magnitude closer to 1.0 for better(?) numerical precision 120 00000095 C4E37D19DD01 vextractf128 xmm5, ymm3, 1 ; extract high lane 121 0000009B C5D159EB vmulpd xmm5, xmm3 122 0000009F 0F12DD movhlps xmm3, xmm5 ; extract high half 123 000000A2 F20F59DD mulsd xmm3, xmm5 ; xmm3 = hproduct(sqrt(xmm0)) 124 ; sqrtsd xmm3, xmm3 ; xmm3 = 4th root = geomean(xmm0) ;deferred until quadratic 125 126 ;;; XMM2 = quadratic mean = max of the means 127 000000A6 C5FD59E8 vmulpd ymm5, ymm0,ymm0 128 000000AA FFD1 call rcx ; arith mean of squares 129 000000AC 0F16EB movlhps xmm5, xmm3 ; [quad^2, geo^2] 130 000000AF 660F51D5 sqrtpd xmm2, xmm5 ; [quad , geo] 131 132 ;;; XMM1 = harmonic mean = min of the means 133 000000B3 C5DD5EE8 vdivpd ymm5, ymm4, ymm0 ; 4/x 134 000000B7 FFD1 call rcx ; arithmetic mean under inversion 135 000000B9 C5DB5ECD vdivsd xmm1, xmm4, xmm5 ; 4/. (the factor of 4 cancels out) 136 137 ;;; XMM5 = arithmetic mean 138 000000BD C5FC28E8 vmovaps ymm5, ymm0 139 000000C1 FFD1 call rcx 140 141 000000C3 0F16E9 movlhps xmm5, xmm1 ; [arith, harm] 142 000000C6 C4E35518C201 vinsertf128 ymm0, ymm5, xmm2, 1 ; x = [arith, harm, quad, geo] 143 144 000000CC C5EB5CD1 vsubsd xmm2, xmm1 ; largest - smallest mean: guaranteed non-negative 145 000000D0 660F2E51F0 ucomisd xmm2, [rcx-16] ; quad - harm > threshold 146 000000D5 77BA ja .loop 147 148 ; vzeroupper ; not needed for correctness, only performance 149 ; return with the arithmetic mean in the low element of xmm0 = scalar return value 150 000000D7 C3 ret 151 152 ; "literal pool" between the function 153 000000D8 95D626E80B2E113E .fpconst_threshold: dq 1e-9 154 000000E0 0000000000001040 .fpconst_4: dq 4.0 ; TODO: golf these zeros? vpbroadcastb and convert? 155 .arith_mean: ; returns YMM5 = hsum(ymm5)/4. 156 000000E8 C4E37D19EF01 vextractf128 xmm7, ymm5, 1 157 000000EE C5D158EF vaddpd xmm5, xmm7 158 000000F2 C5D17CED vhaddpd xmm5, xmm5 ; slow but small 159 000000F6 C5D35EEC vdivsd xmm5, xmm4 ; only low element matters 160 000000FA C3 ret 161 000000FB 7B000000 .size: dd$ - meanmean_double_avx

0x7b = 123 bytes


## C test harness

#include <immintrin.h>
#include <stdio.h>
#include <math.h>

static const struct ab_avg {
double a,b;
double mean;
} testcases[] = {
{1, 1, 1},
{1, 2, 1.45568889},
{100, 200, 145.568889},
{2.71, 3.14, 2.92103713},
{0.57, 1.78, 1.0848205},
{1.61, 2.41, 1.98965438},
{0.01, 100, 6.7483058},
};

__m128 meanmean_float_avx(__m128);       // or float ...
__m256d meanmean_double_avx(__m128d);    // or double ...
int main(void) {
int len = sizeof(testcases) / sizeof(testcases);
for(int i=0 ; i<len ; i++) {
const struct ab_avg *p = &testcases[i];
#if 1
__m128 arg = _mm_set_ps(0,0, p->b, p->a);
double res = meanmean_float_avx(arg);
#else
double res = meanmean_double_avx(arg);
#endif
double allowed_diff = (p->b - p->a) / 100000.0;
double delta = fabs(p->mean - res);
if (delta > 1e-3 || delta > allowed_diff) {
printf("%f %f => %.9f but we got %.9f.  delta = %g allowed=%g\n",
p->a, p->b, p->mean, res, p->mean - res, allowed_diff);
}
}

while(1) {
double a = drand48(), b = drand48();  // range= [0..1)
if (a>b) {
double tmp=a;
a=b;
b=tmp; // sorted
}
//      a *= 0.00000001;
//      b *= 123156;
// a += 1<<11;  b += (1<<12)+1;  // float version gets stuck inflooping on 2048.04, 4097.18 at fpthreshold = 4e-5

// a *= 1<<11 ; b *= 1<<11;   // scaling to large magnitude makes sum of squares loses more precision
//a += 1<<11; b+= 1<<11;   // adding to large magnitude is hard for everything, catastrophic cancellation
#if 1
printf("testing float %g, %g\n", a, b);
__m128 arg = _mm_set_ps(0,0, b, a);
__m128 res = meanmean_float_avx(arg);
double quad = res, harm = res;  // same order as double... for now
#else
printf("testing double %g, %g\n", a, b);
__m128d arg = _mm_set_pd(b, a);
__m256d res = meanmean_double_avx(arg);
double quad = res, harm = res;
#endif
double delta = fabs(quad - harm);
double allowed_diff = (b - a) / 100000.0; // calculated in double even for the float case.
// TODO: use the double res as a reference for float res

if (delta > 1e-3 || delta > allowed_diff) {
printf("%g %g we got q=%g, h=%g, a=%g.  delta = %g,  allowed=%g\n",
}
}

}


Build with:

nasm -felf64 mean-mean.asm &&
gcc -no-pie -fno-pie -g -O2 -march=native mean-mean.c mean-mean.o


Obviously you need a CPU with AVX support, or an emulator like Intel SDE. To compile on a host without native AVX support, use -march=sandybridge or -mavx

Run: passes the hard-coded test cases, but for the float version, random test cases often fail the (b-a)/10000 threshold set in the question.

$./a.out (note: empty output before the first "testing float" means clean pass on the constant test cases) testing float 3.90799e-14, 0.000985395 3.90799e-14 0.000985395 we got q=3.20062e-10, h=3.58723e-05, a=2.50934e-05. delta = -3.5872e-05, allowed=9.85395e-09 testing float 0.041631, 0.176643 testing float 0.0913306, 0.364602 testing float 0.0922976, 0.487217 testing float 0.454433, 0.52675 0.454433 0.52675 we got q=0.48992, h=0.489927, a=0.489925. delta = -6.79493e-06, allowed=7.23169e-07 testing float 0.233178, 0.831292 testing float 0.56806, 0.931731 testing float 0.0508319, 0.556094 testing float 0.0189148, 0.767051 0.0189148 0.767051 we got q=0.210471, h=0.210484, a=0.21048. delta = -1.37389e-05, allowed=7.48136e-06 testing float 0.25236, 0.298197 0.25236 0.298197 we got q=0.274796, h=0.274803, a=0.274801. delta = -6.19888e-06, allowed=4.58374e-07 testing float 0.531557, 0.875981 testing float 0.515431, 0.920261 testing float 0.18842, 0.810429 testing float 0.570614, 0.886314 testing float 0.0767746, 0.815274 testing float 0.118352, 0.984891 0.118352 0.984891 we got q=0.427845, h=0.427872, a=0.427863. delta = -2.66135e-05, allowed=8.66539e-06 testing float 0.784484, 0.893906 0.784484 0.893906 we got q=0.838297, h=0.838304, a=0.838302. delta = -7.09295e-06, allowed=1.09422e-06  FP errors are enough that quad-harm comes out less than zero for some inputs. Or with a += 1<<11; b += (1<<12)+1; uncommented: testing float 2048, 4097 testing float 2048.04, 4097.18 ^C (stuck in an infinite loop).  None of these problems happen with double. Comment out the printf before each test to see that the output is empty (nothing from the if(delta too high) block). TODO: use the double version as a reference for the float version, instead of just looking at how they converging with quad-harm. # Javascript - 186 bytes Takes input as an array of numbers. Uses the mean transformations in J42161217's answer to shorten the code. Try It Online f=(v,l=[m=(w,k=0)=>w.map(x=>k+=x)&&k/w.length,w=>1/m(w.map(x=>1/x)),w=>Math.E**m(w.map(x=>Math.log(x))),w=>m(w.map(x=>x**2))**.5].map(x=>x(v)).sort((a,b)=>a-b))=>l-l>1e-5?f(l):l  ## Explanation f = ( v, l=[ m=(w,k=0)=>w.map(x=>k+=x)&&k/w.length, // m = w => arithmetic mean of values in w w=>1/m(w.map(x=>1/x)), // w => harmonic mean of values in w w=>Math.E**m(w.map(x=>Math.log(x))), // w => geometric mean of values in w w=>m(w.map(x=>x**2))**.5 // w => quadratic mean of values in w ].map(x=>x(v)) // get array of each mean using input v, stored in l .sort((a,b)=>a-b) // sort the outputs ) => l - l > 1e-5 ? // is the difference between the largest // and smallest means > 1/100000? f(l) : // if yes, get the mean mean of the means l // if no, arbitrarily return the smallest value // as close enough  • I thought I was going to be clever and implement the relationship with logarithms but it looks like you and J42161217 got there first! – Pureferret Apr 3 '19 at 11:08 • @Pureferret I take no credit for that,I blatantly stole it :D – asgallant Apr 3 '19 at 15:58 • you wrote it in JavaScript though! – Pureferret Apr 3 '19 at 15:59 • That was the easy part. Golfing it was hard. – asgallant Apr 3 '19 at 16:13 • The TIL was not configured correctly. I added a TIL link to the answer. – asgallant Apr 4 '19 at 16:27 # Perl 5, 92 72 bytes sub f{@_=map{$m=$_;sum(map$_**$m/@_,@_)**(1/$m)}1,1e-9,-1,2for 1..9;pop}


Try it online!

...using some dirty tricks.

# SNOBOL4 (CSNOBOL4), 296 bytes

	X =INPUT
Y =INPUT
A =(X + Y) / 2.
P =X * Y
G =P ^ 0.5
H =P / A
Q =(2 * (A ^ 2) - P) ^ 0.5
O	OUTPUT =EQ(Q,A) Q	:S(END)
M =A
N =G
O =H
P =Q
A =(M + N + O + P) / 4
G =(M * N * O * P) ^ 0.25
H =4 / (1 / M + 1 / N + 1 / O + 1 / P)
Q =((M ^ 2 + N ^ 2 + O ^ 2 + P ^ 2) / 4) ^ 0.5	:(O)
END


Try it online!

Straightforward implementation. Uses a trick from my answer to a related question to golf a bit more.