How to calculate Maximum or Minimum of two numbers without using if?

Question

How to to calculate the maximim or minimum of two numbers without using "if" ( or something equivalant to that manner)?

The above question is often asked in introductory computer science courses and is answered using this method.

Now although it is not obvious, but using absolute value is also equivalant to using an if statement e.g. defining Max(a,b) = a if a>b else b;

Besides using limits, is there another way of finding the maximum or minimum of two numbers?

For me the absolute value is $\sqrt{a^2}$ for real numbers. I don't see an if here. — t.b., Aug 21 '11 at 10:29
Extra points for the interview is to note that sqrt(x*x) is not a numerical overflow safe function — Naomi, Apr 26 '20 at 08:22

score 50 · Answer 1 · edited Oct 08 '19 at 18:14

50

If you let $a$ and $b$ be any two numbers then,

$$\max (a,b) = \frac{1}{2}( a + b + |a-b| )$$.

edited Oct 08 '19 at 18:14

elias

15

answered Sep 25 '11 at 18:43

ginner

509

1

Very nice one... – NoChance Jul 27 '12 at 15:50
3

http://math.stackexchange.com/questions/46939/maximum-and-minimum – jimjim Dec 08 '12 at 03:32
A proof would be awesome. – tomka Feb 25 '19 at 18:38
Those who read Spivak's calculus know this formula :) – KKZiomek Oct 06 '19 at 22:12
1

the abs() function has an implicit if... – Naomi Dec 23 '19 at 15:45
2

@Shloim not necessarily: abs(x) = sqrt(x * x) – wacha Apr 25 '20 at 16:25
sqrt(x*x) is not a numeric overflow safe method. abs() is implemented in GNU as return i < 0 ? -i : i; – Naomi Apr 26 '20 at 08:20

score 11 · Answer 2 · answered Apr 20 '13 at 19:15

11

If $a$ and $b$ are both positive, then $$ \max(a,b) = \lim_{n\to\infty} \left(a^n+b^n\right)^{1/n}. $$

answered Apr 20 '13 at 19:15

John Dawkins

221

1

From where is this formula obtained? – Aatmaj Nov 22 '20 at 08:11
I would also like to know it. – Cesare Apr 14 '21 at 14:40
@Aatmaj : just assume a is grater than and divide and multiply a and take the limit. I was an exercise in the intermediate real analysis. – jimjim Feb 02 '24 at 00:08
@Cesare :check my comment to atmaj. – jimjim Feb 02 '24 at 00:09

score 8 · Accepted Answer · answered Aug 21 '11 at 11:01

8

Whether you need an if statement to take an absolute value depends on the format in which your numbers are stored. If you're using IEEE floating point numbers, you can take the absolute value by masking out the sign bit.

answered Aug 21 '11 at 11:01

joriki

238,052

@AlexM. : this answer reduced a decisional problem into a mechanically computable problem, so that getting absolute value of a number is just process that can be applied using the representation method. – jimjim Nov 09 '16 at 14:29

score 2 · Answer 4 · answered May 09 '19 at 07:28

2

$$ a = \frac{a + b + (a-b)((2(a-b)+1) mod 2)}{2}, a > b$$ or $$b$$ equals the same, if $$ b > a$$ Maximum

answered May 09 '19 at 07:28

Mouvre

193

score 1 · Answer 5 · answered Feb 01 '24 at 18:24

1

Here’s one more formula that came from definition of discriminant of quadratic equation which is valid for real numbers:

minimum(x,y) = (x+y-sqrt(x^2-2xy+y^2))/2

The source of origin: https://github.com/ohhmm/openmind/blob/d24bd9e4c87265038d29eda475d03da391c38fd5/omnn/math/Valuable.cpp#L2449

answered Feb 01 '24 at 18:24

Serg Kryvonos

179

that is same as $\max ( a,b) = \frac{1}{2}( a + b + |a-b| )$ , why they are reimplementing an already existing function? – jimjim Feb 02 '24 at 00:35
1

It is min, not max. May be used for arbitrary differential coding and HPC kernels generation from its formula: https://github.com/ohhmm/generator/blob/main/generator/generator.cpp – Serg Kryvonos Feb 09 '24 at 20:37

How to calculate Maximum or Minimum of two numbers without using if?

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