Rewriting a logarithm (log_b[x]) so it's just written arithmetically

[lapis anvil]

Hi everyone, I've posted a few times in here before.

I'm coding in a very limited language in a Minecraft server, specifically Hypixel Housing, called HTSL, and I'm trying to simulate a Logarithm with only lines of arithmetic calculations.
For example:

test = 1000
test /= 10
test *= 10

So I can only do these types of calculations and no if/else statements or for loops or special recursive functions. With that in mind, I'm trying to find the closest graph that can be done with pure arithmetic math (respectively written as a mathematical formula) to match the actual logarithm function but I'm still very far (starting to think it's impossible).

You can use doubles and integers, as well as floor division and absolute value for this, but that's about it.
No for loops, no math.X functions, no if/else statements, no new functions, just pure, arithmetic code.

Anyone know how to do this? The closest I got to using grok ai is this formula in code skulptor, but it's still very far from the actual logarithm formula.
https://py3.codeskulptor.org/#user310_lWdBgHWwTi_0.py

Someone help me pls!
As title suggests, I'll PayPal you $10 if you help me find the best solution.

[lapis anvil]

($10 BOUNTY) Rewriting a logarithm (log_b[x]) so it's just written arithmetically

If you guys have any questions about the constraints pls dm me, I'll write what's allowed based on what Housing allows me lol

[mellow pelican]

What domain of inputs do you need it to work on

[storm geyser]

,w maclauren expansion ln(x)

[snow ibex]

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[storm geyser]

$$\ln(x) = -\sum_{n = 1}^\infty \frac{(1 - x)^n}{n} \quad \forall x \in (0, 1)$$

[snow ibex]

@storm geyser

[storm geyser]

You can also apply a horizontal stretch:

[storm geyser]

$$\ln(\frac xb) = -\sum_{n = 1}^\infty \frac{\qty(1 - \frac xb)^n}{n}$$

[snow ibex]

@storm geyser

[storm geyser]

Do not offer money for doing homework assignments, and vice versa.

[red lintel]

this doesn't seem like homework

[mellow pelican]

Taylor series is gonna be very bad for values not close to the center

[red lintel]

yeah

but

if you can store log(10) in some variable

and then write the number x as a * 10^n, then you can find the log by doing log(a) + nlog(10)

The challenge comes in finding a and n

[mellow pelican]

Yeah I am thinking you would need loops to do that

[red lintel]

https://en.wikipedia.org/wiki/Newton–Cotes_formulas

might be worth looking at

https://www.desmos.com/calculator/poh6iovy8d

It is kinda accurate

and if you find a way to get a and n, you might be able to find all values very easily

If we can bound all the inputs below some number N

this can become much easier

[lapis anvil]

hey guys, didnt expect to get a response

[lapis anvil]

wdym?

I'll only be passing 2 values, the b in log(b)_x and the x

both values need to be greater than 0

and less than the integer overflow limit which is like 2^32-1

[lapis anvil]

for loops arent allowed, so the sigma notation is omitted

By the way for housing, if statements ARE allowed but for reference I'm trying to fit everything within a single "conditional" which in Housing terms, is a bag with a redstone icon that can carry a max of 25 actions (which in this case, are the arithmetic calculations)

@mellow pelican refer to this code in HTSL (can be unparsed for python code): https://pastebin.com/3AXbeivD

This person made Exponentation in Housing using only arithmetic calculations and floor division, with the exception of if statements

If statements aren't allowed in this case, and I heard logarithms are much harder to be written arithmetically than exponentation, just something to think about

[mellow pelican]

@lapis anvil we need to know the domain though still

[lapis anvil]

how do I find the domain?

[mellow pelican]

Just what range of values for input do you expect

[lapis anvil]

ahhh ok

here i think i had it saved

// precise, range: log_b ∈ ℤ, log_b > 0, log_x ∈ ℤ, log_x > 0
// given "b" and "x", return its nearest integer of log_b(x)

ngl this is probably wrong, this was before I later realized u cant have zero or like 1 in one of the inputs

basically just what the typical range is for when inputting whole numbers in a logarithm, anything thats greater than 0 or 1 i forgot

sorry i dont really know lol

[mellow pelican]

So basically any number greater than 0 and less than the maximum possible double?

[red lintel]

@lapis anvil can you tell me the list of operations you can do

are you able to get the under lying bits of a number

then you can do IEE quake pro shenagins to get the log

is it JUST arithematic

and floor division?

[fresh agate]

Just do e^x and switch the x,y coordinates

[lapis anvil]

hey guys appreciate the responses

[storm geyser]

But only basic arithmetic is allowed.

[lapis anvil]

all arithmetic operations, including truncation/floor division, and absolute value

no if/else statements, no for loops/while loops, no domain range etc, no special functions, no classes

[storm geyser]

@mellow pelican @lapis anvil
What if you calculate the basic Taylor Polynomial for some value like x = 1.5, then use that to construct a new Taylor Series centered at x = 1.5.

[lapis anvil]

hmm could that work?

im not familiar with the taylor polynomial

but ill see

[storm geyser]

    chat {"This message has been sent " + (index + 1) + " time(s)!"}
}```Source: https://hypixel.net/threads/updated-guide-htsl.5555038/

[lapis anvil]

nono

[storm geyser]

Oh, I see.

[lapis anvil]

it will just carry the print message 10 times rather than it being a for loop i think

[storm geyser]

def lnTaylorSeries(x: float, n: int) -> float:
    approximation: float = 1.0
    for _ in range(n):
        approximation *= (1 - x)
    approximation /= n
    return -approximation

def sum(accuracy: int, x: float) -> float:
    sum: float = 0.0
    n: int = 1
    for _ in range(accuracy):
        sum += lnTaylorSeries(x, n)
        n += 1
    return sum

approximation: float = sum(100, 1.5)
print(approximation)```I was hoping something like this would work.

[lapis anvil]

unfortunately housing doesnt have a built in for loop which sucks so it cant be possible

[lapis anvil]

woaaa

nope i wish that worked

lambda expression looks cool but idk how that would be written arithmetically

[storm geyser]

Is the for-loop the only thing that doesn't work?

[storm geyser]

What lambda expression?

[lapis anvil]

is the -> lambda?

i was taught for java it is that symbol

[storm geyser]

That's just a type-hint that indicates what datatype the function will return.

[lapis anvil]

ohhhh ok

mb

[storm geyser]

def function(arguments) -> dataType:``````java
public static dataType function (arguments)```Here's a "translation" of sorts to understand it.

[mellow pelican]

What does this mean

This does not ACTUALLY loop your actions, it will simply duplicate them.

Are they trying to say it doesnt loop forever?

Very next example they give is
loop 10 index {
chat {"This message has been sent " + (index + 1) + " time(s)!"}
}

So you can index the loop

[storm geyser]

@lapis anvil
Why does the for-loop not work? It merely duplicates the action without using the index of the loop.

@lapis anvil
Can you give me your Minecraft name? I'd like to see your House and potentially find a better method than logarithms.

[slim rock]

I think the key is that it forgets any mutation of the state

[storm geyser]

Sorry, I don't understand this.

Is it possible for me to remove the <@&830347221085061141> role?

[slim rock]

If you set a variable in the loop, it will set it 10 times, but it won't update it 10 times, it just does the first step 10 times.

[slim rock]

Ask modmail, that's above my pay grade unfortunately. I don't have access to that

[storm geyser]

Isn't the first step constant though?

[slim rock]

For instance, if you did

out = 0
loop 10 index {
  out += index
}

You would expect to get out set to 55, but I believe it would set it to 10

All of the other changes occur independently and are thrown away.

[storm geyser]

Oh, I see.

[red lintel]

alright

I have an idea

$\operatorname{floor}\left(\frac{\left(n-10^{1+m}\operatorname{floor}\left(\frac{n}{10^{1+m}}\right)\right)}{10^{\left(m\right)}}\right)$

[snow ibex]

TimesZeroed

[red lintel]

this formula allows you to get the mth digit of the number n

now it does use the powers 10^m

but if you only wanted the first digit, then you could just plug in 0

[storm geyser]

Exponentiation is very hard in this HTSL language.

[red lintel]

yeah but you don't need to

for simplicity

let us just say the maximum number we would need to take the log of is

100000

[red lintel]

then we could easily hard code the values of 0,1,...5

and get all those digits

we could then do

y = (10^-1) digit 1 + (10^-2) digit 2...

again, we will hard code 10^-stuff so we don't have to do any exponents

lastly we use some log approximation of log between 0 to 1, such as one from simpsons rule

to get something very accurate

[red lintel]

becuause log(x) = log (y * 10^n) = log(y) + nlog(10)

and log(10) can just become a magic constnat

[slim rock]

@lapis anvil may I request that you retract your bounty offer please?

[red lintel]

not the money 😭

https://www.desmos.com/calculator/u2scrvdhia

[lapis anvil]

hey im back sorry

[red lintel]

alternatively, you can use the pade aproximation

[lapis anvil]

dont worry

[red lintel]

seen in this graph

[red lintel]

nah it's fine, I'm not doing this for money lol

[red lintel]

anyway this pade approximation might work for you

exponents don't need to be coded as you can just do

multiplication

[lapis anvil]

woahhh this is really cool

wait

but im taking 2 inputs though

did i mess u up? lmao

[red lintel]

what

[red lintel]

what are the two inputs

b and x?

[slim rock]

@lapis anvil 2nd request.

[lapis anvil]

yes if thats fine fine

sorry im taking a shower rn half teaponding

[lapis anvil]

okk ill take this biunty to dms then

[red lintel]

just use the change of the base formula

so whatever formula i have

divided by log ( whatever base you want)

I have another idea

if you can get a log formula to work really accurately between 0 and 1

you can do -log(1/x) for x>1

[lapis anvil]

duuude i dont know

check dms

[lapis anvil]

ohhh ok ty

[twilit swift]

Rewriting a logarithm (log_b[x]) so it's just written arithmetically

[red lintel]

are bounties not allowed on this server?

I read through the rules and it seems like it only applies to hw and assignments

[storm geyser]

I guess the idea of monetization at all is banned on this server.

[red lintel]

that makes sense

[slim rock]

It does specifically call out hw and assignments, but it was not intended to be limited to just hw and assignments. Facilitating the transfer of money opens the server up to a lot of potential liability and so on, and imo it would probably be bad to have help threads that promise monetary bounties because they would attract a bunch of (potentially low quality) contributors looking to make a quick buck, and so on.

[red lintel]

oh ok, I think it should be stated in the rules then

[slim rock]

The rules aren't intended to be a complete enumeration of every single thing that you can possibly do wrong

Besides, the only action taken was asking politely that the title of the thread be changed, and then when ignored twice, a senior mod just changing the title.

[red lintel]

alright

[red lintel]

nah it's fine