the most egregious programming tutorial ever

My favorite part is how it still uses a trig function

you are the howtobasic of mathematics. lol

Funnily enough, I wasn’t too hurt when you used curly braces instead of a colon. It’s a common mistake, sometimes it gets hard switching between languages very often. I was hurt when you called that symbol a “hashtag”

Finally, a channel does a better explanation of the Cordic algorithm than just "rotating the vector" to approximate a trig function, When rotations require trig functions. Brilliant video.

This is all well and good. But to any future programmers watching this, please do not use weird Unicode math characters in your code, just use the names of things, like phi, theta, delta, using these symbols will drive anyone that isn’t a heavy math user mad when trying to read your code.

Hey there, awesome video, but I did just want to give a pointer. Using a variable called d next to x, y, or phi is generally considered an abuse of notation since it looks closer to an infinitesimal rather than actual variable.

1) I think even if it's what algorithm is really used, there are some fine details about things regarding precision. Because if it shows 6 decimal places, then all of them should be correct. But error in cycle accumulates 2) Your code won't work for angles > 4pi 3) Question in the beginning was how do you calculate those without calculator. But then you pull out from somewhere: some constant which is limit of product (which is also you need to calculate without calculator), and table of 50 arctan, which you also need to calculate. I think more plausible way to calculate sin/cos without calculator to use angle halving formulas, and rotate by pi/2, pi/4, pi/8, pi/16 and so on.

If someone doesn't want to use pow function, the powers of 2 can be achieved by taking 1 and shifting it a few bits (remembering that 2^(-n) is the same as 1/2^(n))

Didn't understand this, will have to watch again later. But when it comes to approximating sin and cos, I find that this is a good plan: 1) Add or subtract multiples of 2*pi until you're in the range -pi to pi. 2) Map the angle to the first quadrant and remember what that will do to the sign of the final result. 3) If you're dong the sin or cos of an angle greater than pi/4, do the cos or sin of the complementary angle. With those three steps, we've guaranteed that our angle is no more than 0.785 radians. We can Taylor series it and get a good approximation within just a few terms. But we can take it even further: 4) Pre-calculate some sines and cosines of angles like pi/4, pi/8, etc. Save them as constants to whatever arbitrary degree of precision you like. 5) Remember your trig identities, like sin(a+b) = sina*cosb + coa*sinb, and cos(a+b) = cosa*cosb - sina*sinb. With those in mind, suppose you want to calculate sin(3*pi/16). Well, that's sin(pi/8 + pi/16), and if you've precalculated sin(pi/8), then you just have to calculate sin(pi/16) and cos(pi/16) and do the trig identities. And since pi/16 is a little under 0.2, the calculations for sin(pi/16) and cos(pi/16) will converge very quickly.

Instant subscription. Perfect intro into math amd coding combined for oeople unfamiliar with it. Keep it up!

Excellent video mate! However, wouldnt a taylor series be easier for a calculator to deal with?

this guys gonna be huge in the future

From the thumbnail, I thought it would be how modern calculators give symbolic answers for special cases when it recognizes them. IAC, what you described is called the CORDIC algorithm. It needs one iteration per bit of the answer, so 55 iterations seems right as that matches the mantissa of a double precision floating point value. CORDIC _can_ be implemented using only addition, subtraction, bit shifts, and table lookups -- no multiplication or division. Your code doesn't exploit this, and in fact uses division gratuitously. (division being horribly slow even on modern CPUs). This makes it the preferred algorithm for low-end calculators that use 8-bit microcontrollers. For a more capable CPU, the Taylor series takes fewer iterations and will need fewer as the angle is smaller.

Amazing video, I wish you the best your future efforts, and I can only hope you keep this quality up!

thanks a lot. this has been a question at the back of my mind for a lot of time

Pretty cool! Though if we don’t have functions for sine and cosine, shouldn’t we also not have functions for arctangent? Or is this actually the way computers calculate it?

Hello, what app do you use for that blackboard? I thought it looked very cool.

8:08 the ** operator also works. i.e: 2**(-n)

Wow this vedio is very underrated. Excellent subscribed

I remember in the 70's my dad brought home a TI calculator that had trig functions. Being about 8, I had no idea what they mean but I thought it was interesting that the calculator would take a couple of seconds to handle these functions. I made it my goal in life to be able to use all the functions on a calculator (it also had log as well.) But always wondered why it took so long to calculate sin, now I know.

Damn, i didnt know howtobasic was a mathematician 💀

this channel is a gem how i just saw this

Cool video! If you want to make those print statements a little easier to write and more readable, you can put an 'f' before the quotes and use curly brackets to avoid needing the str() functions. As in print(f"sin({θ}) = {y}")

Something worth noting here is, you still need to calculate arctan(2^-n) somehow, which is also a trig function. However, given this is very close to 2^-n, you can simply remove arctan for larger order terms, and perhaps hard-code first few terms to further decrease error.

In degrees, use angle bisection as approximation. In radians, use the power series.

How do you get rid of the atan() function in the code? We're not supposed to use trig functions here, unless there is a video explaining how to approximate atan() without other trig functions!

or simply use binomial expansion of trig functions, define the function, replace the x with the variable name in the function parameter, keep writing as many terms as you can then you will get almost identical results to real values

Imagine if a business major sees this. I think they’ll explode. Math majors may be sad, depressed, lonely, and overworked, but at least we can understand shit like this!

being an actual python programmer, seeing the beginner tactics (like concatenation instead of functional strings or using Unicode characters as variables, or printing instead of returning) made me remind myself that beginners don't need to follow python conventions when their methods work. This was before I noticed you used curly brackets. (no hard feelings, great video)

This means that the calculator needs to have a arctan(x) table in the memory or defined somehow for it to calculate sin(x)?

You can just use tailor series, it works well with small numbers

4:51 I understand how the arctan values can be precomputed, but how do you calculate the cosine?

I love the part where you used wolfram alpha to make you own trigonometric equation

I coded up CORDIC many years ago and was going to implement it in a FPGA but got bogged down with the floating point conversions.

7:54 Python uses the same double-asterisk operator as FORTRAN for exponentiation, so 2ⁿ would be written as `2 ** n`. Math.pow() always returns floating point numbers as a result, whereas the double-star operator will return integer values when appropriate.

Acktually the sin is calculated using multiple techniques. Firstly, you only need to calculate the first quarent of the sin. Since other quarent can be calculate using trig. Secondly, look up table is used for common value like π/12, π/6, π/4, π/3, π/2 and more. Thirdly, values are close to 0 are return without calculation. Depend on how accurate the approximation need to be, cordic and Chebyshev polynomials can be use.

Why does the algorithm for finding trig functions need you calculate arctan? How does it do that?

Great video bruv! Just remember me when you have millions of subscribers😃

why can't we use infinite series expansion of sin, taking the first n terms such that it gives answer within accepted error limit?

What will you do? A B C or D? A: You can always go to the park B: You can always get to work on time C: You can always make a PERFECT triangle D: You go to Paris every year E: you ALWAYS get what you want

I do get some weird values. π/4 stops after 2 iterations, but ends up at the really wrong value (0.6072529350088812 instead of 0.7071067811865475). And cos(0) is really wrong, after 1 iteration. For π/2 the sin and cos are fine, but understandably the tan value is a bit wonky.

This was really a tutoriel that I watched with curiosity until the end. I liked both the math and computer part very much. My only question is, cos(arctan(1)).cos(arctan(2)).cos(arctan(3))... I think it is not appropriate to calculate it on the computer. Because we used trig again? Also i _think_ you can use Taylor Series of sinx , cosx, or tanx for example: sinx ~ x -x^3/3! + x^5/5! -x^7/7!

Nice video. I didn’t understand everything but it was pretty interesting 👍

"But wait, that requires cos and sin." "Aaaarerggghg!!!!!!!!!" Got me dying. 💀 Eggs in a blender.

this is amazing

Good video!

Very nice!

Iteration isn't the fastest method and there's a chance that change never reaches zero because of finite precision. It's better to use a polynomial or rational function. See Computer Approximations by J.F. Hart et al.

instead of math.pow, you can do 2**-n, no idea if it has the same time complexity tho

Confirmed, Wolfram Alpha existed before calculators did.

The most underrated chanell on the platform

I’ve wanted to know this for a while

Fortunate that people were able to use wolfram alpha back in the day, despite not having a calculator

The whole point of the original CORDIC (published by Jack Volder in 1957ish) was to replace computationally heavy/expensive multiplication and division in old memory-poor computers with additions/subtractions and some table lookups. Logs were also possible. Though based on some obscure 17th Century mathematics it was still a damn impressive algorithm. The code here would not have worked efficiently on early computers and calculators. In fact, it would have defeated the whole point of the original CORDIC. Interesting, though.

cool approximation of sin cos and tan, impressively interesting approach to programming it tho

i thought it used a parabolic approximation for 0-pi/2. then reflected and rotated that as necessary

Better to pronounce the sign() function as SIGNUM. Not “sine” - because that would confuse it with sin().

8:03 that's not "to the power of", that's "xor". XOR is a weird binary thingy, if you want "to the power of", use ** instead of ^

calculators actually have tables with all the sin values with the maximum precision they need. they dont directly calculate sin() because of perfomance

And I thought for half a century that mathematicians and programmers have zero emotions ...

Video: How to make a trig function 8:45 : Ok first you have to use a trig function

If you're gonna use a trig function anyway (atan), why not just def trig(theta): return math.sin(theta)

i love this

Imagine being in 1956 without a calculator and having a Python interpreter...xD

How does the calculator display it in a form like √2 /2 or 3π/2?

I noticed the brackets immediately and was very confused by it, I was like: Why didn't we just do this in c or somethin and why did he do that?

Interesting video but I don't like the fact that this algorithm uses a trig function to define other trig functions. I think it's sexier to derive trig functions from lower level math abstractions.

Why are you so fucking funny lmao

11:00 this jump scared me slightly

peak cinema of math

0:15 A

now make an infinite precision pi calculator

The humor isn't quite for me. It's jarring and frankly distracting. Also, why do you use arctan to generate sin? How will you compute it if it is not available, since one should presume it is an equally complicated function? If the answer is taylor expansion, what's wrong with using good old taylor expansion for sin / cost / tan in the first place?

Okay, umm, how do you calculate arctan? You've kinda kicked the can down the road by relying on another function. Obviously you could do numerical integration but that would be slow as balls

Why use this approach over a Taylor / Maclaurin series?

When I saw the brackets I died

I have subscribed

Legend says the CORDIC isn't used anymore.

Why not use Taylor series approximations?

can't you just use maclaurin's expansion for the first couple terms

I had to stop watching a few minutes in because I couldn't focus afraid of a scene with wasted eggs and phone books sudddenly appearing.

I thought calculators use Taylor's series. What's wrong with that?

С) Taylor series

ok but how to calculate the atan then?

8:15 or you can use the ** operator

Watcing this as a programmer hurts.

8:13 Could've just used the ** (double-star) operator. if you're worried about any performance issues.... IDGAF HE'S PROGRAMMING IN PYTHON FOR FUCK'S SAKE

imagine not waiting until deltamath was invented

Also me, who knows what sin 60 degrees is and also knows that 60 degree = 1.047 radian. so i just approx sin of 1 radian as sin of 60 degrees which gives me 0.86. I call that good enough and move on. ᕙ(⇀‸↼‶)ᕗ + 1 sub

I really liked the video but the python part made we want to bang my head

How does a calculate find atan(2^-n)?

How to find sin of whatever? Use tan. But how do I find the tan of whatever?

0:17 ☠️

I guess he used unicode characters for demonstation purposes, but please don’t. Use emojis instead.

ERORR: division by zero line 7 and 13

Imagine if Aliens come down and see us doing this, and just pull out a protractor and say “guys, why aren’t you just using these with scale models?”

I actually asked myself 2 days or so ago

but this isnt how a calculator does it? wheres the half adders? the full adders? the shift register? wheres the decimal to binary conversion? the complements of 9? all i saw was someone write code, that is then compiled by some program to hex, or machine code, and THAT is utterly different to what a calculator is actually doing to perform this, or any other calculation... try busting down the hex into opcodes and then stepping through how the actual processor deals with the code loaded to it...

the frustrated AUURRGHHH 🥚