🧮 Math#

bs.math:_

The beatifull world of mathematics… in Minecraft!

🔧 Functions#

You can find below all the function available in this module.

Arccosine#

bs.math:trgi/arccos

Calculate the arccosinus of a value between -1 and 1

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The value you want to calculate the arccosine of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Output

(score) @s bs.out.0: The result of the calculation, in degrees (not shifted)

Example

Calculate and display the arccos of 0,42

# Once
scoreboard players set @s bs.in.0 420
function bs.math:arccos
tellraw @a [{"text":"arccos(0.42) = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Arcsine#

bs.math:trg/arcsin

Compute the arcsinus of a value between -1 and 1

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The value you want to calculate the arcsine of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Output

(score) @s bs.out.0: The result of the calculation, in degrees (not shifted)

Example

Calculate and display the arcsinus of 0.42

# Once
scoreboard players set @s bs.in.0 420
function bs.math:arcsin
tellraw @a [{"text":"arcsin(0.42) = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Arctangent#

bs.math:arctan

Compute the arctangent of a value between -infinite and +infinite

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The value you want to calculate the arctangent of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Output

(score) @s bs.out.0: The result of the calculation, in degrees (not shifted)

Example

Calculate and display the arctan of 0.42

# Once
scoreboard players set @s bs.in.0 420
function bs.math:arctan
tellraw @a [{"text":"arctan(0.42) = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

How does it work?

This function use two approximations to calculate the arctangent of a value:

\[\begin{split} \begin{cases} \tan(x) = \left(\frac \pi 2 \frac x {|x|} - 4 \frac x {4x^2 + 1}\right)\frac \pi {180} &\forall |x| \geq 0.72\\ \tan(x) = \left( x - \frac {x^3} 3 + \frac {x^5} 5 \right)\frac \pi {180} & \forall |x| < 0.72 \end{cases} \end{split}\]

Basis rotation 3D#

bs.math:basis_rotation_3d

Allows to obtain the equivalent of the vector passed in parameter in a base with a different orientation. Useful to convert an absolute/relative position into a local position for a given entity.

Inputs

(execution) as <entities>: The entities you want to perform the computation on
(scores) @s bs.in.[0,1,2]: Vector coordinates \((X,Y,Z)\) in the starting base
(scores) @s bs.in.[3,4]: Horizontal angle \(\phi\) (along \(\hat{y}\)) and vertical angle \(\theta\) (along \(\hat{\phi}\)) rotation (in degree) from the starting base

Basis system

This system use the Minecraft coordinate system. Thus:

\(\hat{y}\) is the vertical axis.
\(\phi=0\) (starting point of the horizontal angle) is along the \(\hat{z}\) axis.
\(\theta=0\) (starting point of the vertical angle) is on the horizontal plane.

Outputs

(scores) bs.out.[0,1,2]: Vector coordinates \((X',Y',Z')\) in the target base

Examples

A block is in ~2 ~5 ~10 from me, I want to have this position in local coordinate (^? ^? ^?)

# One time

# Relative coordinates (we multiply by 1000 to have more precision on the result, which will also be multiplied by 1000)
scoreboard players set @s bs.in.0 2000
scoreboard players set @s bs.in.1 5000
scoreboard players set @s bs.in.2 10000

# Difference between my orientation (= that of the coondata grid ^X ^Y ^Z) and the orientation of the Minecraft blocks grid (~X ~Y ~Z)
function bs.orientation:get
scoreboard players operation @s bs.in.3 = @s bs.ori.h
scoreboard players operation @s bs.in.4 = @s bs.ori.v

# Perform the basic rotation
function bs.math:basis_rotation_3d

# See the result
tellraw @a [{"text": "X = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"},{"text":", Y = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs. out.1"},"color":"gold"},{"text":", Z = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.2"},"color":"gold"}]

I want to have a vector pointing to where I’m looking, but in relative coordinates ~X ~Y ~Z (also called “classical” vector in this library)

# Once

# Retrieve a vector ^ ^ ^1 corresponding to a vector directed according to the orientation of the entity (we multiply by 1000 to have more precision on the result, which will also be multiplied by 1000)
scoreboard players set @s bs.in.0 0
scoreboard players set @s bs.in.1 0
scoreboard players set @s bs.in.2 1000

# Get the orientation
function bs.orientation:get
scoreboard players operation @s bs.in.3 = @s bs.ori.h
scoreboard players operation @s bs.in.4 = @s bs.ori.v

# Reversal of the orientation since we want to have the relative orientation of the game grid compared to the orientation of the player (unlike the previous example)
scoreboard players operation @s bs.in.3 *= -1 bs.const
scoreboard players operation @s bs.in.4 *= -1 bs.const

# Perform the basic rotation
function bs.math:basis_rotation_3d

# See the result
tellraw @a [{"text": "X = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"},{"text":", Y = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs. out.1"},"color":"gold"},{"text":", Z = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.2"},"color":"gold"}]

Cosine#

bs.math:cos

Compute the cosine of an angle between 0 and 360

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The angle in degrees

Outputs

(score) @s bs.out.0: The cosine of the angle shifted by 3 digits (ex: 0.42 -> 420)

Example

Calculate and display the cosine of 42

# Once
scoreboard players set @s bs.in.0 42
function bs.math:cos
tellraw @a [{"text": "cos(42) = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

How does it work?

This function use the property \(\cos(x) = \sin(x + 90)\) to compute the cosine of an angle. It then uses the sine function to compute the result.

Exponential#

bs.math:exp

Compute the exponential of the number passed in parameter on the score bs.in.0 and return the result on the score bs.out.0

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The number to be exponentiated shifted by two digits (1,2345 -> 123) for better precision in integer scores

Technical limitation

Due to the limit of integers that can be stored in a score, the interval of bs.in.0 is limited to [-600,1200] (i.e. [-6;12] in real value)

Output

(score) @s bs.out.0: The result of the operation shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Example

Calculate \(exp(3)\)

# Once
scoreboard players set @s bs.in.0 300
function bs.math:exp
tellraw @a [{"text":"exp(3)*10^3 = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Note

We are looking for a better implementation of this function. If you have any ideas, please join our Discord server to discuss with us!

Factorial#

bs.math:factorial

Compute the factorial of the number

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The number to be factorialized

Technical limitation

Due to the limit of integers that can be stored in a score, the interval of bs.in.0 is limited to [0,12]

Output

(score) @s bs.out.0: The result of the operation

Example

Compute \(3!\)

# Once
scoreboard players set @s bs.in.0 3
function bs.math:factorial
tellraw @a [{"text": "3! = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Greatest common denominator#

bs.math:gcd

Compute the greatest common denominator of two numbers

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The first number
(score) @s bs.in.1: The second number

Output

(score) @s bs.out.0: The greatest common denominator

Example

Calculate the greatest common denominator between 16 and 12

# Once
scoreboard players set @s bs.in.0 16
scoreboard players set @s bs.in.1 12
function bs.math:gcd
tellraw @a [{"text": "gcd(16,12) = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

Logarithm#

Base e (Neperian)

bs.math:log

Compute the Neperian logarithm (base e) of a number

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The number to be logarithmized, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Output

(score) @s bs.out.0: The result of the operation, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Example

Calculate \(ln(28)\)

# Once
scoreboard players set @s bs.in.0 28000
function bs.math:log
tellraw @a [{"text":"ln(28)*10^3 = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Base 2

bs.math:log2

Compute the logarithm in base 2 of a number

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The number to be logarithmized, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Output

(score) @s bs.out.0: The result of the operation, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Example

Calculate \(log_2(28)\):

# Once
scoreboard players set @s bs.in.0 28000
function bs.math:log2
tellraw @a [{"text":"log2(28)*10^3 = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Base 10

bs.math:log10

Compute the logarithm in base 10 of a number

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The number to be logarithmized, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Output

(score) @s bs.out.0: The result of the operation, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Example

Calculate \(log_{10}(28)\)

# Once
scoreboard players set @s bs.in.0 28000
function bs.math:log10
tellraw @a [{"text":"log10(28)*10^3 = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Base a

bs.math:loga

Computes the logarithm in base a of a number

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The number to be logarithmized, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores
(score) @s bs.in.1: The base of the logarithm (not shifted)

Output

(score) @s bs.out.0: The result of the operation, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores

Example

Calculate \(log_4(28)\)

# Once
scoreboard players set @s bs.in.0 28000
scoreboard players set @s bs.in.1 4
function bs.math:loga
tellraw @a [{"text":"log4(28)*10^3 = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Power#

Normal

bs.math:pow

Compute \(x^y\)

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The base
(score) @s bs.in.1: The exponent

Output

(score) @s bs.out.0: The result of the calculation

Example

Compute \(2^6\)

# Once
scoreboard players set @s bs.in.0 2
scoreboard players set @s bs.in.1 6
function bs.math:pow
tellraw @a [{"text": "2^6 = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

Scale 3

bs.math:pow/scale/3

Compute \(x^y\)

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The base, a number shifted by 3 digits (1,2345 -> 1234)
(score) @s bs.in.1: The exponent, not shifted

Output

(score) @s bs.out.0: The result of the calculation, a number shifted by 3 digits (1,2345 -> 1234)

Example

Compute \(2.345^6\)

# Once
scoreboard players set @s bs.in.0 2345
scoreboard players set @s bs.in.1 6
function bs.math:pow/scale/3
tellraw @a [{"text": "2.345^6 = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

Random#

bs.math:random

Generates a random number

Inputs

(execution) as <entities>: The entities you want to perform the calculation on

Output

(score) @s bs.out.0: An integer random number between \(-2^{31}\) and \(2^{31}-1\)

Tip

To reduce this interval, execute the function then do a “modulo” operation on the result (random % 10 -> the random number will be included in the interval [0;9])

Example

Get and display a random number between 0 and 100:

# Once
function bs.math:random
scoreboard players operation @s bs.out.0 %= 101 bs.const
tellraw @a [{"text": "random() = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

Beware: the score `bs.const` does not contain all possible
values. Make sure the value you want to use exists and initialize it
if necessary.

Retrieve the next power of 2#

bs.math:get_next_pow2

Compute the power of 2 directly superior to the number given in parameter.

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The number from which you want to calculate the next power of 2

Output

(score) @s bs.out.0: The result of the calculation

Example

Find the power of 2 greater than 43

# Once
scoreboard players set @s bs.in.0 43
function bs.math:get_next_pow2
tellraw @a [{"text":"get_next_pow2(43) = ","color":"dark_gray"},{"score":{"name":"@s","objective":"bs.out.0"},"color":"gold"}]

Rounded division#

bs.math:divide

Allows you to divide one number by another by rounding the result to the nearest whole number (where Minecraft rounds down to the next whole number).

Inputs

(execution) as <entities>: The entities you want to perform the operation on
(score) @s bs.in.0: The numerator
(score) @s bs.in.1: The denominator

Output

(score) @s bs.out.0: The result of the division

Example

Calculate \(9/5\)

# Once
scoreboard players set @s bs.in.0 9
scoreboard players set @s bs.in.1 5
function bs.math:divide
tellraw @a [{"text": "9 / 5 = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

Sine#

bs.math:sin

Computes the sine of an angle between 0 and 360

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The angle in degrees

Outputs

(score) @s bs.out.0: The sine of the angle shifted by 3 digits (ex: 0.42 -> 420)

Example

Calculate and display the sine of 42

# Once
scoreboard players set @s bs.in.0 42
function bs.math:sin
tellraw @a [{"text": "sin(42) = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

How does it work?

This function use the Bhaskara approximation which tell us that

\[ \sin(x) \approx \frac{4x(180-x)}{40500-x(180-x)} \quad \forall x \in [0, 180] \]

From this relation, and using the properties

\(\sin(-x) = -\sin(x)\) (antisymetry)
\(\sin(x+360) = \sin(x)\) (periodicity)

We can compute the sine of any angle.

Square root#

bs.math:sqrt

Compute the square root of the number

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The number you want to calculate the square root of

Output

(score) @s bs.out.0: The result of the calculation

Example

Calculate and display \(\sqrt{42}\)

# Once
scoreboard players set @s bs.in.0 42
function bs.math:sqrt
tellraw @a [{"text": "sqrt(42) = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

How does it work?

This system rely on a very simple mathematical concept called dichotomy. As the maximum number that a score can handle is \(2^{31}-1 = 2,147,483,647\), the maximum square root is \(\sqrt{2,147,483,647} \approx 46,340\). Also, we are dealing with only integer number, so we have a finite number of possible square root. The idea is then to take a number at half of the maximum limit and compute the square of this number. If it is upper thant the input, then we decrease our selected number by a quarter of the maximum limit (and if it’s lower, we increase it). We do this operation again and again by increasing/decreasing with \(2^N\) time the maximum numer (\(N\) being the number of iterations) until finding the square root.

As this algorithm converge using a \(2^N\) iterator, the convergeance occure in \(N=\log_2(\text{max limit}) = \log_2(46,340) \approx 16\) iterations.

For a conveniant reason, instead of taking half of the maximum limite, we take the first power of two that is above. In this way, every division by \(2^N\) lead to an integer number.

Tangent#

bs.math:tan

Compute the tangeant of an angle between 0 and 360

Inputs

(execution) as <entities>: The entities you want to perform the calculation on
(score) @s bs.in.0: The angle in degrees (not shifted)

Outputs

(score) @s bs.out.0: The tangeant of the angle shifted by 3 digits (ex: 0.42 -> 420)

Example

Calculate and display the tengeante of 42

# Once
scoreboard players set @s bs.in.0 42
function bs.math:tan
tellraw @a [{"text": "tan(42) = ", "color": "dark_gray"},{"score":{"name":"@s", "objective": "bs.out.0"}, "color": "gold"}]

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