# 🧮 Math

**`#bs.math:help`**

The beatifull world of mathematics... **in Minecraft!**


```{image} /_imgs/modules/math-light.png
:align: center
:class: only-light
```

```{image} /_imgs/modules/math-dark.png
:align: center
:class: only-dark
```

```{epigraph}
"Mathematics has very subtle inventions that can be of great service, both to satisfy the curious and to facilitate all arts and reduce the labor of men."

-- René Descartes
```


```{button-link} https://youtu.be/Bt0HKaOosqU
:color: primary
:align: center
:shadow:

{octicon}`device-camera-video` Watch a demo
```

---

## 🔧 Functions

You can find below all functions available in this module.

---

### Combine

```{function} #bs.math:combine

Compute the combine of 2 numbers.

:Inputs:
  **Scores `$math.combine.[m,n] bs.in`**: Numbers to be combined, the smaller input will be taken from the greater input.

:Outputs:
  **Return | Score `$math.combine bs.out`**: Result of the operation.
```

```{admonition} Technical limitation
:class: important

The value of `bs.out` is incorrect if the result is greater than 2147483647 or `$math.combine.[m,n] bs.in` are not both positive.
```

*Compute $combine(4,2)$:*
```mcfunction
# Once
scoreboard players set $math.combine.m bs.in 4
scoreboard players set $math.combine.n bs.in 2
function #bs.math:combine
tellraw @a [{"text": "combine(4,2) = ","color":"dark_gray"},{"score":{"name":"$math.combine","objective":"bs.out"},"color":"gold"}]
```

> **Credits**: Ethanout

---

### Exponential

```{function} #bs.math:exp

Compute the exponential function.

:Inputs:
  **Storage `bs:in math.exp.value` [number]**: Number to be exponentiated.

:Outputs:
  **Storage `bs:out math.exp` [double]**: Result of the operation.
```

```{admonition} Technical limitation
:class: important
Due to the limit of integers that can be stored in a score, the interval of `bs:in` is limited to `[-6,15[`.
```

*Compute $exp(3)$:*
```mcfunction
# Once
data modify storage bs:in math.exp.value set value 3.0
function #bs.math:exp
data get storage bs:out math.exp
```

![](/_imgs/modules/math/exp.png)

> **Credits**: Aksiome, KubbyDev

---

### Factorial

```{function} #bs.math:factorial

Compute the factorial of the number.

:Inputs:
  **Score `$math.factorial.n bs.in`**: Number to be factorialized.

:Outputs:
  **Return | Score `$math.factorial bs.out`**: Result of the operation.
```

```{admonition} Technical limitation
:class: important

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

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

![](/_imgs/modules/math/factorial.png)

> **Credits**: KubbyDev

---

### Float radix

```{function} #bs.math:frexp

Decompose a floating point number into a normalized fraction and an integral power of two.

:Inputs:
  **Storage `bs:in math.frexp.value` [number]**: Number to be decomposed.

:Outputs:
  **Storage `bs:out math.frexp.e` [int]**: Exponent for the power of 2.

  **Storage `bs:out math.frexp.x` [double]**: Normalized fraction in range  `]-1,-0.5]` or `[0.5,1[`.
```

*Decompose 5.8 into its mantissa and exponent:*
```mcfunction
# Once
data modify storage bs:in math.frexp.value set value 5.8
function #bs.math:frexp
data get storage bs:out math.frexp
```

> **Credits**: Aksiome

---

### Greatest common denominator

```{function} #bs.math:gcd

Compute the greatest common denominator of two numbers.

:Inputs:
  **Scores `$math.gcd.[a,b] bs.in`**: The two numbers.

:Outputs:
  **Return | Score `$math.gcd bs.out`**: The greatest common denominator.
```

*Calculate the greatest common denominator between 16 and 12:*
```mcfunction
# Once
scoreboard players set $math.gcd.a bs.in 16
scoreboard players set $math.gcd.b bs.in 12
function #bs.math:gcd
tellraw @a [{"text": "gcd(16,12) = ", "color": "dark_gray"},{"score":{"name":"$math.gcd", "objective": "bs.out"}, "color": "gold"}]
```

![](/_imgs/modules/math/gcd.png)

> **Credits**: Aksiome, Leirof

---

### Inverse trigonometry

::::{tab-set}
:::{tab-item} Acos

```{function} #bs.math:acos

Compute the arc cosine of a value between -1 and 1.

:Inputs:
  **Score `$math.acos.value bs.in`**: Value you want to compute the arccosine of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores.

:Outputs:
  **Return | Score `$math.acos bs.out`**: Result of the operation in degrees, shifted by 2 digits.
```

*Compute and display the arccosine of 0.42:*
```mcfunction
# Once
scoreboard players set $math.acos.value bs.in 420
function #bs.math:acos
tellraw @a [{"text":"acos(0.42) = ","color":"dark_gray"},{"score":{"name":"$math.acos","objective":"bs.out"},"color":"gold"}]
```

![](/_imgs/modules/math/arccos.png)

:::
:::{tab-item} Asin

```{function} #bs.math:asin

Compute the arc sine of a value between -1 and 1.

:Inputs:
  **Score `$math.asin.value bs.in`**: Value you want to compute the arcsine of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores.

:Outputs:
  **Return | Score `$math.asin bs.out`**: Result of the operation in degrees, shifted by 2 digits.
```

*Compute and display the arcsine of 0.42:*
```mcfunction
# Once
scoreboard players set $math.asin.value bs.in 420
function #bs.math:asin
tellraw @a [{"text":"asin(0.42) = ","color":"dark_gray"},{"score":{"name":"$math.asin","objective":"bs.out"},"color":"gold"}]
```

![](/_imgs/modules/math/arcsin.png)

:::
:::{tab-item} Atan

```{function} #bs.math:atan

Compute the arc tangent of a value between -infinite and +infinite.

:Inputs:
  **Score `$math.atan.value bs.in`**: Value you want to compute the arctangent of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores.

:Outputs:
  **Return | Score `$math.atan bs.out`**: Result of the operation in degrees, shifted by 2 digits.
```

*Compute and display the arctan of 0.42:*
```mcfunction
# Once
scoreboard players set $math.atan.value bs.in 420
function #bs.math:atan
tellraw @a [{"text":"atan(0.42) = ","color":"dark_gray"},{"score":{"name":"$math.atan","objective":"bs.out"},"color":"gold"}]
```

![](/_imgs/modules/math/arctan.png)

:::
:::{tab-item} Atan2

```{function} #bs.math:atan2

Compute the 2-argument arctangent of y and x.

:Inputs:
  **Scores `$math.atan2.[y,x] bs.in`**: Values you want to compute the arctangent of, shifted by 3 digits (1,2345 -> 1234) for better precision in integer scores.

:Outputs:
  **Return | Score `$math.atan2 bs.out`**: Result of the operation in degrees, shifted by 2 digits.
```

*Compute and display the atan2 of (0.42, 0.8):*
```mcfunction
# Once
scoreboard players set $math.atan2.y bs.in 420
scoreboard players set $math.atan2.x bs.in 800
function #bs.math:atan2
tellraw @a [{"text":"atan2(0.42, 0.8) = ","color":"dark_gray"},{"score":{"name":"$math.atan2","objective":"bs.out"},"color":"gold"}]
```

:::
::::


> **Credits**: Aksiome, KubbyDev, Leirof

---

### Logarithm

::::{tab-set}

:::{tab-item} Base e (Neperian)

```{function} #bs.math:log

Compute the Neperian logarithm (base e) of a number.

:Inputs:
  **Storage `bs:in math.log.value` [number]**: Number to be logarithmized.

:Outputs:
  **Storage `bs:out math.log` [double]**: Result of the operation.
```

*Calculate $ln(28)$:*
```mcfunction
# Once
data modify storage bs:in math.log.value set value 28.0
function #bs.math:log
data get storage bs:out math.log
```

:::
:::{tab-item} Base 2

```{function} #bs.math:log2

Compute the logarithm in base 2 of a number.

:Inputs:
  **Storage `bs:in math.log2.value` [number]**: Number to be logarithmized.

:Outputs:
  **Storage `bs:out math.log2` [double]**: Result of the operation.
```

*Calculate $log_2(28)$:*
```mcfunction
# Once
data modify storage bs:in math.log2.value set value 28.0
function #bs.math:log2
data get storage bs:out math.log2
```
:::
:::{tab-item} Base 10

```{function} #bs.math:log10

Compute the logarithm in base 10 of a number.

:Inputs:
  **Storage `bs:in math.log10.value` [number]**: Number to be logarithmized.

:Outputs:
  **Storage `bs:out math.log10` [double]**: Result of the operation.
```

*Calculate $log_{10}(28)$:*
```mcfunction
# Once
data modify storage bs:in math.log10.value set value 28.0
function #bs.math:log10
data get storage bs:out math.log10
```

:::
:::{tab-item} Base a

```{function} #bs.math:loga

Compute the logarithm in base a of a number.

:Inputs:
  **Storage `bs:in math.loga.value` [number]**: Number to be logarithmized.

  **Storage `bs:in math.loga.a` [number]**: Base of the logarithm.

:Outputs:
  **Storage `bs:out math.loga` [double]**: Result of the operation.
```

*Calculate $log_4(28)$:*
```mcfunction
# Once
data modify storage bs:in math.loga.a set value 4
data modify storage bs:in math.loga.value set value 28.0
function #bs.math:loga
data get storage bs:out math.loga
```

:::
::::

> **Credits**: Aksiome, KubbyDev

---

### Power

::::{tab-set}

:::{tab-item} Power

```{function} #bs.math:pow {scaling:<scaling>}

Compute $x^y$.

:Inputs:
  **Score `$math.pow.base bs.in`**: The base.

  **Score `$math.pow.exp bs.in`**: The exponent.

  **Macro Var `scaling` [number]**: Scalar for the function’s input base and the output.
:Outputs:
  **Return | Score `$math.pow bs.out`**: Result of the operation.
```

*Compute $2.245^6$:*
```mcfunction
# Once
scoreboard players set $math.pow.base bs.in 2245
scoreboard players set $math.pow.exp bs.in 6
function #bs.math:pow {scale:1000}
tellraw @a [{"text": "(2.245^6)*(1000) = ", "color": "dark_gray"},{"score":{"name":"$math.pow", "objective": "bs.out"}, "color": "gold"}]
```

:::
:::{tab-item} Power of 2

```{function} #bs.math:pow2

Compute $2^n$.

:Inputs:
  **Score `$math.pow2.exp bs.in`**: The exponent.

:Outputs:
  **Return | Score `$math.pow2 bs.out`**: Result of the operation.
```

*Compute $2^6$:*
```mcfunction
# Once
scoreboard players set $math.pow2.exp bs.in 6
function #bs.math:pow2
tellraw @a [{"text": "2^6 = ", "color": "dark_gray"},{"score":{"name":"$math.pow2", "objective": "bs.out"}, "color": "gold"}]
```

:::
::::

![](/_imgs/modules/math/power.png)

> **Credits**: Aksiome, Leirof

---

### Rounded division

```{function} #bs.math:divide

Divide a number by another then round the result to the nearest integer (Minecraft rounds down to the next integer).

:Inputs:
  **Score `$math.divide.num bs.in`**: The numerator.

  **Score `$math.divide.den bs.in`**: The denominator.
:Outputs:
  **Return | Score `$math.divide bs.out`**: Result of the division.
```

*Calculate $9/5$:*
```mcfunction
# Once
scoreboard players set $math.divide.num bs.in 9
scoreboard players set $math.divide.den bs.in 5
function #bs.math:divide
tellraw @a [{"text": "9 / 5 = ", "color": "dark_gray"},{"score":{"name":"$math.divide", "objective": "bs.out"}, "color": "gold"}]
```

![](/_imgs/modules/math/divide.png)

> **Credits**: Aksiome, theogiraudet

---

### Square root

::::{tab-set}

:::{tab-item} Integer

```{function} #bs.math:isqrt

Compute the square root of an int number.

:Inputs:
  **Score `$math.isqrt.value bs.in`**: Number you want to calculate the square root of.

:Outputs:
  **Return | Score `$math.isqrt bs.out`**: Floored result of the square root.
```

*Calculate and display $\sqrt{42}$:*
```mcfunction
# Once
scoreboard players set $math.isqrt.value bs.in 42
function #bs.math:isqrt
tellraw @a [{"text": "sqrt(42) = ", "color": "dark_gray"},{"score":{"name":"$math.isqrt", "objective": "bs.out"}, "color": "gold"}]
```

:::
:::{tab-item} Decimal

```{function} #bs.math:sqrt

Compute the square root of a floating number.

:Inputs:
  **Storage `bs:in math.sqrt.value` [number]**: Number you want to calculate the square root of.

:Outputs:
  **Storage `bs:out math.sqrt` [float]**: Result of the operation.
```

*Calculate and display $\sqrt{42}$:*
```mcfunction
# Once
data modify storage bs:in math.sqrt.value set value 42
function #bs.math:sqrt
tellraw @a [{"text": "sqrt(42) = ", "color": "dark_gray"},{"nbt": "math.sqrt", "storage": "bs:out", "color": "gold"}]
```
:::
::::

![](/_imgs/modules/math/sqrt.png)

> **Credits**: Ethanout

---

### Trigonometry

::::{tab-set}
:::{tab-item} Cos

```{function} #bs.math:cos

Compute the cosine of an angle between 0 and 360.

:Inputs:
  **Score `$math.cos.angle bs.in`**: Angle in degrees shifted by 2 digits (ex: 90.15 -> 9015).

:Outputs:
  **Return | Score `$math.cos bs.out`**: Cosine of the angle shifted by 3 digits (ex: 0.42 -> 420).
```

*Compute and display the cosine of 42:*
```mcfunction
# Once
scoreboard players set $math.cos.angle bs.in 4200
function #bs.math:cos
tellraw @a [{"text": "cos(42) = ", "color": "dark_gray"},{"score":{"name":"$math.cos", "objective": "bs.out"}, "color": "gold"}]
```

![](/_imgs/modules/math/cosine.png)

```{admonition} How does it work?
:class: dropdown

This functions 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 and thus the cosine.

![](/_imgs/modules/math/bhaskara.png)

```

:::
:::{tab-item} Sin

```{function} #bs.math:sin

Compute the sine of an angle between 0 and 360.

:Inputs:
  **Score `$math.sin.angle bs.in`**: Angle in degrees shifted by 2 digits (ex: 90.15 -> 9015).

:Outputs:
  **Return | Score `$math.sin bs.out`**: Sine of the angle shifted by 3 digits (ex: 0.42 -> 420).
```

*Compute and display the sine of 42:*
```mcfunction
# Once
scoreboard players set $math.sin.angle bs.in 4200
function #bs.math:sin
tellraw @a [{"text": "sin(42) = ", "color": "dark_gray"},{"score":{"name":"$math.sin", "objective": "bs.out"}, "color": "gold"}]
```

![](/_imgs/modules/math/sine.png)

```{admonition} How does it work?
:class: dropdown

This functions 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.

![](/_imgs/modules/math/bhaskara.png)

```

:::
:::{tab-item} Tan

```{function} #bs.math:tan

Compute the tangent of an angle between 0 and 360.

:Inputs:
  **Score `$math.tan.angle bs.in`**: Angle in degrees shifted by 2 digits (ex: 90.15 -> 9015).

:Outputs:
  **Return | Score `$math.tan bs.out`**: Tangent of the angle shifted by 3 digits (ex: 0.42 -> 420).
```

*Compute and display the tangent of 42:*
```mcfunction
# Once
scoreboard players set $math.tan.angle bs.in 4200
function #bs.math:tan
tellraw @a [{"text": "tan(42) = ", "color": "dark_gray"},{"score":{"name":"$math.tan", "objective": "bs.out"}, "color": "gold"}]
```

![](/_imgs/modules/math/tangent.png)

:::
:::{tab-item} Sincos

```{function} #bs.math:sincos

Compute both the sine and cosine of an angle between 0 and 360 in a single operation.

:Inputs:
  **Score `$math.sincos.angle bs.in`**: Angle in degrees shifted by 2 digits (ex: 90.15 -> 9015).

:Outputs:
  **Score `$math.sincos.cos bs.out`**: Cosine of the angle shifted by 3 digits (ex: 0.42 -> 420).

  **Score `$math.sincos.sin bs.out`**: Sine of the angle shifted by 3 digits (ex: 0.42 -> 420).
```

*Compute and display the sine and cosine of 42:*
```mcfunction
# Once
scoreboard players set $math.sincos.angle bs.in 4200
function #bs.math:sincos
tellraw @a [{"text": "cos(42) = ", "color": "dark_gray"},{"score":{"name":"$math.sincos.cos", "objective": "bs.out"}, "color": "gold"}]
tellraw @a [{"text": "sin(42) = ", "color": "dark_gray"},{"score":{"name":"$math.sincos.sin", "objective": "bs.out"}, "color": "gold"}]
```

:::
::::

> **Credits**: Aksiome, Leirof

---

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