# Formulas

Profectus utilizes formulas for various features, such as increasing requirements for repeatables and challenges or determining resource gains in conversions. These formulas often need to be inverted or integrated to enable features like buying multiple levels of a repeatable at once or determining when a conversion will increase resource gains. The Formula class can handle these operations, supporting every function Decimal does, while tracking the operations internally.

For example, a cost function like `Decimal.pow(this.amount, 1.05).times(100)`

can be represented using a Formula: `Formula.variable(this.amount).pow(1.05).times(100)`

.

```
const myRepeatable = createRepeatable(() => ({
requirements: createCostRequirement(() => ({
resource: points,
cost: Formula.variable(myRepeatable.amount).pow_base(1.05).times(100)
})),
maximize: true
}));
```

## Softcaps

Often games incorporate "softcaps" on formulas - that means making the formula step-wise, where everything above a certain value has an additional operation on it that makes it scale differently. Formulas support this via `Formula.step`

, which takes a threshold value and a function to modify the formula appropriately. The function gets a formula that will contain a variable for the amount *above* the threshold of the original value. Whatever result is returned will then be added back onto the threshold to achieve the softcapped value. Take, for example, a formula `Formula.variable().div(10).sqrt()`

that we want to apply a softcap to - let's say it should square root everything after 1e100. The softcapped formula will look like this: `Formula.variable().div(10).sqrt().step(1e100, f => f.sqrt())`

.

## Limitations

### Single Variable

For inverting and integrating, formulas should have a single variable, which serves as the input. Although other parts of the formula can be computed values, they must not be dependent on the input variable for inverting or integrating purposes. Formulas work correctly as long as changing other parts of the formula doesn't affect the input variable.

The variable is defined by wrapping the input to the formula in `Formula.variable`

. In the cost requirement example above, the repeatable's `amount`

is the input to the variable. That means inverting the function would calculate the amount based on the cost, and integrating would be finding the area under the curve where the repeatable's `amount`

is the x axis and the output of the formula is the y axis.

If the cost formula had not marked a variable, e.g. `Formula.pow(1.05, myRepeatable.amount).times(100)`

then the formula would not be invertible or integrable, and would throw an error if either operation was tried. While `evaluate()`

will work as expected, passing a value to `evaluate()`

will not, as it will not know what value to override.

### Invertibility and Integrability

Certain operations may not support inverting or integrating. Functions such as rounding or clamping are non-invertible, while others like super-log are non-integrable. You may only use a single complex operation in addition to add, sub, mult, and div operations. For formulas with two complex operations, a custom formula is required.

### Spending Resources

When working with formulas, the `spendResources`

property determines whether the formula needs to be invertible or integrable. The property is used in two utilities: calculateMaxAffordable and calculateCost. These utilities are ultimately employed wherever formulas are used, such as in cost requirements or conversions.

Spending resources refers to whether max affordability and cost calculations should account for resources spent. If spending resources is set to true, the formula will be integrated and inverted, albeit with certain limitations. For example, exponential modifiers and soft caps can make a formula non-integrable.

Consider a simple formula that returns the input value as the cost. For example, to make three sequential purchases, the costs would be 1, 2, and 3, totaling 6. When resources are spent, making three purchases in a single operation will cost a total of 6. However, if resources are not spent, the cost would only be 3, the cost of the last purchase. Accounting for resources being spent means that the overall cost calculation reflects the cumulative costs of multiple purchases.

Integration in this context is an estimation, as it calculates the area under a continuous curve, while actual purchases occur at integer values. This means that calculating max affordable might overshoot and the cost at that amount might undershoot. To help address this, the `summedPurchases`

property can be used, which manually sums the last X purchases, reducing the error margin. This also ensures higher accuracy when the player makes fewer purchases at once, which is when accuracy is most relevant. For larger numbers of purchases, a slight overshoot or undershoot is typically acceptable.

### Modifiers

When applying changes to a formula using modifiers, use the `modifierToFormula`

utility to apply the modifier while preserving invertibility if the modifier is invertible.

### Custom Formulas

If the existing functions are insufficient for your needs, create a custom formula by implementing your own evaluation, inversion, and integration functions. Use the operations in `src/game/formulas/operations.ts`

as reference. For spending resources, the integral formula must be invertible.