experience/level growth system formula based on a plot chart, what kind of equation suits this one?

Question

Hi I am trying to build a experience/level growth system based on the following plot chart, however I can't think of a proper formula or equation to this chart. Can anyone lend me a hand, please? The x-axis grow significant bigger very fast but the y-axis grow very slow after it reaches 200.

I have tried log and power function but can't make it work. What kind of equation should I use to make it conform to this chart?

Answer 1

I can't resist a good formula puzzle. Here's one with a decent fit, though it's less elegant than I'd like...

float PowerCurveValueAt(float x)
{
    // Compute a root function with a steep rise in the low range.
    float low = pow(x, 0.55f);

    // Compute a second root function that's shallower in the high range.
    float high = 5f * pow(y, 0.2f);

    // Compute a blend factor that smoothly transitions between low & high.
    float blend = 1.0f / (x/30000f + 1);

    // Blend the two together, and scale to the desired output range.
    return 1.28f * lerp(high, low, blend);
} 

This formula is a bit neater, though it's slightly steeper at the far end...

float LogCurveValueAt(float x)
{
     return 6.66f * log(pow((x+300f)/500f, 8f));
}

Answer 2

DMGregory did good work at approximating the curve you posted as closely as possible.

But to provide a more general and more widely applicable answer:

• When you want fast growth at first which then gets slower but never caps, usey = sqrt(x)
• When you want to approach a constant cap value c but never reach it, use y = c - c / x
• When you really want the hard cap, but the latter function has too fast growth in the beginning but then plateaus too fast, you can substitute x with a square-root function (c - c / sqrt(x)) or even logarithmic function (c - c / log(x)).

I personally find a spreadsheet application like Excel very useful in trying out my formulas and quickly plotting them into a diagram. When you want to get really serious about designing and plotting your game formulas, you might want to give Wolfram Mathematica a try.