> For AI agents: the complete documentation index is available at https://open-spaced-repetition.github.io/ts-fsrs/llms.txt, the full documentation bundle is available at https://open-spaced-repetition.github.io/ts-fsrs/llms-full.txt.

# FSRS-5

FSRS-5 has 19 trainable weights. It keeps the fixed FSRS-4.5 forgetting curve and adds a grade-dependent same-day stability update.

See the [Model overview](https://open-spaced-repetition.github.io/ts-fsrs/guide/model.md) for the shared symbols and Model contract.

:::info Formula source
Reviewed against [awesome-fsrs Wiki — FSRS-5](https://github.com/open-spaced-repetition/awesome-fsrs/wiki/The-Algorithm#fsrs-5).
:::

## Forgetting curve and interval

```math
R(t,S)=\left(1+\frac{19t}{81S}\right)^{-1/2}
```

```math
I(r,S)=\frac{81S}{19}\left(r^{-2}-1\right)
```

## Initial memory

```math
S_0(G)=\operatorname{clamp}\left(w_{G-1},S_{min},S_{max}\right)
```

```math
D_0(G)=w_4-e^{w_5(G-1)}+1,\qquad
D_{initial}=\operatorname{clamp}\left(D_0(G),1,10\right)
```

## Difficulty

```math
\Delta D=-w_6(G-3),\qquad
D^*=D+\Delta D\frac{10-D}{9}
```

```math
D'=\operatorname{clamp}\left(w_7D_0(4)+(1-w_7)D^*,1,10\right)
```

## Stability after review

For `math:t > 0`, or when `enableShortTerm` is disabled, the rating selects the following branch.

`math:H_G = w_{15}` for Hard and `math:1` otherwise; `math:E_G = w_{16}` for Easy and `math:1` otherwise.

When `enableShortTerm` is enabled, `math:C` is the previous stability multiplied by `math:\exp(-w_{17}w_{18})`; otherwise, `math:C = S`:

```math
S_f=\min\left(w_{11}D^{-w_{12}}\left((S+1)^{w_{13}}-1\right)e^{w_{14}(1-R)},C\right)
```

```math
S_r=S\left[1+e^{w_8}(11-D)S^{-w_9}\left(e^{w_{10}(1-R)}-1\right)H_GE_G\right]
```

```math
S'_{raw}(G)=
\begin{cases}
S_f,&G=1\\
S_r,&G\in\{2,3,4\}
\end{cases}
```

`math:C` is an implementation cap for post-lapse stability, not a separately configured stability floor. The model then clamps `math:S'_{\mathrm{raw}}` to the range from `math:S_{\min}` to `math:S_{\max}`.

## Same-day update

Here, “same-day” means the model receives `elapsedDays = 0`; the model does not inspect calendar dates. Direct callers and the scheduler’s `chrono` determine the elapsed value.

When `enableShortTerm` is enabled and `math:t = 0`:

```math
S'=\operatorname{clamp}\left(
S e^{w_{17}(G-3+w_{18})},S_{min},S_{max}
\right)
```