> 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-4.5

FSRS-4.5 has 17 trainable weights. It changes the forgetting curve from FSRS-4 while retaining the same difficulty and successful-recall structure.

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-4.5](https://github.com/open-spaced-repetition/awesome-fsrs/wiki/The-Algorithm#fsrs-45).
:::

## 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)=\operatorname{clamp}\left(w_4-(G-3)w_5,D_{min},D_{max}\right)
```

## Difficulty

```math
D^*=D-w_6(G-3)
```

```math
D'=\operatorname{clamp}\left(w_7w_4+(1-w_7)D^*,D_{min},D_{max}\right)
```

## Stability after review

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

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

The Again branch prevents post-lapse stability from exceeding the previous `math:S`. The model then clamps `math:S'_{\mathrm{raw}}` to the range from `math:S_{\min}` to `math:S_{\max}`.