> 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-3

FSRS-3 has 13 trainable weights and uses an exponential forgetting curve. Its stability updates use the newly calculated difficulty `math:D'`.

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-3](https://github.com/open-spaced-repetition/awesome-fsrs/wiki/The-Algorithm#fsrs-v3).
:::

## Forgetting curve and interval

```math
R(t,S)=0.9^{t/S}
```

```math
I(r,S)=S\frac{\ln r}{\ln 0.9}
```

## Initial memory

```math
S_0(G)=\operatorname{clamp}\left(w_0+(G-1)w_1,S_{min},S_{max}\right)
```

```math
D_0(G)=\operatorname{clamp}\left(w_2+(G-3)w_3,D_{min},D_{max}\right)
```

## Difficulty

```math
D^*=D+w_4(G-3)
```

```math
D'=\operatorname{clamp}\left(w_5w_2+(1-w_5)D^*,D_{min},D_{max}\right)
```

## Stability after review

```math
S_f=w_9(D')^{w_{10}}S^{w_{11}}e^{w_{12}(1-R)}
```

```math
S_r=S\left[1+e^{w_6}(11-D')S^{w_7}\left(e^{w_8(1-R)}-1\right)\right]
```

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

The model then clamps `math:S'_{\mathrm{raw}}` to the range from `math:S_{\min}` to `math:S_{\max}`.