Softmax Function

The softmax function turns a list of scores (logits) into probabilities. Language models use it to choose the next word:

softmax(x)_i = e^(x_i) / (e^(x_1) + e^(x_2) + … + e^(x_n))

Large scores make e^x overflow, so first subtract the largest score from every score. The result is mathematically the same.

Implement softmax(logits).

Input

logits: A list of scores.

Output

Return the list of probabilities, in the same order. They are all between 0 and 1 and add up to 1.

Constraints

1 ≤ length ≤ 1,000

-1,000 ≤ each score ≤ 1,000

Examples

Example 1

Input:
logits = [1.0, 2.0, 3.0]

Expected output:
[0.090031, 0.244728, 0.665241]

Explanation

e¹ ≈ 2.718, e² ≈ 7.389 and e³ ≈ 20.086 add up to 30.193, so the probabilities are about [0.090031, 0.244728, 0.665241].

Example 2

Input:
logits = [1000.0, 1000.0]

Expected output:
[0.5, 0.5]

Explanation

Computing e^1000 directly overflows. Subtracting the maximum gives [0, 0], and e⁰ / (e⁰ + e⁰) = 0.5 for both.

References

Run the sample tests to check your code without submitting.

Example 1 Not run

Input:

logits = [1.0, 2.0, 3.0]

Expected output:

[0.090031, 0.244728, 0.665241]

Your output

Example 2 Not run
Input:
```
logits = [1000.0, 1000.0]
```
Expected output:
```
[0.5, 0.5]
```
Your output

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