THE TEMPERATURE CALIBRATOR

Difficulty: Medium Tags: LLMs, Sampling, Softmax, Temperature

Story

A generation service exposes a temperature parameter. A client bug multiplies logits by temperature instead of dividing by it, making high temperature more deterministic (the opposite of its intent). Return the next-token distribution for the specified positive temperature.

The Math

For logits z and temperature tau > 0:

p = softmax(z / tau)

Use a numerically stable softmax after dividing each logit by tau. This returns probabilities only; it does not sample a token.

Input Format

z (length d), tau.

Output Format

d probabilities, 6 decimal places.

Constraints

• 1 <= d <= 10^5, 10^-3 <= tau <= 10^3, |z_i| <= 10^4.
• Time limit: 1.0 second. Memory: 64 MB.

Example

Input

z = [2, 0, -2], tau = 2

Output

0.665241 0.244728 0.090031

Explanation: Dividing logits by 2 gives (1, 0, -1), whose softmax is approximately the displayed distribution.