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Hello Dflash team,
I was testing DFlash on a Blackwell 6000 Pro to evaluate how speculative decoding affects output generation. I observed that it performs worse on MMLU math questions than the vanilla meta-llama/Llama-3.1-8B-Instruct when using SGLang.
Question: Find the product of the given polynomials in the given polynomial ring. f(x) = 4x - 5, g(x) = 2x^2 - 4x + 2 in Z_8.
Answer using dflash draft decoding: To meta-llama/Llama-3.1-8B-Instruct - Not correct answer
To find the product of the given polynomials in Z_8[x], we need to multiply them as we would in a regular polynomial ring, but we need to perform the operations modulo 8.
f(x) = 4x - 5
g(x) = 2x^2 - 4x + 2
First, we multiply each term of f(x) by each term of g(x):
(4x)(2x^2) = 8x^3 (mod 8) = 0x^3
(4x)(-4x) = -16x^2 (mod 8) = 0x^2
(4x)(2) = 8x (mod 8) = 0x
(-5)(2x^2) = -10x^2 (mod 8) = 6x^2
(-5)(-4x) = 20x (mod 8) = 4x
(-5)(2) = -10 (mod 8) = 6
Now, we combine like terms:
0x^3 + 0x^2 + 0x + 6x^2 + 4x + 6
Combine like terms:
-6x^2 + 4x + 6
So, the product of f(x) and g(x) in Z_8[x] is -6x^2 + 4x + 6.
Answer without dflash (meta-llama/Llama-3.1-8B-Instruct) - Correct
To find the product of the given polynomials in the polynomial ring Z_8[x], we'll follow the standard multiplication process for polynomials, but we'll perform the arithmetic operations in Z_8.
f(x) = 4x - 5
g(x) = 2x^2 - 4x + 2
First, multiply each term of f(x) by each term of g(x):
(4x * 2x^2) + (4x * (-4x)) + (4x * 2) + ((-5) * 2x^2) + ((-5) * (-4x)) + ((-5) * 2)
Now, simplify the expressions using Z_8 arithmetic:
(8x^3) mod 8 = 0 (since 8 mod 8 = 0)
(-16x^2) mod 8 = 0 (since 16 mod 8 = 0)
(8x) mod 8 = 0 (since 8 mod 8 = 0)
(-10x^2) mod 8 = -2x^2 (since 10 mod 8 = 2)
(20x) mod 8 = 4x (since 20 mod 8 = 4)
(-10) mod 8 = 6 (since 10 mod 8 = 2)
Now, combine like terms:
0 + 0 + 0 - 2x^2 + 4x + 6
The resulting polynomial is:
-2x^2 + 4x + 6
However, this polynomial is in Z[x] (the ring of polynomials with integer coefficients). Since we're working in Z_8[x], we need to express the coefficients in Z_8.
In Z_8, the coefficients -2 and 6 can be expressed as 6 and 6, respectively.
So, the product of the given polynomials in Z_8[x] is:
6x^2 + 4x + 6
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