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\log_{3}\left(8 + 4 c\right) = 3 \implies c = \frac{19}{4}
1 + a = 0 \implies a = -1
-6 + v = 0 \implies v = 6
40 - 8 = 32
\int \left(4 z^3\right) dz = z^4 + C
t v \left(-9 - v\right) = -9 t v - t v^2
-5 b + b^2 = 0 \implies b \in \{ 0, 5 \}
\frac{d}{dz}\left(7 - z - \left(9 + z\right)\right) = -2
\log_{2}\left(8 + 3 x\right) = 3 \implies x = 0
f(w) = 3 + 4 w + w^2, \quad g(c) = 2 + c \implies f(g(c)) = 15 + 8 c + c^2
\frac{d}{dv}\left(v^4\right) = 4 v^3
\frac{-7}{25} = \frac{-7}{25}
z \left(3 + z\right) = 3 z + z^2
25 + 4 = 29
(2, -1, 3) \times (2, -3, -1) = (10, 8, -4)
\int \left(-\sin{\left(u\right)}\right) du = -1 + \cos{\left(u\right)} + C
\int \left(2\right) dv = 2 v + C
33 - 16 = 17
7 - 8 v + v^2 = 0 \implies v \in \{ 1, 7 \}
f(y) = 4 + 4 y + y^2, \quad g(a) = 1 + a \implies f(g(a)) = 9 + 6 a + a^2
f(w) = 3 + 2 w + w^2, \quad g(u) = 2 + 3 u \implies f(g(u)) = 11 + 18 u + 9 u^2
-47 - 33 = -80
\log_{5}\left(8 + y\right) = 1 \implies y = -3
f(z) = 1 + 3 z + z^2, \quad g(v) = 1 + v \implies f(g(v)) = 5 + 5 v + v^2
-4 - 31 = -35
-70 + 11 c + 10 c^2 + c^3 = 0 \implies c \in \{ -7, -5, 2 \}
\lim_{u \to -5} \left(2 u\right) = -10
\int \left(\cos{\left(y\right)}\right) dy = \sin{\left(y\right)} + C
\frac{32}{47} = \frac{32}{47}
(-1, 0, 0) \times (-4, -3, 5) = (0, 5, 3)
8 - 9 w + w^2 = 0 \implies w \in \{ 1, 8 \}
28 \cdot 39 = 1092
\log_{2}\left(1 + 2 y\right) = 3 \implies y = \frac{7}{2}
\lim_{c \to -1} \left(c\right) = -1
-48 - 2 w + w^2 = 0 \implies w \in \{ -6, 8 \}
35 \cdot 24 = 840
\frac{-38}{38} = -1
\frac{d}{dx}\left(x^2 - \sin{\left(x\right)}\right) = 2 x - \cos{\left(x\right)}
\frac{d}{dx}\left(x \sin{\left(x\right)}\right) = x \cos{\left(x\right)} + \sin{\left(x\right)}
\frac{44}{12} = \frac{11}{3}
(-4, -5, -5) \times (-1, 2, 0) = (10, 5, -13)
\log_{3}\left(4 + 2 a\right) = 1 \implies a = \frac{-1}{2}
84 - 44 z - z^2 + z^3 = 0 \implies z \in \{ -7, 2, 6 \}
\log_{4}\left(7 + 2 v\right) = 1 \implies v = \frac{-3}{2}
5 + b = 0 \implies b = -5
-20 + y + y^2 = 0 \implies y \in \{ -5, 4 \}
\lim_{w \to -2} \left(4 + w\right) = 2
6 \cdot 1 = 6
-40 a - 3 a^2 + a^3 = 0 \implies a \in \{ -5, 0, 8 \}
u = 0 \implies u = 0
\lim_{z \to -3} \left(-2\right) = -2
\int \left(-1 + 2 z\right) dz = -z + z^2 + C
\int \left(1\right) dv = v + C
8 + t = 0 \implies t = -8
f(x) = 1 + 3 x + x^2, \quad g(w) = 2 + w \implies f(g(w)) = 11 + 7 w + w^2
\int \left(1\right) dw = w + C
\frac{d}{dy}\left(-2 y + y^4\right) = -2 + 4 y^3
\frac{2}{10} = \frac{1}{5}
\frac{d}{dw}\left(\cos{\left(w\right)}\right) = -\sin{\left(w\right)}
-20 - c + c^2 = 0 \implies c \in \{ -4, 5 \}
\lim_{b \to 3} \left(0\right) = 0
\frac{d}{dx}\left(-4 x + x^2\right) = -4 + 2 x
\log_{2}\left(6 + 4 t\right) = 2 \implies t = \frac{-1}{2}
\log_{2}\left(7 + 4 c\right) = 2 \implies c = \frac{-3}{4}
6 z + z^2 = 0 \implies z \in \{ -6, 0 \}
\log_{3}\left(6 + 4 x\right) = 3 \implies x = \frac{21}{4}
-1 + w = 0 \implies w = 1
\int \left(1 - \sin{\left(b\right)}\right) db = -1 + b + \cos{\left(b\right)} + C
-41 - 10 = -51
\frac{d}{dy}\left(y\right) = 1
(-1, 5, 5) \times (4, 4, 3) = (-5, 23, -24)
f(t) = 3 + 5 t + t^2, \quad g(c) = 4 + 2 c \implies f(g(c)) = 39 + 26 c + 4 c^2
3 + z = 0 \implies z = -3
\frac{d}{dc}\left(c^3\right) = 3 c^2
-64 + u^2 = 0 \implies u \in \{ -8, 8 \}
5 \left(-2 - w\right) = -10 - 5 w
-32 - 21 = -53
f(c) = 3 + c + c^2, \quad g(z) = 4 + z \implies f(g(z)) = 23 + 9 z + z^2
\frac{d}{dw}\left(0\right) = 0
\lim_{b \to 3} \left(b^4\right) = 81
\lim_{u \to -4} \left(\sin{\left(u\right)}\right) = -\sin{\left(4\right)}
\int \left(3 w^2\right) dw = w^3 + C
\frac{43}{43} = 1
f(v) = 3 + 3 v + v^2, \quad g(c) = 4 + c \implies f(g(c)) = 31 + 11 c + c^2
\lim_{c \to 3} \left(8\right) = 8
\frac{d}{da}\left(a\right) = 1
(2, -5, -3) \times (-2, -3, -4) = (11, 14, -16)
f(y) = 4 + 3 y + y^2, \quad g(x) = 4 + x \implies f(g(x)) = 32 + 11 x + x^2
-144 + 6 z + 11 z^2 + z^3 = 0 \implies z \in \{ -8, -6, 3 \}
-42 + t + t^2 = 0 \implies t \in \{ -7, 6 \}
\log_{3}\left(5 + 2 y\right) = 1 \implies y = -1
-16 - 10 w + 7 w^2 + w^3 = 0 \implies w \in \{ -8, -1, 2 \}
\lim_{u \to 3} \left(\sin{\left(u\right)}\right) = \sin{\left(3\right)}
\log_{5}\left(1 + 3 u\right) = 2 \implies u = 8
\log_{4}\left(6 + 3 a\right) = 3 \implies a = \frac{58}{3}
\frac{d}{dt}\left(2 t + t^2\right) = 2 + 2 t
-23 - 19 = -42
38 - 31 = 7
34 + 13 = 47
\lim_{v \to 5} \left(-10 - v\right) = -15

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HALF Mathematic Database

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Feature	HALF	GSM8K	MATH
Primary Focus	Pre-training / Synthetic Symbolic Pre-training	Grade school math word problems	Competition-level mathematics
Dataset Size	~53.9 Million examples (4 shards)	~8.5K examples (7.5K train / 1K test)	12.5K examples (7.5K train / 5K test)
Format	Single-line LaTeX equations (`A \implies B`)	Natural language word problems	LaTeX & step-by-step reasoning proofs
Natural Language	None (pure LaTeX math expressions)	High (English narrative word problems)	High (English proofs + LaTeX notation)
Data Generation	Programmatically generated / Synthetic	Human-written / Human-verified	Human-compiled (AMC, AIME, HMMT)
Target Task	Symbolic identity training, syntax alignment	Multi-step arithmetic reasoning	Deep conceptual problem-solving

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