ddrg/math_structure_deberta
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32,460 | 30 = 2220422932^3 + \left(-2218888517\right)^3 + (-283059965) \cdot (-283059965) \cdot (-283059965) |
9,583 | \left(\sqrt{z + d}\right)^2 = \sqrt{G + m} * \sqrt{G + m}\Longrightarrow G + m = z + d |
-6,693 | 80/100 + 3/100 = 8/10 + \dfrac{1}{100}\times 3 |
-14,550 | 1 + 5 \cdot 7 = 1 + 35 = 1 + 35 = 36 |
15,023 | \dfrac{x - y}{x^2 - y \cdot y} = \frac{1}{y + x} |
-5,133 | \dfrac{66.6}{1000} = \frac{66.6}{1000} |
26,406 | \tfrac{6}{35} = 3/5*\frac{2}{7} |
20,175 | \frac{1}{54}\cdot 4 = \frac{\dfrac{1}{2}}{2}\cdot 8\cdot \frac{1}{27} |
17,157 | i^{-1} = i^{1 + 2 \times (-1)} = \dfrac{i}{i^2} = i/(-1) = -i |
6,302 | 0 = k\cdot 3 + (-1) \Rightarrow k = 1/3 |
1,448 | m = 3/2*y rightarrow 2/3*m = y |
2,443 | \left\lfloor{\frac{90000}{35}}\right\rfloor = 2571 |
2,919 | \frac{1}{20} = \frac14 - \frac{1}{5} |
40,555 | \frac{1}{3}*3 = 1 = 3 + 2\left(-1\right) |
10,843 | y\cos(a) = \sin(a) x \Rightarrow -x^2 \sin^2(a)*2 = -y^2 \cos^2(a)*2 |
17,016 | \mathbb{E}[\sum_{l=1}^x X_l] = \sum_{l=1}^x \mathbb{E}[X_l] |
13,601 | 0 = x^3 + b^3 + h \cdot h \cdot h - 3 \cdot x \cdot b \cdot h = (x + b + h) \cdot \left(x^2 + b^2 + h^2 - x \cdot b - b \cdot h - h \cdot x\right) |
7,202 | 2 + 2 + 2 + 2 + \dotsm = -\frac12 |
-7,793 | \frac{1}{-4}\cdot (20\cdot i - 20) = -\frac{20}{-4} + i\cdot 20/\left(-4\right) |
28,915 | z^6 + z^5 + z^4 + z^3 = (z^3 + z) \cdot \left(z^2 + z^3\right) |
7,100 | 90/100 \cdot \left(1 - \frac{75}{100}\right) = \frac{25}{100} \cdot 90 \cdot \frac{1}{100} = 9/40 |
27,794 | m \cdot 0 = m + (-1)^m \cdot 0 = m = 0 + \left(-1\right)^0 \cdot m = 0 \cdot m |
-26,432 | 16 = 24 \times 2/3 |
45,561 | 195 = 3*5*13 |
37,180 | t^2 + 2\cdot y\cdot t + \left(-1\right) = 0 \Rightarrow -y ± \sqrt{1 + y^2} = t |
11,270 | 1 = (1^{1.5})^{\frac{1}{2}} |
46,411 | \sin{4 \cdot x}/\sin{x} = \dfrac{1}{e^{i \cdot x} - e^{-i \cdot x}} \cdot \left(e^{4 \cdot i \cdot x} - e^{-4 \cdot i \cdot x}\right) = e^{3 \cdot i \cdot x} + e^{i \cdot x} + e^{-i \cdot x} + e^{-3 \cdot i \cdot x} = 2 \cdot \cos{x} + 2 \cdot \cos{3 \cdot x} |
-21,055 | 6/8 = \frac34\cdot 2/2 |
3,552 | E(T_1) + \cdots + E(T_x) = E(T_1 + \cdots + T_x) |
6,773 | \cos{6z} = \cos(5z + z) |
-30,916 | 3 \cdot b + 6 = 3 \cdot b + 6 |
21,852 | \alpha \times |Z|^2 + \beta \times |Z^2| = \alpha \times |Z|^2 + \beta \times |Z|^2 = (\alpha + \beta) \times |Z|^2 |
18,533 | \dfrac{D\cdot x}{D} = 1 \Rightarrow x\cdot D = D |
-18,319 | \frac{42 + y^2 - y\cdot 13}{y^2 - 7\cdot y} = \frac{(y + 6\cdot \left(-1\right))\cdot (y + 7\cdot (-1))}{(y + 7\cdot (-1))\cdot y} |
-4,552 | -\frac{5}{x + 2} - \frac{1}{x + (-1)} = \frac{3 - 6*x}{x^2 + x + 2*\left(-1\right)} |
-19,342 | \frac48\tfrac13 = \tfrac{1}{\dfrac143 \cdot 8} |
1,050 | 21 \cdot z/20 = \frac{z}{5} \cdot 4 + \dfrac{z}{4} |
23,389 | Y_2\cdot A_1 = A_1\cdot Y_2 |
23,629 | (\tfrac{u}{2} + w/2)*2 = u + w |
-20,651 | \frac{t\cdot (-18)}{3\cdot t + 30\cdot (-1)} = \tfrac33\cdot \frac{(-6)\cdot t}{t + 10\cdot (-1)} |
15,532 | b = \arcsin{h} rightarrow \sin{b} = h |
-15,685 | \frac{1}{s^{12}*x^{15}*(\frac{1}{x^5*s^4})^4} = \frac{\tfrac{1}{x^{15}}*\frac{1}{s^{12}}}{\dfrac{1}{x^{20}}*\frac{1}{s^{16}}} |
-19,488 | \phantom{\dfrac{1}{9} \times \dfrac{8}{5}} = \dfrac{1 \times 8}{9 \times 5} = \dfrac{8}{45} |
24,708 | 1 - \sin^2{z} = \cos^2{z} = (1 + \cos{2*z})/2 |
-12,428 | \frac{1}{2}*116 = 58 |
-12,117 | 1/6 = \frac{x}{6*\pi}*6*\pi = x |
-10,267 | -\tfrac{45 \cdot (-1) + 45 \cdot y}{y \cdot 15 + 60 \cdot (-1)} = 15/15 \cdot \left(-\frac{1}{4 \cdot (-1) + y} \cdot (3 \cdot y + 3 \cdot (-1))\right) |
-154 | \frac{10!}{(3(-1) + 10)!} = 10\cdot 9\cdot 8 |
29,559 | -1/24 = \dfrac{1}{2} + 2/2 + \dots |
23,145 | 1! + 2! + \dots + n! + (n + 1)! \leq 2n! + (n + 1)! = \left(n + 3\right) n! \leq 2(n + 1) n! |
11,212 | \binom{-1}{q} = (\left(-1\right)*\left(-2*\dotsm*(-1 - q + 1)\right))/q! = (-1)^q |
-20,622 | 5/5 \cdot \frac{1}{i \cdot 6} \cdot (-8 \cdot i + 4 \cdot (-1)) = \dfrac{1}{30 \cdot i} \cdot (20 \cdot (-1) - i \cdot 40) |
2,647 | x*x^N = x^{1 + N} |
18,294 | -z^2 \cdot 2 + (z^2 + 1) \cdot (z^2 + 1) = z^4 + 1 |
1,039 | (1/4)^{1/2} = \dfrac{1}{2} |
-19,575 | 7\cdot \frac{1}{3}/9 = 7/(9\cdot 3) = 7/27 |
-20,825 | \frac{1}{\left(-1\right) + n} \cdot ((-4) \cdot n) \cdot 7/7 = \frac{1}{7 \cdot n + 7 \cdot (-1)} \cdot (n \cdot (-28)) |
-11,646 | 22\cdot i - 15 + 8 = -7 + 22\cdot i |
6,332 | n = n + (-1) + 1 = n + 2\cdot (-1) + 2 = \dotsm = (n + 1)/2 + (n + (-1))/2 |
-16,690 | -5\cdot y = -5\cdot y\cdot \left(-5\cdot y\right) + -5\cdot y\cdot \left(-7\right) = 25\cdot y^2 + 35\cdot y = 25\cdot y^2 + 35\cdot y |
-29,144 | 18 = 4\cdot 4 + 1\cdot 2 |
-19,683 | 20/9 = \frac{4}{9} \cdot 5 |
8,496 | 4 \lt z \cdot z \implies 0 \lt (2 + z) \cdot (z + 2 \cdot \left(-1\right)) |
51,751 | 1 = i^0 |
29,559 | 1/2 + \frac{2}{2} + ... = -1/24 |
-12,841 | 3/4 = 18/24 |
28,679 | 2\cdot 5\cdot \dfrac{5!}{2!} = 600 |
-20,286 | \frac13 \cdot 3 \cdot (-10/7) = -30/21 |
7,844 | -\frac{1}{1 + x} + 1/x = \dfrac{1}{x + x \times x} |
-3,338 | 6\times \sqrt{2} = \sqrt{2}\times \left(4 + 3 + (-1)\right) |
-1,423 | -\frac{8}{3}\cdot 3/5 = ((-8)\cdot \frac13)/(5\cdot 1/3) |
-2,886 | 7*\sqrt{2} = (5 + 4 + 2*\left(-1\right))*\sqrt{2} |
808 | \cos(x) = (e^{ix} + e^{-ix})/2 = \overline{\cos(x)} = \frac{1}{2}\left(e^{-ix} + e^{ix}\right) |
20,064 | (a - b)/(b*a) = -1/a + 1/b |
29,492 | 4! \binom{5}{4}*4! = 5!*4! |
-20,665 | \frac33 \frac16(y + 9(-1)) = \frac{1}{18}(y*3 + 27 \left(-1\right)) |
-10,491 | \dfrac{20}{z \cdot z\cdot 16} = 2/2\cdot \dfrac{10}{8\cdot z^2} |
1,688 | (x + (-1))*\left(x + 2*\left(-1\right)\right)*(x + 3*\left(-1\right)) = x^3 - 6*x^2 + x*11 + 6*(-1) |
1,332 | \tfrac{\tfrac{F'}{F}\cdot \frac{K}{F}}{K \cap F'/F} = K\cdot F'/F |
22,322 | 4\cdot3/2=6 |
13,205 | x + 4x = 1x + 4x = (1+4)x = 5x |
5,313 | 1/2 + \tfrac{1}{3} + 1/3 = 7/6 |
-20,083 | \frac{1}{4(-1) + z*4}\left(-14 z + 10 (-1)\right) = \frac122 \frac{5\left(-1\right) - 7z}{2(-1) + 2z} |
26,714 | |c| \cdot c/|c| = c |
29,605 | (2^d)^3 = 8^d |
17,826 | \sin(x + g + d) = \sin(d + g + x) |
-15,507 | \frac{1}{k^{20} \times (\dfrac{q}{k^4})^2} = \frac{1}{k^{20} \times \dfrac{1}{k^8} \times q^2} |
9,579 | 5/27 = 6/27\cdot 5/6 |
-5,785 | \frac{2}{20 + z \cdot 4} = \frac{1}{(5 + z) \cdot 4} \cdot 2 |
6,255 | \sqrt{7}\cdot 2 = \sqrt{7} + \sqrt{2} + \sqrt{7} - \sqrt{2} |
26,707 | 128 = (1 + 1) \cdot (1 + 1) \cdot (1 + 1) \cdot (3 + 1) \cdot (1 + 3) |
-9,354 | 2*3*11 + x*11 = 11 x + 66 |
23,744 | g\cdot p = g\cdot p\cdot g = p\cdot g |
14,441 | y^{b + c} = y^b*y^c |
-1,560 | \frac{9}{4} = \frac14 \cdot 9 |
-18,264 | \tfrac{1}{(k + 6)\times k}\times (k + 6)\times (9\times \left(-1\right) + k) = \frac{54\times \left(-1\right) + k^2 - 3\times k}{k\times 6 + k^2} |
15,028 | ((x - b)^2 + (-d + b)^2 + (-x + d)^2)/2 = -d*x + x^2 + b^2 + d^2 - b*x - b*d |
8,916 | 14.5 = \cos{z} + 5 \implies 9.5 = \cos{z} |
-13,377 | 2 + \frac{1}{8}72 = 2 + 9 = 2 + 9 = 11 |
274 | \binom{1/2}{m} = \frac{1}{2 \cdot (\frac12 + \left(-1\right)) \cdot (\frac{1}{2} + 2 \cdot (-1)) \cdot \dots \cdot (1/2 - m + 1) \cdot m!} |
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Dataset Card for "math_formulas"
Mathematical dataset containing formulas based on the AMPS Khan dataset and the ARQMath dataset V1.3. Based on the retrieved LaTeX formulas, more equivalent versions have been generated by applying randomized LaTeX printing with this SymPy fork. The formulas are intended to be well applicable for MLM. For instance, a masking for a formula like (a+b)^2 = a^2 + 2ab + b^2 makes sense (e.g., (a+[MASK])^2 = a^2 + [MASK]ab + b[MASK]2 -> masked tokens are deducable by the context), in contrast, formulas such as f(x) = 3x+1 are not (e.g., [MASK](x) = 3x[MASK]1 -> [MASK] tokens are ambigious).
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