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int64 -30,985
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28,585 | \mathbb{E}\left((A - \mu)^2\right) = (\mathbb{E}\left(A\right) - \mu)^2 + \mathbb{Var}\left(A\right) |
22,216 | -0.618 = \frac12 (1 - \sqrt{5}) \approx -0.618 |
9,789 | (1 + x)\cdot (x^2 + 1)\cdot (1 + x^4)\cdot \dotsm\cdot (x^{2^n} + 1) = \frac{1}{-x + 1}\cdot (-x^{2^{n + 1}} + 1) |
21,927 | l^2 = b_{l + 1} - b_l = \frac{1}{l + 1 - l} \cdot (b_{l + 1} - b_l) |
16,267 | 2 (-1) + x = -(2 - x) |
17,192 | x\beta n = nx \beta |
-20,436 | \frac{1}{x + 10 \left(-1\right)} \left(x + 10 (-1)\right)/9 = \frac{1}{90 \left(-1\right) + x*9}(x + 10 \left(-1\right)) |
29,374 | (1 + n)^2 = 1 + n^2 + 2*n |
32,367 | 0 \cdot z = (0 + 0) \cdot z = 0 \cdot z + 0 \cdot z |
32,027 | y r x = r y x |
16,943 | 2/10\cdot \frac{3}{10} = 6/100 |
-5,933 | \frac{5}{j \cdot 5 + 45 \cdot \left(-1\right)} = \frac{5}{5 \cdot (j + 9 \cdot (-1))} |
12,677 | y^2 + 8\cdot y + 14 - q = 0 rightarrow y = (-8 \pm (10 + 4\cdot q)^{1/2})/2 |
11,040 | -h^t + f^t = \left(-h + f\right) \cdot \left(f^{t + (-1)} + f^{2 \cdot (-1) + t} \cdot h + h^2 \cdot f^{3 \cdot (-1) + t} + \dots + f \cdot h^{t + 2 \cdot \left(-1\right)} + h^{t + \left(-1\right)}\right) |
37,633 | 12 = \frac{18}{3} \cdot (5 + 4 \cdot \left(-1\right) + 1) |
-26,575 | \left(8 + x\right) (-x + 8) = 8 \cdot 8 - x^2 |
-2,690 | 7^{1/2} + 25^{1/2} \cdot 7^{1/2} = 7^{1/2} \cdot 5 + 7^{1/2} |
32,987 | \tfrac{45}{216} = \frac{1}{24}\cdot 5 |
20,697 | \operatorname{acos}(-q) = \operatorname{acos}(\cos(\pi - \operatorname{acos}(q)))\Longrightarrow -\operatorname{acos}(q) + \pi = \operatorname{acos}(-q) |
-10,262 | -\frac{20}{15\cdot z + 5} = -\frac{1}{1 + z\cdot 3}\cdot 4\cdot 5/5 |
3,446 | \dfrac52 \cdot \dfrac{1}{9} = 5/18 |
25,714 | 0 - z^3\cdot 2 + 9\cdot z^2 - z\cdot 12 + 5 = -2\cdot z^2 \cdot z + 9\cdot z^2 - z\cdot 12 + 5 |
28,282 | \frac{\text{d}z}{\text{d}y} = \frac{1}{2 \cdot z \cdot y - y^2} \cdot (z^2 - 2 \cdot z \cdot y) = \frac{1}{2 \cdot y/z - (\frac{y}{z})^2} \cdot \left(1 - 2 \cdot y/z\right) |
30,843 | 48 = 2! \cdot 2! \cdot 2! \cdot 3! |
36,090 | \dfrac{1}{8} = (\dfrac12)^3 |
31,164 | 22 = 8 \cdot (-1) + 30 |
-1,773 | \frac{1}{3}\pi = -\pi \frac135 + 2\pi |
7,249 | 17^5\cdot 13^4\cdot 3 \cdot 3\cdot 7 \cdot 7 = 17^5 (3\cdot 7\cdot 13^2)^2 |
-26,542 | (10 - 3 \cdot x) \cdot \left(10 + 3 \cdot x\right) = -9 \cdot x^2 + 100 |
-15,771 | -\frac{1}{10}\cdot 44 = -8\cdot 8/10 + 10\cdot \frac{1}{10}\cdot 2 |
-30,160 | \frac{\mathrm{d}}{\mathrm{d}z} z^9 = 9\cdot z^{9 + (-1)} = 9\cdot z^8 |
1,409 | \frac{1}{\sqrt{n} + \sqrt{n + 1}} = -\sqrt{n} + \sqrt{1 + n} |
-4,733 | \frac{1}{x^2 + 3*x + 2}*(5*(-1) - x*2) = -\frac{1}{x + 1}*3 + \frac{1}{x + 2} |
4,806 | t\cdot y^i = t\cdot y^i |
32,515 | 9 = -8\cdot 2 + 25 |
11,192 | 6/11 \cdot 7/12 = \frac{1}{132} \cdot 42 |
-29,571 | (7 + 5 \cdot z^2 + z)/z = 5 \cdot z^2/z + z/z + 7/z |
-4,182 | \frac{48\cdot z^5}{24\cdot z} = \frac{48}{24}\cdot z^5/z |
12,880 | 1 + 6\xi + 1 = \xi*6 + 2 |
24,108 | (11 + 3) \cdot (11 + 7 \cdot (-1)) = 56 |
-5,182 | 4.92\cdot 10 = \frac{49.2}{10}\cdot 1 = 4.92\cdot 10^0 |
15,482 | k + k + k = 3\cdot k |
6,015 | y + y^2 = -\frac{1}{4} + (y + \dfrac{1}{2})^2 |
-713 | e^{5 \cdot i \cdot \pi/3 \cdot 15} = (e^{\frac53 \cdot i \cdot \pi})^{15} |
1,192 | \sin(x) \sin(\alpha) + \cos(\alpha) \cos(x) = \cos(-x + \alpha) |
3,978 | \tfrac{2}{4 + 7 + 2} = \dfrac{2}{13} |
21,617 | -8 + x\cdot 4 + 2\cdot h = (2\cdot h^2 - 8\cdot h + 4\cdot x\cdot h)/h |
18,674 | E^c - D = D^c \cap E^c |
12,180 | -b^2\cdot 3 + 66\cdot b + 315\cdot (-1) = 0 \implies b^2 - 22\cdot b + 105 = \left(b + 15\cdot (-1)\right)\cdot (b + 7\cdot (-1)) = 0 |
-22,993 | 21/27 = \frac{7 \cdot 3}{3 \cdot 9} |
3,006 | -u^3 + x \cdot x \cdot x = \left(-u + x\right) (u \cdot u + x^2 + xu) |
22,683 | -b^2 + a^2 - ab + ba = \left(a + b\right) (-b + a) |
-6,147 | \dfrac{1}{2*i + 18*(-1)}*5 = \frac{5}{(i + 9*(-1))*2} |
-9,926 | 0.01 (-28) = -28/100 = -\frac{1}{25}7 |
-13,128 | -22.5/\left(-0.5\right) = 45 |
3,415 | 64 S^4 r r + 4 = (r^2 S^4\cdot 16 + 1)\cdot 4 |
17,337 | b = db = bd |
20,805 | π/6 = \arccos(\sqrt{3}/2) |
33,820 | \frac{3 - x}{(x + 3*(-1))^2} = -\frac{1}{(x + 3*(-1))^2}*(x + 3*\left(-1\right)) = -\frac{1}{x + 3*(-1)} |
34,616 | 1 + \cos{x} = 1 + \cos{2 x/2} = 2 \cos^2{x/2} |
28,720 | 2^{1/2} \approx 1 + \frac12 - 1/8 + \dfrac{1}{16} = 23/16 \approx 1.4375 |
21,342 | \sin(y\cdot 2) = \cos(y) \sin\left(y\right)\cdot 2 |
26,899 | \cos\left(x \cdot 2\right) = -2 \cdot \sin^2(x) + 1 \implies \sin(x) = \sqrt{\frac12 \cdot (1 - \cos\left(2 \cdot x\right))} |
-29,374 | (x - g) \cdot (x + g) = -g \cdot g + x^2 |
11,972 | 545140134^2*12^2 = (12^3 + 640320^3)*163 |
-20,336 | 2/2 \cdot \tfrac{1}{-x \cdot 7 + 5 \cdot (-1)} \cdot 2 = \frac{1}{10 \cdot \left(-1\right) - 14 \cdot x} \cdot 4 |
1,922 | 7 = 6 \cdot (-1) + 18 + 5 \cdot \left(-1\right) |
34,072 | 512^2 = 64^2 64 |
14,850 | (1 + 2\cdot (-1))^2 = (-1) \cdot (-1) = 1 \leq 2\cdot \left(1 + (-2)^2\right) = 2\cdot \left(1 + 4\right) = 10 |
4,912 | -x_2 + x_1 = x_0 - x_1 \implies x_2\cdot ... = -x_1\cdot 2 + x_0 |
33,256 | 73 = 145 + 72 \times (-1) |
1,632 | \sin(\alpha) \cdot \cos(J) - \cos(\alpha) \cdot \sin\left(J\right) = \sin\left(-J + \alpha\right) |
717 | 1/z=\bar{z}=z^n\implies z^{n+1}=1 |
-9,413 | 2*2*7*x = 28*x |
3,974 | x := x_1 x_2 \cdots x_n := x_1 x_2 \cdots x_n |
47,991 | 2 + 8 (-1) = -6 = 3 (-2) |
14,747 | \dfrac{117.7}{1 + 0.07} \frac{1}{1 + 0.1} = 100 |
1,961 | \frac{1}{-y + 1} = \frac{1 + 0\cdot (-1)}{-y + 1} |
9,038 | \frac{1}{x + 2\left(-1\right)}\left(4 + 2x^3 - 10 x\right) = 2(-1) + 2x^2 + 4x |
-20,472 | \frac{q\cdot 9 + 9(-1)}{q\cdot 10}\cdot 6/6 = \frac{1}{60 q}\left(54 q + 54 (-1)\right) |
9,484 | 5 + \left(1 + n\right)^2 = n \cdot n + 2 \cdot n + 6 |
48,701 | x^2 + x + 1 = (x^2 + x + \tfrac 1 4) + \frac 3 4 = \left(x + \frac 1 2 \right)^2 + \frac 3 4 |
41,398 | \cos{\frac{\pi}{4}} = \sin{\frac{1}{4}\cdot \pi} = 1 |
-12,190 | 1/2 = \frac{q}{16 \pi} \cdot 16 \pi = q |
2,804 | t^3-3t^2+t+1=(t-1)(t^2-2t-1) |
-7,145 | 3/13\cdot \frac{5}{12} = 5/52 |
14,115 | \frac{100}{6} = \frac{1}{x}\cdot 40 rightarrow 2.4 = x |
-11,621 | i\cdot 4 + 0 + 20 (-1) = i\cdot 4 - 20 |
6,649 | 16 = y^2 \cdot 9 + z^2 - 6yz \Rightarrow (z - 3y) \cdot (z - 3y) = 16 |
15,999 | z^4 - z^2 + 1 = (1 + z * z - \sqrt{3}*z)*(z * z + z*\sqrt{3} + 1) |
29,059 | cb = b = bc |
-29,802 | d/dx (5 + x^2*2 - x*6) = 6\left(-1\right) + x*4 |
-9,675 | \dfrac{2}{2} = 1 |
16,160 | c = a/b \Rightarrow a = b \times c = 0 \times c = 0 |
15,823 | \sin{\frac{1}{5}\cdot \pi/2} = \sin{\pi/10} |
12,958 | d + x + c = x + c + d = c + d + x |
-27,702 | \frac{\text{d}}{\text{d}y} \cos(y) = -\sin\left(y\right) |
15,520 | \frac16(1 + 2 + 3 + 4 + 5 + 6) = \frac{6}{2*6}7 = \tfrac{7}{2} |
10,616 | x - b = (\sqrt{x} - \sqrt{b})\cdot (\sqrt{x} + \sqrt{b}) \geq (\sqrt{x} - \sqrt{b}) \cdot (\sqrt{x} - \sqrt{b}) |
-22,102 | \dfrac{1}{20}30 = \dfrac123 |
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