kreasof-ai/SPL-1M-math-formula · Datasets at Hugging Face

text stringlengths 35 651	corrupted_text stringlengths 36 616	input_ids sequencelengths 2.05k 2.05k	attention_mask sequencelengths 2.05k 2.05k	labels sequencelengths 2.05k 2.05k
[CLS]Second Binomial Formula: Considering real numbers $h$ and $g$, we can state that $(h - g) \cdot (h - g)$ is equal to $g^2 + h^2 - 2\cdot h\cdot g$.[SEP]	[CLS]Second bbfo Formula: adding real numbersgh$ and $ ($, we can state that $(rh -tg), (\ two g(h - g)$ is equal to $ (},\2 + h^2 - Ge_{\cdot h\cd g$.[SEP]	[ 50281, 10951, 48557, 4786, 26658, 27, 6240, 1524, 3904, 72, 73, 5, 285, 370, 313, 1366, 359, 476, 1375, 326, 3019, 9492, 428, 32304, 582, 5081, 767, 305, 9, 73, 428, 305, 1009, 310, 4503, 281, 370, 313, 5548, 19, 559, 288, 63, 19, 428, 3096, 1126, 3830, 288, 61, 2428, 305, 1352, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[CLS]Gamma Function: For all complex numbers $F$ with $F\not\in\{0, -1, -2, \dots\}$, we have $\int\limits_0^\infty x\,\mathrm{d}x = \Gamma(F)$[SEP]	[CLS]Gamma Function: For aw complex ty $wf$ot $ fr\not\in\{0, -018, GB2, \ discrete\}$, we he $\int\limits]\}}}\ \\|xt x\,\mathrm{ d}xy = ...Gamma}{\FS)$[SEP]	[ 50281, 4220, 12917, 27, 1198, 3768, 2570, 963, 370, 48808, 5, 302, 370, 1315, 61, 1439, 61, 249, 6921, 17, 13, 428, 12058, 13, 20514, 19, 13, 393, 13358, 22805, 359, 344, 669, 565, 61, 10423, 9955, 12191, 11045, 633, 1269, 12577, 2690, 92, 277, 94, 5246, 426, 3346, 4220, 2704, 7864, 1009, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[CLS]Integral of a Function: For any integrable function $f(\chi)$ defined on a real interval $(c, b)$, the integral of $f(\chi)$ over the interval $(c,b)$, denoted as $\int\limits_c^b f(\chi)\,\mathrm{d}\chi$, is equal to $-G(c) + G(b)$, where $G(\chi)$ is the antiderivative of $f(\chi)$.[SEP]	[CLS]Integral of a Function: For any integrable function $f(\chi)$ defined on a real interval $(c, b)$, the integral of $f(\chi)$ over the interval $(c,b)$, denoted as $\int\limits_c^b f(\chi)\,\mathrm{d}\chi$, is equal to $-G(c) + G(b)$, where $G(\chi)$ is the antiderivative of $f(\chi)$.[SEP]	[ 50281, 29653, 267, 273, 247, 12917, 27, 1198, 667, 40369, 1159, 370, 71, 1035, 4635, 1009, 2931, 327, 247, 1524, 7726, 3019, 68, 13, 270, 4244, 253, 9909, 273, 370, 71, 1035, 4635, 1009, 689, 253, 7726, 3019, 68, 13, 67, 4244, 17007, 347, 669, 565, 61, 10423, 64, 68, 63, 67, 269, 1035, 4635, 43790, 2690, 92, 69, 889, 4635, 1366, 310, 4503, 281, 9929, 40, 9, 68, 10, 559, 443, 9, 67, 4244, 835, 370, 40, 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[CLS]Derivative Quotient Rule: The derivative of the product of two functions, $u$ and $v$, with respect to $y$, denoted as $\frac{d}{dy} (h(y)/v\left(y\right))$, is given by the expression $\frac{1}{v^2(y)}\cdot \left(-h(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} v(y) + v(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} h(y)\right)$.[SEP]	[CLS]Derivative Quotient Rule: The derivative of the product of two functions, $u$ and $v$, with respect to $y$, denoted as $\frac{d}{dy} (h(y)/v\left(y\right))$, is given by the expression $\frac{1}{v^2(y)}\cdot \left(-h(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} v(y) + v(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} h(y)\right)$.[SEP]	[ 50281, 19932, 400, 800, 3277, 302, 850, 7804, 27, 380, 4309, 273, 253, 1885, 273, 767, 3470, 13, 370, 86, 5, 285, 370, 87, 1366, 342, 1675, 281, 370, 90, 1366, 17007, 347, 669, 1124, 92, 69, 1217, 6421, 94, 313, 73, 9, 90, 1933, 87, 61, 1274, 9, 90, 61, 918, 44418, 310, 1677, 407, 253, 2048, 669, 1124, 92, 18, 1217, 87, 63, 19, 9, 90, 7398, 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[CLS]difference of squares: For all real numbers $a,f$ we have that $a^2 - f^2 = (a + f) \cdot (-f + a)$.[SEP]	[CLS]difference of sa: For all realinfty ${-,-fe$ dl ha GD $(-a^2 ( f^2 ((- gla + f) \zt (-f ( a)$.[SEP]	[ 50281, 40651, 273, 618, 27, 1198, 512, 1524, 3259, 370, 10241, 6567, 453, 5, 45439, 419, 30019, 49960, 66, 63, 19, 313, 269, 63, 19, 12374, 1289, 66, 559, 269, 10, 393, 15701, 3383, 71, 313, 247, 3822, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[CLS]Eulers Complex Exponential Identity: If we have a complex number $y$, the expression $e^{\mathrm{i} \cdot y}$ can be represented as $\sin{y} i + \cos{y}$.[SEP]	[CLS]Eulers Complex Exponential Identity: If we have a complex number $y$, the expression "!"e^{\mathrm{i} \cdot y}$ can be represented as $\sin{y} i + gycos{y}$.[SEP]	[ 50281, 38, 335, 398, 24154, 17702, 45426, 35981, 27, 1310, 359, 452, 247, 2570, 1180, 370, 90, 1366, 253, 2048, 31487, 70, 1743, 2690, 92, 74, 94, 393, 3830, 340, 724, 476, 320, 6607, 347, 669, 7432, 92, 90, 94, 891, 559, 19859, 4752, 92, 90, 3363, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[CLS]Normal Distribution: f(z) = e^{\frac{1}{\sigma^2*2}((-1) (z - \mu)^2)}/(\sigma \sqrt{2\pi})[SEP]	[CLS]Normal Distribution: f( Numbers) ((- e^{\frac{1}{\sigma^2)^{\2}((-1}\ (z - ghmu)^2)})]}{\sigma \\|sqrt{2\pi})[SEP]	[ 50281, 20167, 30313, 27, 269, 9, 35306, 10, 12374, 299, 1743, 1124, 92, 18, 2704, 2592, 63, 19, 15207, 19, 1603, 1490, 18, 889, 313, 91, 428, 32798, 1906, 4800, 19, 3117, 9102, 2704, 2592, 11045, 2609, 92, 19, 61, 2059, 2311, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[CLS]Binomial Theorem: Considering real numbers $c$ and $f$, and any natural number $l$, the expression $(c + f)^l$ can be expanded using the summation notation $\sum_{x=0}^l \binom{l}{x} \cdot f^{l - x} \cdot c^x$.[SEP]	[CLS]Binomial Theorem: Considering real> $ cs$ and $nf$,dn any obtain number $l$), the expression $c + f)^l$ can be denote using the \ notation $(\sum_{x=\0}^Long \binom{l})}x} \cdot f^{l - x} =cdot c^x757[SEP]	[ 50281, 38601, 28261, 6899, 27, 24258, 1524, 31, 370, 29180, 5, 285, 370, 35478, 1366, 17915, 667, 4044, 1180, 370, 77, 17673, 253, 2048, 370, 68, 559, 269, 4800, 77, 5, 476, 320, 9173, 970, 253, 393, 14951, 9722, 2204, 578, 89, 2029, 17, 2306, 12486, 393, 23275, 92, 77, 22613, 89, 94, 393, 3830, 269, 768, 77, 428, 1269, 94, 426, 3830, 260, 63, 89, 31203, 50282, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 50283, 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[CLS]Taylor Series: g(z) = \sum_{n=0}^\infty g^{(n)}(b)/n! (-b + z)^n[SEP]	[CLS]Taylor Series: g(z) = \sum_{n=0}^\infty g^{( not)}(b)/n! (-b + z)^n[SEP]	[50281,37979,12234,27,305,9,91,10,426,393,2204,578,79,30,17,9616,3259,305,4932,417,14411,67,1933,79,(...TRUNCATED)	[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED)	[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED)

[CLS]Second Binomial Formula: Considering real numbers $h$ and $g$, we can state that $(h - g) \cdot (h - g)$ is equal to $g^2 + h^2 - 2\cdot h\cdot g$.[SEP]

[CLS]Second bbfo Formula: adding real numbersgh$ and $ ($, we can state that $(rh -tg), (\ two g(h - g)$ is equal to $ (},\2 + h^2 - Ge_{\cdot h\cd g$.[SEP]

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[CLS]Gamma Function: For all complex numbers $F$ with $F\not\in\{0, -1, -2, \dots\}$, we have $\int\limits_0^\infty x\,\mathrm{d}x = \Gamma(F)$[SEP]

[CLS]Gamma Function: For aw complex ty $wf$ot $ fr\not\in\{0, -018, GB2, \ discrete\}$, we he $\int\limits]\}}}\ \|xt x\,\mathrm{ d}xy = ...Gamma}{\FS)$[SEP]

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[CLS]Integral of a Function: For any integrable function $f(\chi)$ defined on a real interval $(c, b)$, the integral of $f(\chi)$ over the interval $(c,b)$, denoted as $\int\limits_c^b f(\chi)\,\mathrm{d}\chi$, is equal to $-G(c) + G(b)$, where $G(\chi)$ is the antiderivative of $f(\chi)$.[SEP]

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[CLS]Derivative Quotient Rule: The derivative of the product of two functions, $u$ and $v$, with respect to $y$, denoted as $\frac{d}{dy} (h(y)/v\left(y\right))$, is given by the expression $\frac{1}{v^2(y)}\cdot \left(-h(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} v(y) + v(y)\cdot \frac{\mathrm{d}}{\mathrm{d}y} h(y)\right)$.[SEP]

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[CLS]Law of Cosines: -b\cdot \cos(C)\cdot a\cdot 2 + a \cdot a + b \cdot b = c^2[SEP]

[CLS]Law of Cosines: -xb!(cdot \cos(C\,\cdot a\Dist ng + a \ cd a + b \cdot BE { c^2[SEP]

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[CLS]difference of squares: For all real numbers $a,f$ we have that $a^2 - f^2 = (a + f) \cdot (-f + a)$.[SEP]

[CLS]difference of sa: For all realinfty ${-,-fe$ dl ha GD $(-a^2 ( f^2 ((- gla + f) \zt (-f ( a)$.[SEP]

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[CLS]Eulers Complex Exponential Identity: If we have a complex number $y$, the expression $e^{\mathrm{i} \cdot y}$ can be represented as $\sin{y} i + \cos{y}$.[SEP]

[CLS]Eulers Complex Exponential Identity: If we have a complex number $y$, the expression "!"e^{\mathrm{i} \cdot y}$ can be represented as $\sin{y} i + gycos{y}$.[SEP]

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[CLS]Normal Distribution: f(z) = e^{\frac{1}{\sigma^2*2}((-1) (z - \mu)^2)}/(\sigma \sqrt{2\pi})[SEP]

[CLS]Normal Distribution: f( Numbers) ((- e^{\frac{1}{\sigma^2)^{\2}((-1}\ (z - ghmu)^2)})]}{\sigma \|sqrt{2\pi})[SEP]

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[CLS]Binomial Theorem: Considering real numbers $c$ and $f$, and any natural number $l$, the expression $(c + f)^l$ can be expanded using the summation notation $\sum_{x=0}^l \binom{l}{x} \cdot f^{l - x} \cdot c^x$.[SEP]

[CLS]Binomial Theorem: Considering real> $ cs$ and $nf$,dn any obtain number $l$), the expression $c + f)^l$ can be denote using the \ notation $(\sum_{x=\0}^Long \binom{l})}x} \cdot f^{l - x} =cdot c^x757[SEP]

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[CLS]Taylor Series: g(z) = \sum_{n=0}^\infty g^{(n)}(b)/n! (-b + z)^n[SEP]

[CLS]Taylor Series: g(z) = \sum_{n=0}^\infty g^{( not)}(b)/n! (-b + z)^n[SEP]

[50281,37979,12234,27,305,9,91,10,426,393,2204,578,79,30,17,9616,3259,305,4932,417,14411,67,1933,79,(...TRUNCATED)

[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED)

[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED)