7.8

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.8.E1	${\displaystyle{\displaystyle\mathsf{M}\left(x\right)=\frac{\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}{e^{-x^{2}}}}}$ \MillsM@{x} = \frac{\int_{x}^{\infty}e^{-t^{2}}\diff{t}}{e^{-x^{2}}}		Error	Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] == Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}, GenerateConditions->None],Exp[- (x)^(2)]]	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Plus[-0.2849976548947546, Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]} Result: Plus[-0.545641360765047, Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E1	${\displaystyle{\displaystyle\frac{\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}{e^{-x^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}}$ \frac{\int_{x}^{\infty}e^{-t^{2}}\diff{t}}{e^{-x^{2}}} = e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\diff{t}		(int(exp(- (t)^(2)), t = x..infinity))/(exp(- (x)^(2))) = exp((x)^(2))*int(exp(- (t)^(2)), t = x..infinity)	Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}, GenerateConditions->None],Exp[- (x)^(2)]] == Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)], {t, x, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 3]
7.8.E2	${\displaystyle{\displaystyle\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right)}}$ \frac{1}{x+\sqrt{x^{2}+2}} < \MillsM@{x}	${\displaystyle{\displaystyle x\geq 0}}$	Error	Divide[1,x +Sqrt[(x)^(2)+ 2]] < Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}]	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Less[0.28077640640441515, Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]} Result: Less[0.5, Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E2	${\displaystyle{\displaystyle\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}+(4/\pi)}}}}$ \MillsM@{x} \leq \frac{1}{x+\sqrt{x^{2}+(4/\pi)}}	${\displaystyle{\displaystyle x\geq 0}}$	Error	Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] <= Divide[1,x +Sqrt[(x)^(2)+(4/Pi)]]	Missing Macro Error	Failure	-	Failed [3 / 3] Result: LessEqual[Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.2961182351849971], {Rule[x, 1.5]} Result: LessEqual[Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.5766361194388748], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E3	${\displaystyle{\displaystyle\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}\left(x\right)}}$ \frac{\sqrt{\pi}}{2\sqrt{\pi}x+2} \leq \MillsM@{x}	${\displaystyle{\displaystyle x\geq 0}}$	Error	Divide[Sqrt[Pi],2*Sqrt[Pi]*x + 2] <= Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}]	Missing Macro Error	Failure	-	Failed [3 / 3] Result: LessEqual[0.24222581297045487, Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]} Result: LessEqual[0.46984109573138116, Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E3	${\displaystyle{\displaystyle\mathsf{M}\left(x\right)<\frac{1}{x+1}}}$ \MillsM@{x} < \frac{1}{x+1}	${\displaystyle{\displaystyle x\geq 0}}$	Error	Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] < Divide[1,x + 1]	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Less[Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.4], {Rule[x, 1.5]} Result: Less[Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.6666666666666666], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E4	${\displaystyle{\displaystyle\mathsf{M}\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}}}}$ \MillsM@{x} < \frac{2}{3x+\sqrt{x^{2}+4}}	${\displaystyle{\displaystyle x>-\tfrac{1}{2}\sqrt{2}}}$	Error	Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] < Divide[2,3*x +Sqrt[(x)^(2)+ 4]]	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Less[Times[9.487735836358526, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.2857142857142857], {Rule[x, 1.5]} Result: Less[Times[1.2840254166877414, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.5615528128088303], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E5	${\displaystyle{\displaystyle\frac{x^{2}}{2x^{2}+1}\leq\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}}}$ \frac{x^{2}}{2x^{2}+1} \leq \frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}	${\displaystyle{\displaystyle x\geq 0}}$	((x)^(2))/(2*(x)^(2)+ 1) <= ((x)^(2)*(2*(x)^(2)+ 5))/(4*(x)^(4)+ 12*(x)^(2)+ 3)	Divide[(x)^(2),2*(x)^(2)+ 1] <= Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
7.8.E5	${\displaystyle{\displaystyle\frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}\leq x\mathsf{M}\left(x\right)}}$ \frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3} \leq x\MillsM@{x}	${\displaystyle{\displaystyle x\geq 0}}$	Error	Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3] <= x*Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Failed [3 / 3] Result: LessEqual[0.4253731343283582, Times[14.23160375453779, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]]], {Rule[x, 1.5]} Result: LessEqual[0.22, Times[0.6420127083438707, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]]], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E5	${\displaystyle{\displaystyle x\mathsf{M}\left(x\right)<\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}}}$ x\MillsM@{x} < \frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}	${\displaystyle{\displaystyle x\geq 0}}$	Error	x*Exp[Power[x,2]] Int[Exp[-t^2], {t, x, Infinity}] < Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Failed [3 / 3] Result: Less[Times[14.23160375453779, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 1.5, DirectedInfinity[1]}]], 0.42834890965732086], {Rule[x, 1.5]} Result: Less[Times[0.6420127083438707, Int[Power[2.718281828459045, Times[-1.0, Power[t, 2]]] Test Values: {t, 0.5, DirectedInfinity[1]}]], 0.31481481481481477], {Rule[x, 0.5]} ... skip entries to safe data
7.8.E5	${\displaystyle{\displaystyle\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^{2}+1}{2x^{2}+3}}}$ \frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15} < \frac{x^{2}+1}{2x^{2}+3}	${\displaystyle{\displaystyle x\geq 0}}$	(2*(x)^(4)+ 9*(x)^(2)+ 4)/(4*(x)^(4)+ 20*(x)^(2)+ 15) < ((x)^(2)+ 1)/(2*(x)^(2)+ 3)	Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15] < Divide[(x)^(2)+ 1,2*(x)^(2)+ 3]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
7.8.E6	${\displaystyle{\displaystyle\int_{0}^{x}e^{at^{2}}\mathrm{d}t<\frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2\right)}}$ \int_{0}^{x}e^{at^{2}}\diff{t} < \frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2\right)	${\displaystyle{\displaystyle a>0,x>0}}$	int(exp(a*(t)^(2)), t = 0..x) < (1)/(3*a*x)*(2*exp(a*(x)^(2))+ a*(x)^(2)- 2)	Integrate[Exp[a*(t)^(2)], {t, 0, x}, GenerateConditions->None] < Divide[1,3*a*x]*(2*Exp[a*(x)^(2)]+ a*(x)^(2)- 2)	Error	Failure	-	Successful [Tested: 9]
7.8.E7	${\displaystyle{\displaystyle\int_{0}^{x}e^{t^{2}}\mathrm{d}t<\frac{e^{x^{2}}-1}{x}}}$ \int_{0}^{x}e^{t^{2}}\diff{t} < \frac{e^{x^{2}}-1}{x}	${\displaystyle{\displaystyle x>0}}$	int(exp((t)^(2)), t = 0..x) < (exp((x)^(2))- 1)/(x)	Integrate[Exp[(t)^(2)], {t, 0, x}, GenerateConditions->None] < Divide[Exp[(x)^(2)]- 1,x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
7.8.E8	${\displaystyle{\displaystyle\operatorname{erf}x<\sqrt{1-{\mathrm{e}^{-4x^{2}/\pi}}}}}$ \erf@@{x} < \sqrt{1-\expe^{-4x^{2}/\cpi}}	${\displaystyle{\displaystyle x>0}}$	erf(x) < sqrt(1 - exp(- 4*(x)^(2)/Pi))	Erf[x] < Sqrt[1 - Exp[- 4*(x)^(2)/Pi]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]