4.5: Difference between revisions

Latest revision as of 11:05, 28 June 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.5.E1	${\displaystyle{\displaystyle\frac{x}{1+x}<\ln\left(1+x\right)}}$ `\frac{x}{1+x} < \ln@{1+x}`	${\displaystyle{\displaystyle x>-1,x\neq 0}}$	(x)/(1 + x) < ln(1 + x)	Divide[x,1 + x] < Log[1 + x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.5.E1	${\displaystyle{\displaystyle\ln\left(1+x\right)<x}}$ `\ln@{1+x} < x`	${\displaystyle{\displaystyle x>-1,x\neq 0}}$	ln(1 + x) < x	Log[1 + x] < x	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.5.E2	${\displaystyle{\displaystyle x<-\ln\left(1-x\right)}}$ `x < -\ln@{1-x}`	${\displaystyle{\displaystyle x<1,x\neq 0}}$	x < - ln(1 - x)	x < - Log[1 - x]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E2	${\displaystyle{\displaystyle-\ln\left(1-x\right)<\frac{x}{1-x}}}$ `-\ln@{1-x} < \frac{x}{1-x}`	${\displaystyle{\displaystyle x<1,x\neq 0}}$	- ln(1 - x) < (x)/(1 - x)	- Log[1 - x] < Divide[x,1 - x]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E3	${\displaystyle{\displaystyle\|\ln\left(1-x\right)\|<\tfrac{3}{2}x}}$ `\|\ln@{1-x}\| < \tfrac{3}{2}x`	${\displaystyle{\displaystyle 0<x,x\leq 0.5828\dots}}$	abs(ln(1 - x)) < (3)/(2)*x	Abs[Log[1 - x]] < Divide[3,2]*x	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E4	${\displaystyle{\displaystyle\ln x\leq x-1}}$ `\ln@@{x} \leq x-1`	${\displaystyle{\displaystyle x>0}}$	ln(x) <= x - 1	Log[x] <= x - 1	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.5.E5	${\displaystyle{\displaystyle\ln x\leq a(x^{1/a}-1)}}$ `\ln@@{x} \leq a(x^{1/a}-1)`	${\displaystyle{\displaystyle a>0,x>0}}$	ln(x) <= a*((x)^(1/a)- 1)	Log[x] <= a*((x)^(1/a)- 1)	Error	Failure	-	Successful [Tested: 9]
4.5.E6	${\displaystyle{\displaystyle\|\ln\left(1+z\right)\|\leq-\ln\left(1-\|z\|\right)}}$ `\|\ln@{1+z}\| \leq -\ln@{1-\|z\|}`	${\displaystyle{\displaystyle\|z\|<1}}$	abs(ln(1 + z)) <= - ln(1 -abs(z))	Abs[Log[1 + z]] <= - Log[1 -Abs[z]]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
4.5.E7	${\displaystyle{\displaystyle e^{-x/(1-x)}<1-x}}$ `e^{-x/(1-x)} < 1-x`	${\displaystyle{\displaystyle x<1}}$	exp(- x/(1 - x)) < 1 - x	Exp[- x/(1 - x)] < 1 - x	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E7	${\displaystyle{\displaystyle 1-x<e^{-x}}}$ `1-x < e^{-x}`	${\displaystyle{\displaystyle x<1}}$	1 - x < exp(- x)	1 - x < Exp[- x]	Error	Failure	-	Successful [Tested: 1]
4.5.E8	${\displaystyle{\displaystyle 1+x<e^{x}}}$ `1+x < e^{x}`	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	1 + x < exp(x)	1 + x < Exp[x]	Skipped - no semantic math	Failure	-	Successful [Tested: 3]
4.5.E9	${\displaystyle{\displaystyle e^{x}<\frac{1}{1-x}}}$ `e^{x} < \frac{1}{1-x}`	${\displaystyle{\displaystyle x<1}}$	exp(x) < (1)/(1 - x)	Exp[x] < Divide[1,1 - x]	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E10	${\displaystyle{\displaystyle\frac{x}{1+x}<1-e^{-x}}}$ `\frac{x}{1+x} < 1-e^{-x}`	${\displaystyle{\displaystyle x>-1}}$	(x)/(1 + x) < 1 - exp(- x)	Divide[x,1 + x] < 1 - Exp[- x]	Skipped - no semantic math	Failure	-	Successful [Tested: 3]
4.5.E10	${\displaystyle{\displaystyle 1-e^{-x}<x}}$ `1-e^{-x} < x`	${\displaystyle{\displaystyle x>-1}}$	1 - exp(- x) < x	1 - Exp[- x] < x	Error	Failure	-	Successful [Tested: 3]
4.5.E11	${\displaystyle{\displaystyle x<e^{x}-1}}$ `x < e^{x}-1`	${\displaystyle{\displaystyle x<1}}$	x < exp(x)- 1	x < Exp[x]- 1	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E11	${\displaystyle{\displaystyle e^{x}-1<\frac{x}{1-x}}}$ `e^{x}-1 < \frac{x}{1-x}`	${\displaystyle{\displaystyle x<1}}$	exp(x)- 1 < (x)/(1 - x)	Exp[x]- 1 < Divide[x,1 - x]	Error	Failure	-	Successful [Tested: 1]
4.5.E12	${\displaystyle{\displaystyle e^{x/(1+x)}<1+x}}$ `e^{x/(1+x)} < 1+x`	${\displaystyle{\displaystyle x>-1}}$	exp(x/(1 + x)) < 1 + x	Exp[x/(1 + x)] < 1 + x	Skipped - no semantic math	Failure	-	Successful [Tested: 3]
4.5.E13	${\displaystyle{\displaystyle e^{xy/(x+y)}<\left(1+\frac{x}{y}\right)^{y}}}$ `e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}`	${\displaystyle{\displaystyle x>0,y>0}}$	exp(x*y/(x + y)) < (1 +(x)/(y))^(y)	Exp[x*y/(x + y)] < (1 +Divide[x,y])^(y)	Skipped - no semantic math	Failure	-	Successful [Tested: 9]
4.5.E13	${\displaystyle{\displaystyle\left(1+\frac{x}{y}\right)^{y}<e^{x}}}$ `\left(1+\frac{x}{y}\right)^{y} < e^{x}`	${\displaystyle{\displaystyle x>0,y>0}}$	(1 +(x)/(y))^(y) < exp(x)	(1 +Divide[x,y])^(y) < Exp[x]	Error	Failure	-	Successful [Tested: 9]
4.5.E14	${\displaystyle{\displaystyle e^{-x}<1-\tfrac{1}{2}x}}$ `e^{-x} < 1-\tfrac{1}{2}x`	${\displaystyle{\displaystyle 0<x,x\leq 1.5936\dots}}$	exp(- x) < 1 -(1)/(2)*x	Exp[- x] < 1 -Divide[1,2]*x	Skipped - no semantic math	Failure	-	Successful [Tested: 2]
4.5.E15	${\displaystyle{\displaystyle\tfrac{1}{4}\|z\|<\|e^{z}-1\|}}$ `\tfrac{1}{4}\|z\| < \|e^{z}-1\|`	${\displaystyle{\displaystyle 0<\|z\|,\|z\|<1}}$	(1)/(4)*abs(z) < abs(exp(z)- 1)	Divide[1,4]*Abs[z] < Abs[Exp[z]- 1]	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E15	${\displaystyle{\displaystyle\|e^{z}-1\|<\tfrac{7}{4}\|z\|}}$ `\|e^{z}-1\| < \tfrac{7}{4}\|z\|`	${\displaystyle{\displaystyle 0<\|z\|,\|z\|<1}}$	abs(exp(z)- 1) < (7)/(4)*abs(z)	Abs[Exp[z]- 1] < Divide[7,4]*Abs[z]	Error	Failure	-	Successful [Tested: 1]
4.5.E16	${\displaystyle{\displaystyle\|e^{z}-1\|\leq e^{\|z\|}-1}}$ `\|e^{z}-1\| \leq e^{\|z\|}-1`		abs(exp(z)- 1) <= exp(abs(z))- 1	Abs[Exp[z]- 1] <= Exp[Abs[z]]- 1	Skipped - no semantic math	Failure	-	Successful [Tested: 1]
4.5.E16	${\displaystyle{\displaystyle e^{\|z\|}-1\leq\|z\|e^{\|z\|}}}$ `e^{\|z\|}-1 \leq \|z\|e^{\|z\|}`		exp(abs(z))- 1 <= abs(z)*exp(abs(z))	Exp[Abs[z]]- 1 <= Abs[z]*Exp[Abs[z]]	Error	Failure	-	Successful [Tested: 1]

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+{{DISPLAYTITLE:Elementary Functions - 4.5 Inequalities}}
 <div style="width: 100%; height: 75vh; overflow: auto;">
 {| class="wikitable sortable" style="margin: 0;"
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 ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
 |-
-| [https://dlmf.nist.gov/4.5.E1 4.5.E1] || [[Item:Q1554|<math>\frac{x}{1+x} < \ln@{1+x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x}{1+x} < \ln@{1+x}</syntaxhighlight> || <math>x > -1, x \neq 0</math> || <syntaxhighlight lang=mathematica>(x)/(1 + x) < ln(1 + x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[x,1 + x] < Log[1 + x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E1 4.5.E1] || <math qid="Q1554">\frac{x}{1+x} < \ln@{1+x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x}{1+x} < \ln@{1+x}</syntaxhighlight> || <math>x > -1, x \neq 0</math> || <syntaxhighlight lang=mathematica>(x)/(1 + x) < ln(1 + x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[x,1 + x] < Log[1 + x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E1 4.5.E1] || [[Item:Q1554|<math>\ln@{1+x} < x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{1+x} < x</syntaxhighlight> || <math>x > -1, x \neq 0</math> || <syntaxhighlight lang=mathematica>ln(1 + x) < x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[1 + x] < x</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E1 4.5.E1] || <math qid="Q1554">\ln@{1+x} < x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{1+x} < x</syntaxhighlight> || <math>x > -1, x \neq 0</math> || <syntaxhighlight lang=mathematica>ln(1 + x) < x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[1 + x] < x</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E2 4.5.E2] || [[Item:Q1555|<math>x < -\ln@{1-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x < -\ln@{1-x}</syntaxhighlight> || <math>x < 1, x \neq 0</math> || <syntaxhighlight lang=mathematica>x < - ln(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x < - Log[1 - x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E2 4.5.E2] || <math qid="Q1555">x < -\ln@{1-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x < -\ln@{1-x}</syntaxhighlight> || <math>x < 1, x \neq 0</math> || <syntaxhighlight lang=mathematica>x < - ln(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x < - Log[1 - x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E2 4.5.E2] || [[Item:Q1555|<math>-\ln@{1-x} < \frac{x}{1-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\ln@{1-x} < \frac{x}{1-x}</syntaxhighlight> || <math>x < 1, x \neq 0</math> || <syntaxhighlight lang=mathematica>- ln(1 - x) < (x)/(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- Log[1 - x] < Divide[x,1 - x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E2 4.5.E2] || <math qid="Q1555">-\ln@{1-x} < \frac{x}{1-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\ln@{1-x} < \frac{x}{1-x}</syntaxhighlight> || <math>x < 1, x \neq 0</math> || <syntaxhighlight lang=mathematica>- ln(1 - x) < (x)/(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- Log[1 - x] < Divide[x,1 - x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E3 4.5.E3] || [[Item:Q1556|<math>|\ln@{1-x}| < \tfrac{3}{2}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\ln@{1-x}| < \tfrac{3}{2}x</syntaxhighlight> || <math>0 < x, x \leq 0.5828\dots</math> || <syntaxhighlight lang=mathematica>abs(ln(1 - x)) < (3)/(2)*x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Log[1 - x]] < Divide[3,2]*x</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E3 4.5.E3] || <math qid="Q1556">|\ln@{1-x}| < \tfrac{3}{2}x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\ln@{1-x}| < \tfrac{3}{2}x</syntaxhighlight> || <math>0 < x, x \leq 0.5828\dots</math> || <syntaxhighlight lang=mathematica>abs(ln(1 - x)) < (3)/(2)*x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Log[1 - x]] < Divide[3,2]*x</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E4 4.5.E4] || [[Item:Q1557|<math>\ln@@{x} \leq x-1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{x} \leq x-1</syntaxhighlight> || <math>x > 0</math> || <syntaxhighlight lang=mathematica>ln(x) <= x - 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[x] <= x - 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E4 4.5.E4] || <math qid="Q1557">\ln@@{x} \leq x-1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{x} \leq x-1</syntaxhighlight> || <math>x > 0</math> || <syntaxhighlight lang=mathematica>ln(x) <= x - 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[x] <= x - 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E5 4.5.E5] || [[Item:Q1558|<math>\ln@@{x} \leq a(x^{1/a}-1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{x} \leq a(x^{1/a}-1)</syntaxhighlight> || <math>a > 0,  x > 0</math> || <syntaxhighlight lang=mathematica>ln(x) <= a*((x)^(1/a)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[x] <= a*((x)^(1/a)- 1)</syntaxhighlight> || Error || Failure || - || Successful [Tested: 9]
+| [https://dlmf.nist.gov/4.5.E5 4.5.E5] || <math qid="Q1558">\ln@@{x} \leq a(x^{1/a}-1)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{x} \leq a(x^{1/a}-1)</syntaxhighlight> || <math>a > 0,  x > 0</math> || <syntaxhighlight lang=mathematica>ln(x) <= a*((x)^(1/a)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[x] <= a*((x)^(1/a)- 1)</syntaxhighlight> || Error || Failure || - || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/4.5.E6 4.5.E6] || [[Item:Q1559|<math>|\ln@{1+z}| \leq -\ln@{1-|z|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\ln@{1+z}| \leq -\ln@{1-|z|}</syntaxhighlight> || <math>|z| < 1</math> || <syntaxhighlight lang=mathematica>abs(ln(1 + z)) <= - ln(1 -abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Log[1 + z]] <= - Log[1 -Abs[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E6 4.5.E6] || <math qid="Q1559">|\ln@{1+z}| \leq -\ln@{1-|z|}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\ln@{1+z}| \leq -\ln@{1-|z|}</syntaxhighlight> || <math>|z| < 1</math> || <syntaxhighlight lang=mathematica>abs(ln(1 + z)) <= - ln(1 -abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Log[1 + z]] <= - Log[1 -Abs[z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E7 4.5.E7] || [[Item:Q1560|<math>e^{-x/(1-x)} < 1-x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-x/(1-x)} < 1-x</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>exp(- x/(1 - x)) < 1 - x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- x/(1 - x)] < 1 - x</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E7 4.5.E7] || <math qid="Q1560">e^{-x/(1-x)} < 1-x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-x/(1-x)} < 1-x</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>exp(- x/(1 - x)) < 1 - x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- x/(1 - x)] < 1 - x</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E7 4.5.E7] || [[Item:Q1560|<math>1-x < e^{-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1-x < e^{-x}</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>1 - x < exp(- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 - x < Exp[- x]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E7 4.5.E7] || <math qid="Q1560">1-x < e^{-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1-x < e^{-x}</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>1 - x < exp(- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 - x < Exp[- x]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E8 4.5.E8] || [[Item:Q1561|<math>1+x < e^{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1+x < e^{x}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>1 + x < exp(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 + x < Exp[x]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E8 4.5.E8] || <math qid="Q1561">1+x < e^{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1+x < e^{x}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>1 + x < exp(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 + x < Exp[x]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E9 4.5.E9] || [[Item:Q1562|<math>e^{x} < \frac{1}{1-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x} < \frac{1}{1-x}</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>exp(x) < (1)/(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x] < Divide[1,1 - x]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E9 4.5.E9] || <math qid="Q1562">e^{x} < \frac{1}{1-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x} < \frac{1}{1-x}</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>exp(x) < (1)/(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x] < Divide[1,1 - x]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E10 4.5.E10] || [[Item:Q1563|<math>\frac{x}{1+x} < 1-e^{-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x}{1+x} < 1-e^{-x}</syntaxhighlight> || <math>x > -1</math> || <syntaxhighlight lang=mathematica>(x)/(1 + x) < 1 - exp(- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[x,1 + x] < 1 - Exp[- x]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E10 4.5.E10] || <math qid="Q1563">\frac{x}{1+x} < 1-e^{-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x}{1+x} < 1-e^{-x}</syntaxhighlight> || <math>x > -1</math> || <syntaxhighlight lang=mathematica>(x)/(1 + x) < 1 - exp(- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[x,1 + x] < 1 - Exp[- x]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E10 4.5.E10] || [[Item:Q1563|<math>1-e^{-x} < x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1-e^{-x} < x</syntaxhighlight> || <math>x > -1</math> || <syntaxhighlight lang=mathematica>1 - exp(- x) < x</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 - Exp[- x] < x</syntaxhighlight> || Error || Failure || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E10 4.5.E10] || <math qid="Q1563">1-e^{-x} < x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1-e^{-x} < x</syntaxhighlight> || <math>x > -1</math> || <syntaxhighlight lang=mathematica>1 - exp(- x) < x</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 - Exp[- x] < x</syntaxhighlight> || Error || Failure || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E11 4.5.E11] || [[Item:Q1564|<math>x < e^{x}-1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x < e^{x}-1</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>x < exp(x)- 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>x < Exp[x]- 1</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E11 4.5.E11] || <math qid="Q1564">x < e^{x}-1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x < e^{x}-1</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>x < exp(x)- 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>x < Exp[x]- 1</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E11 4.5.E11] || [[Item:Q1564|<math>e^{x}-1 < \frac{x}{1-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x}-1 < \frac{x}{1-x}</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>exp(x)- 1 < (x)/(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x]- 1 < Divide[x,1 - x]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E11 4.5.E11] || <math qid="Q1564">e^{x}-1 < \frac{x}{1-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x}-1 < \frac{x}{1-x}</syntaxhighlight> || <math>x < 1</math> || <syntaxhighlight lang=mathematica>exp(x)- 1 < (x)/(1 - x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x]- 1 < Divide[x,1 - x]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E12 4.5.E12] || [[Item:Q1565|<math>e^{x/(1+x)} < 1+x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x/(1+x)} < 1+x</syntaxhighlight> || <math>x > -1</math> || <syntaxhighlight lang=mathematica>exp(x/(1 + x)) < 1 + x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x/(1 + x)] < 1 + x</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/4.5.E12 4.5.E12] || <math qid="Q1565">e^{x/(1+x)} < 1+x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x/(1+x)} < 1+x</syntaxhighlight> || <math>x > -1</math> || <syntaxhighlight lang=mathematica>exp(x/(1 + x)) < 1 + x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x/(1 + x)] < 1 + x</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/4.5.E13 4.5.E13] || [[Item:Q1566|<math>e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}</syntaxhighlight> || <math>x > 0, y > 0</math> || <syntaxhighlight lang=mathematica>exp(x*y/(x + y)) < (1 +(x)/(y))^(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x*y/(x + y)] < (1 +Divide[x,y])^(y)</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 9]
+| [https://dlmf.nist.gov/4.5.E13 4.5.E13] || <math qid="Q1566">e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{xy/(x+y)} < \left(1+\frac{x}{y}\right)^{y}</syntaxhighlight> || <math>x > 0, y > 0</math> || <syntaxhighlight lang=mathematica>exp(x*y/(x + y)) < (1 +(x)/(y))^(y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x*y/(x + y)] < (1 +Divide[x,y])^(y)</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/4.5.E13 4.5.E13] || [[Item:Q1566|<math>\left(1+\frac{x}{y}\right)^{y} < e^{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(1+\frac{x}{y}\right)^{y} < e^{x}</syntaxhighlight> || <math>x > 0, y > 0</math> || <syntaxhighlight lang=mathematica>(1 +(x)/(y))^(y) < exp(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 +Divide[x,y])^(y) < Exp[x]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 9]
+| [https://dlmf.nist.gov/4.5.E13 4.5.E13] || <math qid="Q1566">\left(1+\frac{x}{y}\right)^{y} < e^{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(1+\frac{x}{y}\right)^{y} < e^{x}</syntaxhighlight> || <math>x > 0, y > 0</math> || <syntaxhighlight lang=mathematica>(1 +(x)/(y))^(y) < exp(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 +Divide[x,y])^(y) < Exp[x]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 9]
 |-
-| [https://dlmf.nist.gov/4.5.E14 4.5.E14] || [[Item:Q1567|<math>e^{-x} < 1-\tfrac{1}{2}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-x} < 1-\tfrac{1}{2}x</syntaxhighlight> || <math>0 < x, x \leq 1.5936\dots</math> || <syntaxhighlight lang=mathematica>exp(- x) < 1 -(1)/(2)*x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- x] < 1 -Divide[1,2]*x</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 2]
+| [https://dlmf.nist.gov/4.5.E14 4.5.E14] || <math qid="Q1567">e^{-x} < 1-\tfrac{1}{2}x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-x} < 1-\tfrac{1}{2}x</syntaxhighlight> || <math>0 < x, x \leq 1.5936\dots</math> || <syntaxhighlight lang=mathematica>exp(- x) < 1 -(1)/(2)*x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- x] < 1 -Divide[1,2]*x</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 2]
 |-
-| [https://dlmf.nist.gov/4.5.E15 4.5.E15] || [[Item:Q1568|<math>\tfrac{1}{4}|z| < |e^{z}-1|</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{4}|z| < |e^{z}-1|</syntaxhighlight> || <math>0 < |z|, |z| < 1</math> || <syntaxhighlight lang=mathematica>(1)/(4)*abs(z) < abs(exp(z)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,4]*Abs[z] < Abs[Exp[z]- 1]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E15 4.5.E15] || <math qid="Q1568">\tfrac{1}{4}|z| < |e^{z}-1|</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{4}|z| < |e^{z}-1|</syntaxhighlight> || <math>0 < |z|, |z| < 1</math> || <syntaxhighlight lang=mathematica>(1)/(4)*abs(z) < abs(exp(z)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,4]*Abs[z] < Abs[Exp[z]- 1]</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E15 4.5.E15] || [[Item:Q1568|<math>|e^{z}-1| < \tfrac{7}{4}|z|</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|e^{z}-1| < \tfrac{7}{4}|z|</syntaxhighlight> || <math>0 < |z|, |z| < 1</math> || <syntaxhighlight lang=mathematica>abs(exp(z)- 1) < (7)/(4)*abs(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Exp[z]- 1] < Divide[7,4]*Abs[z]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E15 4.5.E15] || <math qid="Q1568">|e^{z}-1| < \tfrac{7}{4}|z|</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|e^{z}-1| < \tfrac{7}{4}|z|</syntaxhighlight> || <math>0 < |z|, |z| < 1</math> || <syntaxhighlight lang=mathematica>abs(exp(z)- 1) < (7)/(4)*abs(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Exp[z]- 1] < Divide[7,4]*Abs[z]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E16 4.5.E16] || [[Item:Q1569|<math>|e^{z}-1| \leq e^{|z|}-1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|e^{z}-1| \leq e^{|z|}-1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(exp(z)- 1) <= exp(abs(z))- 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Exp[z]- 1] <= Exp[Abs[z]]- 1</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E16 4.5.E16] || <math qid="Q1569">|e^{z}-1| \leq e^{|z|}-1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|e^{z}-1| \leq e^{|z|}-1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>abs(exp(z)- 1) <= exp(abs(z))- 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Exp[z]- 1] <= Exp[Abs[z]]- 1</syntaxhighlight> || Skipped - no semantic math || Failure || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/4.5.E16 4.5.E16] || [[Item:Q1569|<math>e^{|z|}-1 \leq |z|e^{|z|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{|z|}-1 \leq |z|e^{|z|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(abs(z))- 1 <= abs(z)*exp(abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[Abs[z]]- 1 <= Abs[z]*Exp[Abs[z]]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/4.5.E16 4.5.E16] || <math qid="Q1569">e^{|z|}-1 \leq |z|e^{|z|}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{|z|}-1 \leq |z|e^{|z|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(abs(z))- 1 <= abs(z)*exp(abs(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[Abs[z]]- 1 <= Abs[z]*Exp[Abs[z]]</syntaxhighlight> || Error || Failure || - || Successful [Tested: 1]
 |}
 </div>

4.5: Difference between revisions

Latest revision as of 11:05, 28 June 2021

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