ClearAll[sTarget,integralInterior,integralExterior,sInverted]
(*Setextremelyhighworkingprecisionbuffer*)wp=120;pg=100;
(*1.High-PrecisionTargetBenchmark*)sTarget=2*NIntegrate[Log[(Pi*x)/Sin[Pi*x]]/x,{x,0,1},Method->"DoubleExponential",WorkingPrecision->wp,PrecisionGoal->pg];
(*2.InteriorIntegral[0,1]*)integralInterior=NIntegrate[Log[(Pi*x)/Sin[Pi*x]]/x,{x,0,1},Method->"DoubleExponential",WorkingPrecision->wp,PrecisionGoal->pg];
(*3.InvertedExteriorIntegral[1,Infinity]*)integralExterior=NIntegrate[Log[(Pi/t)/Sin[Pi/t]]/t,{t,1,Infinity},Method->"DoubleExponential",WorkingPrecision->wp,PrecisionGoal->pg];
(*ReconstructedS1usingInversionDuality*)sInverted=integralInterior+integralExterior;
(*4.ExactSymbolicVerification(ZeroNoise)*)exactInterior=Integrate[Log[(Pi*x)/Sin[Pi*x]]/x,{x,0,1}];exactExterior=Integrate[Log[(Pi/t)/Sin[Pi/t]]/t,{t,1,Infinity}];symbolicDiff=Simplify[(exactInterior+exactExterior)-2*exactInterior];
Print["============================================================"];Print[" EXTREME PRECISION INVERSION DUALITY TEST "];Print["============================================================"];Print["Target Direct Value (100 digits) : ",N[sTarget,100]];Print["Reconstructed Sum (100 digits) : ",N[sInverted,100]];Print["------------------------------------------------------------"];Print["Numerical Absolute Difference : ",ScientificForm[N[Abs[sInverted-sTarget],10]]];Print["Exact Symbolic Residual : ",symbolicDiff];Print["============================================================"];
============================================================
EXTREMEPRECISIONINVERSIONDUALITYTEST
============================================================
TargetDirectValue(100digits):2.312678950427516318486944583815615630360507295938872973700906561186830487867162412038245480159774207
ReconstructedSum(100digits):2.312678950427516318486944583815615630360507295938872973700906561186830487867162412038245480159774207
------------------------------------------------------------
NumericalAbsoluteDifference:5.*10^(-119)
ExactSymbolicResidual:t-x
∞
∫
1
Log
πCsc
π
t
t
t
1
∫
0
Log[πxCsc[πx]]
x
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