FCCompareNumbers[x, y] compares two purely numerical or semi-numerical expressions x and y and returns the number of agreeing significant digits calculated from the relative differences.
These two numbers disagree in their 6th significant digit
FCCompareNumbers[5.123542342 + 1.23145 I, 5.123542324 + 1.23146 I]\text{FCCompareNumbers: Minimal number of significant digits to agree in: }6
\text{FCCompareNumbers: Chop is set to }\;\text{1.$\grave{ }$*${}^{\wedge}$-10}
\text{FCCompareNumbers: No number is set to 0. by Chop at this stage. }
\text{FCGV}(\text{I}) \;\text{FCGV}(\text{CommonDigits})(5.09042)
Requiring agreement only in the first 5 significant digits returns 0
FCCompareNumbers[5.123542342 + 1.23145 I, 5.123542324 + 1.23146 I, DigitCount -> 5]\text{FCCompareNumbers: Minimal number of significant digits to agree in: }5
\text{FCCompareNumbers: Chop is set to }\;\text{1.$\grave{ }$*${}^{\wedge}$-10}
\text{FCCompareNumbers: No number is set to 0. by Chop at this stage. }
0
Here even the first significant digit doesn’t agree (obviously)
FCCompareNumbers[5, 6]\text{FCCompareNumbers: Minimal number of significant digits to agree in: }6
\text{FCCompareNumbers: Chop is set to }\;\text{1.$\grave{ }$*${}^{\wedge}$-10}
\text{FCCompareNumbers: No number is set to 0. by Chop at this stage. }
\text{FCGV}(\text{CommonDigits})(0.778151)
lhs = (0. + 0.*I) - (0.20132103165327941 - 0.00043434443313399246*I)*coeffO^2*parX^2 + (0.047227066764317975)*coeffO^2*parX^2*parY - (0.00005403882927314103)*coeffO^2*parX^2*parY^2 + (1.4588597782189382*^-6 - 4.06569606476957*^-13*I)*coeffO^2*parX^2*parY^3 + (0.03841797609570242 + 0.000028403733516153446*I)*coeffO^2*parX^2*parZ(0.038418\, +0.0000284037 i) \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parZ}-(0.201321\, -0.000434344 i) \;\text{coeffO}^2 \;\text{parX}^2+(0.\, +0. i)+(\text{1.4588597782189382$\grave{ }$*${}^{\wedge}$-6}-\text{4.06569606476957$\grave{ }$*${}^{\wedge}$-13} i) \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}^3-0.0000540388 \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}^2+0.0472271 \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}
rhs = (-0.20132103165327922 + 0.0004343444331339952*I)*coeffO^2*parX^2 + (0.0472270672349811)*coeffO^2*parX^2*parY - (0.00005403887000187252)*coeffO^2*parX^2*parY^2 + 1.4588601127764193*^-6*coeffO^2*parX^2*parY^3 + (0.038417976095702376 + 0.000028403733516153537*I)*coeffO^2*parX^2*parZ(0.038418\, +0.0000284037 i) \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parZ}-(0.201321\, -0.000434344 i) \;\text{coeffO}^2 \;\text{parX}^2+\text{1.4588601127764193$\grave{ }$*${}^{\wedge}$-6} \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}^3-0.0000540389 \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}^2+0.0472271 \;\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}
Here the two above expressions agree in their first 6 significant digits. Notice that the number of the size 10^{-13} is treated as a numerical fluctuation and consequently removed by Chop
FCCompareNumbers[lhs, rhs]\text{FCCompareNumbers: Minimal number of significant digits to agree in: }6
\text{FCCompareNumbers: Chop is set to }\;\text{1.$\grave{ }$*${}^{\wedge}$-10}
\text{FCCompareNumbers: Following numbers on the l.h.s. are set to 0. by Chop: }\{-\text{4.06569606476957$\grave{ }$*${}^{\wedge}$-13}\}
0
The application of Chop can be of course disabled
FCCompareNumbers[lhs, rhs, Chop -> False]
\text{FCCompareNumbers: Minimal number of significant digits to agree in: }6
\text{coeffO}^2 \;\text{parX}^2 \;\text{parY}^3 \;\text{FCGV}(\text{I}) \;\text{FCGV}(\text{Unmatched})(-\text{4.06569606476957$\grave{ }$*${}^{\wedge}$-13})