Incorrect calculations: csc(x) near x=k*π

In this blog post, we explore incorrect calculations of the cosecant function csc(x) in MATLAB near numbers around


$k\pi .$

Example 1. Given that

x1=0.942477796123e1≈3 π
x_1=0.942477796123\textup{e}1 \approx 3\,\pix1​=0.942477796123e1≈3π

and



x2=0.7853981633974e2≈25 π ,x_2=0.7853981633974\textup{e}2 \approx 25\,\pi\,,
x2​=0.7853981633974e2≈25π,

calculate

csc⁡(x1)
\csc(x_1)
csc(x1​)

and

csc⁡(x2)
\csc(x_2)
csc(x2​)

in MATLAB.

Using MATLAB, we computed the values of

csc⁡(x1)
\csc(x_1)
csc(x1​)

and

csc⁡(x2).
\csc(x_2).
csc(x2​).

The result is shown in the following screenshot.

).

Thus, the output of MATLAB contains only 6 and 4 correct digits, with error rates of (16-6)/16 = 62.5% and (16-4)/16 = 75% respectively for the significant figures.

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