Editing Exsecant (section)

== Catastrophic cancellation for small angles ==

Naïvely evaluating the expressions <math>1 - \cos \theta</math> (versine) and <math>\sec \theta - 1</math> (exsecant) is problematic for small angles where <math>\sec \theta \approx \cos \theta \approx 1.</math> Computing the difference between two approximately equal quantities results in [[catastrophic cancellation]]: because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.

For example, the secant of {{math|1°}} is approximately {{math|1={{val|1.000152}}}}, with the leading several digits wasted on zeros, while the [[common logarithm]] of the exsecant of {{math|1°}} is approximately {{math|{{val|-3.817220}}}},{{r|log exsec}} all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place [[trigonometric table]] and then subtracting {{math|1}}, the difference {{math|1=sec&thinsp;1° &minus;&thinsp;1 ≈&nbsp;}}{{wbr}}{{math|1={{val|0.000152}}}} has only 3 [[significant digits]], and after computing the logarithm only three digits are correct, {{math|1=log(sec&thinsp;1° &minus;&thinsp;1) ≈&nbsp;}}{{wbr}}{{math|1=<span class="nowrap">−3.81<span style="color:#a00">8<span style="margin-left:.25em;">156</span></span></span>}}.<ref>The incorrect digits are highlighted in red.</ref> For even smaller angles loss of precision is worse.

If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as <math display=inline>\operatorname{exsec} \theta = \tan \theta \, \tan \tfrac12\theta\vphantom\Big|,</math> or using versine, <math display=inline>\operatorname{exsec} \theta = \operatorname{vers} \theta \, \sec \theta,</math> which can itself be computed as <math display=inline>\operatorname{vers} \theta  = 2 \bigl({\sin \tfrac12\theta}\bigr)\vphantom)^2\vphantom\Big| = {}</math>{{wbr}}{{nobr|<math>\sin \theta \, \tan \tfrac12\theta\,\vphantom\Big|</math>;}} Haslett used these identities to compute his 1855 exsecant and versine tables.{{r|Haslett summary}}{{r|nagle}}

For a sufficiently small angle, a circular arc is approximately shaped like a [[parabola]], and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength.{{r|shunk}}