Torture tests: what do they mean?

[Manolo Sobrino]

A machine that can give you x accurate digits for all the domain is not working internally with just x digits. As was discussed in another thread, the HPs are likely using 2x significant digits in their trigonometrics. The bizarre thing here is that rounding to x for the input data and the arithmetics and then performing the cordic calculations with 2x tables is considered consistent.

In this case, the Mathematica arbitrary precision results are:

12 significant digits:

N[355/226, 12] = 1.57079646018

Tan[1.57079646018`12] = -7.4971*10^6

Now, 355/226 rounded to 12 and then calculating the tangent with 24:

Tan[1.57079646018`24] = -7.4970890650760106*10^6

Rounded to 12 for output:

N[%, 12] = -7.49708906508*10^6

There you go. You need at least 19 significant digits to get that.

I don't know what Casios do here. They should get -7.4972582*10^6, yet they are calculating Tan[1.5707961934128066`16]. There must be a small flaw in their implementation of the tangent.