Accuracy and the power function

[Werner]

(02-18-2014 08:28 PM)Dieter Wrote:  However, in the example with 4 ULP error the internally calculated exponent is 372*ln(2*pi) = 683,69... One ULP more or less equals an absolute error of 1E-12 or an (acceptable) error of 1 ULP in the returned result. If larger errors occur, one may conclude that 15 digit internal precision does not mean that all 15 are exact.

There are two things at work here.
First, the 15-digit routines indeed are not exact to the last digit. In your example, %%LN(6.28318530718) is off by one ULP.
Secondly, the 15-digit routines (%%*, %%+, %%-, %%/) do not round, they truncate. They are, after all, meant as intermediates to produce a correctly rounded 12-digit result (for the elementary operations).

Code:


          %%LN                    %% 372 %%*              %%EXP
exact   1.8378770664094112978.. 683.69026870430100..    8.37357721132000..e296
HP48    1.83787706640940        683.690268704296        8.37357721127809e296
trunc15 1.83787706640941        683.690268704300        8.37357721131159e296
round15                         683.690268704301        8.37357721131998e296

Not only is the %%LN one ULP off, but the subsequent multiplication by 372 actually ends in ..968, but is truncated to ..96. The last line demonstrates what a 15-digit %%LN, 15-digit %%EXP and 15-digit rounding arithmetic would produce.

Werner