HHC 2015 - Savage benchmark curiosity

[Dieter]

(10-07-2015 03:18 PM)Claudio L. Wrote:  The value in the stack was exactly 1469.99992223, and the difference came from ATAN, which returns 1.57011605476 on the 50g, but 1.57011605475 in newRPL, and values diverged from there.
The actual result for ATAN(1469.99992223) is:
1.5701160547549999736...

So it's clearly a case of double-rounding. I think the internal 15-digit precision would round 49999 to 500, and therefore round the last digit to 6 instead of 5.

Now I forced the loop to start on the 50g from the corrected value, and this happened again for number 1647.99989715, also in the ATAN function.
In this case ATAN gives 1.57018953071499665..., which again causes the double rounding and values diverge again.

I do not think this is caused by double rounding. The internal extended precision routines usually truncate the result after 13 resp. 15 digits. This would yield the correct result here: 1.57011605475499 => 1.57011605475.

But all this will only work as expected if (!) the internal result is exact in all 13 resp. 15 digits. Well... it isn't. If it was, there was not need for guard digits. ;-)

The internal routines for the transcendental functions actually are not that exact, that's why the final 12-digit result can be off by one ULP (or even more in some special cases). I cannot say much about the internal 15-digit routines of the 50G, but for the 41C the MCode console shows slight errors in the last (13th) digit even for simple cases like e¹ = 2,718281828458.

That's why in close cases (e.g. the one you mentioned) the last digit may be off by one. I don't think it's a rounding issue, it's simply a slight error in the 15-digit result.

Dieter