19 Oct
2019
19 Oct
'19
5:20 a.m.
On 10/18/2019 6:04 PM, Ken Thompson via TUHS wrote:
while writing "space travel,"
i could not get the space ship integration
around a planet to keep from either gaining or
losing energy due to floating point errors.
i asked dick hamming if he could help. after
a couple hours, he came back with a formula.
i tried it and it worked perfectly. it was some
weird simple double integration that self
corrected for fp round off. as near as i can
ascertain, the formula was never published
and no one i have asked (including me) has
been able to recreate it.
This reminds me of an experiment I performed comparing speeds of x86
assembler, and C, when I first started working in it. The experiment was
to generate a 3D cube wireframe, and rotate it about any (or all three)
of it's axes, moving it X degrees at a time. Simple vector math, really.
Included perspective, so that the back-end of the wireframe cube would
look smaller than the front side.
I had been an assembler snob, having started in MACRO-10 on a KA10
PDP-10 in 9th grade. I always assumed assembler would be faster for
anything, given the right amount of optimization. The assembler side of
the experiment, I did the CGA graphics directly. I think in the C
version, I used the built-in library that came with it. I no longer have
the source for the C version, but I do, for the assembler version. I
didn't have an 8087 floating point accelerator, so I wrote my assembler
example to use two 16-bit words of integers, combining them for a 31-bit
integer value with sign.
Timing 1000 rotations, the assembler version took X amount of time. The
C version took X*1.5
Now mind you, the C version used real floating point, and a software
floating point library with no hardware accelerator. At that point, I
realized C was the way to go. It had passed my experiment with flying
colors. The C compiler, I believe, was from Computer Innovations,
Copyright (c)1981,82,83,84,85
The reason this is similar to Ken's statement above: In the assembler
version, the cube would deform quite a bit before the run would finish.
A 31-bit integer didn't accurately reflect the result of the math. Over
time, that slight inaccuracy really added up. The accuracy of the C
version using floats was spot on. So while I basically cheated for the
assembler version, causing the deformation of the cube over time, the C
version was 100% accurate even though it was slower.
I wonder, is there something inherently different between PDP-11/7
floats and Intel's leading to the inaccuracy Ken mentions? Was the
PDP-11 (or the -7) floating point that much different than IEEE-754 ?
art k.