= GPSD Numbers Matter
Gary E. Miller <gem@rellim.com>
14 July 2022
:author: Gary E. Miller
:description: How and why  GPSD tortures numbers.
:email: <gem@rellim.com>
:keywords: gpsd, NaN, precision
:robots: index,follow
:sectlinks:
:source-highlighter: rouge
:toc: macro

include::inc-menu.adoc[]

== ABSTRACT

Geodesy tortures numbers harder than almost any other discipline.  It
requires that very large numbers be known to very small precision.  This
leads to some unexpected, sometimes perplexing, choices in how *gpsd*
handles numbers.  This white paper will explore many of those choices.

== Numeric Precision

Computers work with either floating point numbers in <<IEEE754>> format or
binary numbers.  Binary numbers are exact, floating point numbers are not.
For example, the quantity 1/10 (one tenth) can not be represented exactly
in IEEE 754.

An IEEE 754 single precision (float) has 7.22 decimal digits of precision (24
binary bits of precision).

A 32 bit integer has 9.33 deciaml digits of precision. That is 31 binary
digits of precision, plus a sign bit.

An IEEE 754 double precision (double) has 15.95 decimal digits of
precision (53 binary bits of precision).

A 64 bit integer has 18.96 digits of decimal precision. That is 63
binary digits of precision, plus a sign bit.

printf() format %f defaults to %.6f.  Below you will see how that may
cause problems.

== Latitude and Longitude

Your GNSS receiver starts with really big, numbers. Like the Earth's
polar radius: 6356752.314245 m. Then with the help of a lot of math,
computes your position to a high precision. The u-blox ZED-F9P reports
0.1 mm (1e-9 or 0.000000001 degree) precision. That is about 12
decimal digits of precision. It is certainly not that accurate, maybe
soon. No matter, *gpsd* wants to not lose the precision of the data it
is given.

printf() format %f defaults to %.6f, which will truncate a latitude or
longitude. so print with %.7f, or even %9f, if you have a survey grade
GPS. Here is a rough idea of how degrees relate to distance, at the
equator:

|====
|Degrees|Resolution|DMS|Distance at equator

|0.0001|1e-4|0° 00′ 0.36″|11.132 m
|0.000001|1e-6|0° 00′ 0.0036″|11.132 cm
|0.0000001|1e-7|0° 00′ 0.00036″|11.132 mm
|0.00000001|1e-8|0° 00′ 0.000036″|1.1132 mm
|0.000000001|1e-9|0° 00′ 0.0000036″|0.1113 mm
|====
Source: <<DD>>

u-blox firmware since at least protocol version 4 (Antaris 4)
has reported latitude and longitude to 0.0000001 (1e-7) with the
UBX-NAV-POSLLH message. At that time, 1e-7 was wildly optimistic.

Starting with protocol version 20, those u-blox with High Precision
firmware supports the UBX-NAV-HPPOSLLH message. That message reports to
0.00000000001 (1e-9) precision, about 0.1 mm.

Python floats are very similar to C doubles, plus some annoying bugs
related to <<NaN>>.

See <<DD>> for more information on Decimal Degrees and precision.

== Altitude

Altitude, once you decide which altitude is which, is numerically easy
for *gpsd*. The u-blox F9P can report altitudes (HAE and MSL) to a
precision of 0.1 mm. <<CoCom>> limits the maximum altitude of civilian
GNSS receivers to a maximum altitude of 18,000 m. That fits comfortably
in 28 binary digits. 29 if you want to go to below Sea Level. This will
not fit in a "float", but fits in a 32 integer (scaled) or a "double".

If you are a rocket scientis, and can get a GNSS receiver that works in
Geostationary orbit (35,786 km) then you would need 39 binary digits of
precision. This will not fit in a "float" or a 32 integer but fits
in a 64 bit integer (scaled) or a "double".

== Time

In the "Latitude and Longitude" section above we learned that C doubles
are just fine for holding position information.  The same can not be said
for "Time". There is loss of precision when storing time as a double!

 * A double is 53 significant bits.
 * POSIX time to nanoSec precision is 62 significant bits
 * POSIX time to nanoSec precision after 2038 is 63 bits
 * POSIX time as a double is only microSec precision

That is why POSIX time as a double and PPS do not play well together.

WARNING:: Loss of precision telling time as a double!

That is why *gpsd* tells time using *struct timespec*.  That look like this:

[source,c]
----
  struct timespec {
      time_t  tv_sec;  /* Seconds */
      long    tv_nsec; /* Nanoseconds */
  };
----

*time_t* is usually a 64-bit integer. Older systems, and some 32-bit
systems, define *time_t* as a 32-bit integer, which is deprecated. A
32-bit integer will overflow at: 03:14:07 UTC on 19 January 2038. Plan
for that apocalypse now.  Source: <<Y2038>>

In 2021 cosmologists believe the age of the universe is just under
14 billion years.  That is 4.4 e17 seconds, which fits comfortably
into a 59 bit unsigned integer.  A 64-bit *time_t* will be good enough
for a while longer.

The other part of *timespec_t* is a long, carrying the nanosecond part
of the time stamp.  In 2021, a GNSS receiver can tell you the start of
every second to about 1 nano second (1 e-9 seconds) accuracy.  That fits
comfortably into a 30 bit unsigned integer.  As long integer in C is
always at least 32 bits.

A *timespec_t* fails when you need to measure time to better than 1 nano
second.  The SI second is defined as 9,192,631,770 cycles of radiation
from a caesium-133 (Cs) atom.  Close to 0.1 nano seconds.  That requires
a 34 bit unsigned integer.

In 2021 the smallest frequency difference that can be measured is about
1 second in 400 million years, one part in about 1.23 e16.  That involves
clocks composed of strontium atoms, and measuring time differences with
optical combs.  The time difference between those two is thus 7.9 e-17
seconds per second.  Needing a 54 bit unsigned integer fraction of a
second to hold it.

=== Time Accuracy

Just because gpsd can represent a number does not mean the number means
what you think it does.  The u-blox ZED-F9T data sheet says it can
output absolute PPS to 5ns.  But the fine print says: "1-sigma, fixed
position mode, depends on temperature, atmospheric conditions, baseline
length, GNSS antenna, multipath conditions, satellite visibility and
geometry".

There are many distinct embodiments of Universal Coordinated Time
(UTC).  In the USA there are two, one kept by the National Institute
of Standards and Technology (NIST] and one by the US Naval Observatory
(USNO).  These are referred to as UTC(NIST) and UTC(USNO).  The primary
source for UTC(NIST) is in Fort Collins Colorado.  Their secondary
(backup) source is in Gaithersburg Maryland.  According to <<NIST2187>>,
in 2021, the secondary UTC(NIST) site is only plus/minus 25 nano seconds
aligned with the primary source.  Don't expect to tell time better than
the NIST.

UTC(USNO) supplies the master clock for the GPS system.  In 2020, NIST
said that UTC(USNO) differed from UTC(NIST) by plus/minus 20 nano
seconds. See <<NIST-USNO>>.  Even if you could track GPS time perfectly,
and it tracked UTC(USNO) perfectly, you are still off by plus/minus 20
nano seconds.

The biggest obstacle to *gpsd* and *ntpd* keeping accurate time is the
granularity of the local host clock.  The *gpsd* release includes a program
called *clock_test*, and the NTPsec release includes a program in the
attic called *clocks*.  Both can characterize your system clock.

Using these programs you can determine the granularity of you system
clock.  Some examples:

|====
|CPU |Clock speed|Clock granularity|Standard deviation
|Raspberry Pi 3B|1.2GHz|155 ns|120 ns
|Raspberry Pi 4B|1.5GHz|56 ns|90 ns
|Xeon E5-1620 v3|3.50GHz|14 ns|46 ns
|Core i5-4570|3.20GHz|11 ns|231 ns
|Core i7-8750H|2.2GHz|18 ns|19 ns
|Ryzen 5 3600|3.6 GHz|10 ns|60 ns
|====

Consider these best cases.  Any load, reduced clock speed, I/O
interrupts, interrupt latency, etc. will reduce the accuracy with which
he system clock can be read or set.  Your goal, and that of NIST stated
in <<NIST2187>>, is that you can tell time to less than 1 micro second.

== NaN ain't your Nana

The most obviously confounding choice is the use in *gpsd* of *NaNs*
(Not A Number). *gpsd* keeps track of a large number of individual
numbers, most of them are invalid at any one time. To keep track of
which integers are valid, a bit field is used. When an integer is
valid, a corresponding bit is set. Keeping track of which bit matches
which integer is challenging. <<IEEE754>> eliminates that problem with
floating point numbers.

When *gpsd* marks a floating point number invalid, it sets the value to
<<NaN>>. So before using any *gpsd* floating point number, check that
it is valid. C obeys <<IEEE754>>. Python sort of does, enough for our
purposes.

=== C NaN

A little C program will make the behavior of <<NaN>> easy to see:

[source%nowrap,c,numbered]
----
// Compile with: gcc nan.c -o nan
#include <stdio.h>     // for printf()

int main(int argc, char **argv)
{
    double a = 1.0 / 0.0;
    double b = -1.0 / 0.0;
    printf("a: %f b: %f\n", a, b);
}
----

What do you expect to see whan that program is run?  Try it:

----
~ $ gcc nan.c -o nan
~ $ ./nan
a: inf b: -inf
----

1.0 divided by 0.0 is infinity.  -1.0 divided by 0.0 is negative infinity.

Any program that printed out a lot of "inf" or -inf" would annoy the users
and they would complain.  To avoid that, *gpsd* clients check, and print
out "n/a" instead.

Here is a common solution:

[source%nowrap,c,numbered]
----
// Compile with: gcc nan.c -o nan
#include <math.h>      // for isnan()
#include <stdio.h>     // for printf()
  
int main(int argc, char **argv)
{
    double a = 1.0 / 0.0;
    if (isnan(a)) {
        printf("a: n/a\n");
    } else {
        printf("a: %f\n", a);
    }
}
----

What do you expect to see whan that program is run?  Try it:

----
~ $ gcc  nan.c -o nan
~ $ ./nan
a: inf
----

Whoops.  All <<NaN>>s are not <<NaN>>s!  Very confusing, rather than try to
explain, I'll send you to the Wikipedia explanation: <<NaN>>.  But there
is a simple solution.  We do not really care if a number if <<NaN>>, or if it
is infinity.  We care that it is finite, and that is easy to test for:

[source%nowrap,c,numbered]
----
// Compile with: gcc nan.c -o nan
#include <math.h>      // for isfinite()
#include <stdio.h>     // for printf()
  
int main(int argc, char **argv)
{
    double a = 1.0 / 0.0;
    if (isfinite(a)) {
        printf("a: %f\n", a);
    } else {
        printf("a: n/a\n");
    }
}
----

What do you expect to see whan that program is run?  Try it:

----
~ $ gcc  nan.c -o nan
~ $ ./nan
a: n/a
----

Exactly the desired result.  Now you know why *isfinite()* is all over
*gpsd* client code.

=== Python NaN

Python is similar, it almost follows <<IEEE754>>, but has many undocumented
"features" that conflict with <<IEEE754>>:

[source%nowrap,numbered]
----
# python
>>> a = 1.0 / 0.0
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ZeroDivisionError: float division by zero
----

For shame.  It does provide a sideways method to set a variable to
various <<NaN>>s:

----
~ $ python
>>> Inf = float('inf')
>>> Ninf = float('-inf')
>>> NaN = float('NaN')
>>> print("Inf: %f Ninf: %f NaN: %f" % (Inf, Ninf, NaN))
Inf: inf Ninf: -inf NaN: nan
----

And *math.isnan()* and *math.isfinite()* work as expected.  Continuing
the previous example:

----
>>> import math
>>> math.isnan(Inf)
False
>>> math.isnan(NaN)
True
>>> math.isfinite(NaN)
False
>>> math.isfinite(Inf)
False
----

And that is why *gpsd* uses *math.isfinite()* instead of *math.isnan()*.

<<NaN>>s have many other interesting properties, be sure to read up on
the subject. The <<IEEE754>> document is a closed source standard. For a
public description look at the Wikipedia <<NaN>> article.

== REFERENCES

* *GPSD Project web site:* {gpsdweb}

[bibliography]
* [[[IEEE754]]] https://standards.ieee.org/standard/754-2019.html

* [[[DD]]] https://en.wikipedia.org/wiki/Decimal_degrees

* [[[CoCom]]] https://en.wikipedia.org/wiki/Coordinating_Committee_for_Multilateral_Export_Controls

* [[[Y2038]]] https://en.wikipedia.org/wiki/Year_2038_problem

* [[[NaN]]] https://en.wikipedia.org/wiki/NaN

* [[[NIST2187]]] https://nvlpubs.nist.gov/nistpubs/TechnicalNotes/NIST.TN.2187.pdf

* [[[NIST-USNO]]] https://www.nist.gov/pml/time-and-frequency-division/time-services/nist-usno/nist-usno-2020-archive

== COPYING

This file is Copyright 2021 by the GPSD project +
SPDX-License-Identifier: BSD-2-clause
