#Python tip: Given inexact data, subtracting nearly equal
values increases relative error significantly more
than absolute error.

4.6 ± 0.2 Age of Earth (4.3%)
4.2 ± 0.1 Age of Oceans (2.4%)
___
0.4 ± 0.3 Huge relative error (75%)

This is called “catastrophic cancellation”.

1/
The subtractive cancellation issue commonly arises in floating point arithmetic. Even if the inputs are exact, intermediate values may not be exactly representable and will have an error bar. Subsequent operations can amplify the error.

1/7th is inexact but within ± ½ ulp.

2/
A commonly given example arises when estimating derivatives with f′(x) ≈ Δf(x) / Δx.

Intuitively, the estimate improves as Δx approaches zero, but in practice, the large relative error from the deltas can overwhelm the result.

Make Δx small, but not too small. 😟

3/
While people tend to think of this as a floating point arithmetic problem, it arises anytime data has a range of uncertainty.

Example with integer arithmetic:
(50 ± 2) - (45 ± 2) ⟶ (5 ± 4)

4% relative errors turn into 80%.

4/
Part of the art of numeric computing is algebraically rearranging calculations to avoid subtracting nearly equal values.

A famous example is rewriting the quadratic equation as:

𝑥 = 2𝑐 ÷ (𝑏∓ √(𝑏²−4𝑎𝑐))

This is used when |4ac| is small relative to |b²|.

5/
The “loss of significance” problem also arises with computing time differences:

Bad:
start = time()
delta = time() - start

Better way:
start = perf_counter()
delta = perf_counter() - start

The former measures from 1970. The latter measures from the start of the process.

6/
If you only have 16 decimal places of precision in a floating point number, why waste 9 of them by measuring from 1970?

>>> time()
1593041920.6464949
>>> time()
1593041922.272326
>>> len('159304192')
9

>>> perf_counter()
8.867890209
>>> perf_counter()
10.422069702

7/

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More from @raymondh

3 Jan
#Python factlet: The len() function insists that the corresponding __len__() method return a value x such that:

0 ≤ x.__index__() ≤ sys.maxsize

* 3.0 and '3' don't have an __index__ method.
* -1 is too small.
* sys.maxsize+1 is too big.

1/
You could call __len__() successfully, but the len() function fails:

class A:
def __len__(self):
return -1

>>> a = A()

>>> a.__len__()
-1

>>> len(a)
...
ValueError: __len__() should return >= 0

2/
Classes written in C typically implement __len__() with the mp_length or sq_length slot. That constrains them to sys.maxsize limits:

>>> r = range(10**100)
>>> r.__len__()
Traceback (most recent call last):
...
OverflowError: Python int too large to convert to C ssize_t

3/
Read 4 tweets
2 Jan
@yera_ee Each way has its advantages.

With dataclasses, you get nice attribute access, error checking, a name for the aggregate data, and a more restrictive equality test. All good things.

Dicts are at the core of the language and are interoperable with many other tools: json, **kw, …
@yera_ee Dicts have a rich assortment of methods and operators.
People learn to use dicts on their first day.
Many existing tools accept or return dicts.
pprint() knows how to handle dicts.
Dicts are super fast.
JSON.
Dicts underlie many other tools.
@yera_ee Embrace dataclasses but don't develop an aversion to dicts.

Python is a very dict centric language.

Mentally rejecting dicts would be like developing an allergy to the language itself. It leads to fighting the language rather than working in harmony with it.
Read 4 tweets
27 Dec 20
1/ #Python tip: Override the signature for *args with the __text_signature__ attribute:

def randrange(*args):
'Choose a random value from range(start[, stop[, step]]).'
return random.choice(range(*args))

randrange.__text_signature__ = '(start, stop, step, /)'
2/ The attribute is accessed by the inspect module:

>>> inspect.signature(randrange)
<Signature (start, stop, step, /)>
3/ Tooltips and help() will now be more informative:

>>> help(randrange)
randrange(start, stop, step, /)
Choose a random value from range(start[, stop[, step]]).
Read 4 tweets
25 Oct 20
1/ #Python tip: The functools.cache() decorator is astonishingly fast.

Even an empty function that returns None can be sped-up by caching it. 🤨

docs.python.org/3/library/func…
2/ Here are the timings:

$ python3.9 -m timeit -r11 -s 'def s(x):pass' 's(5)'
5000000 loops, best of 11: 72.1 nsec per loop

$ python3.9 -m timeit -r11 -s 'from functools import cache' -s 'def s(x):pass' -s 'c=cache(s)' 'c(5)'
5000000 loops, best of 11: 60.6 nsec per loop
3/ Behind the scenes, there is a single dictionary lookup and a counter increment.

Some kinds arguments will take longer to hash, but you get the idea, @cache has very little overhead.
Read 5 tweets
6 Sep 20
1/ #Python data science tip: To obtain a better estimate (on average) for a vector of multiple parameters, it is better to analyze sample vectors in aggregate than to use the mean of each component.

Surprisingly, this works even if the components are unrelated to one another.
2/ One example comes from baseball.

Individual batting averages near the beginning of the season aren't as good of a performance predictor as individual batting averages that have been “shrunk” toward the collective mean.

Shockingly, this also works for unrelated variables.
3/ If this seems unintuitive, then you're not alone. That is why it is called Stein's paradox 😉

The reason it works is that errors in one estimate tend to cancel the errors in the other estimates.

statweb.stanford.edu/~ckirby/brad/o…
Read 5 tweets
31 Aug 20
Another building block for a #Python floating point ninja toolset:

def veltkamp_split(x):
'Exact split into two 26-bit precision components'
t = x * 134217729.0
hi = t - (t - x)
lo = x - hi
return hi, lo

csclub.uwaterloo.ca/~pbarfuss/dekk…
Input: one signed 53-bit precision float

Output: two signed 26-bit precision floats

Invariant: x == hi + lo

Constant: 134217729.0 == 2.0 ** 27 + 1.0
Example:

>>> hi, lo = veltkamp_split(pi)
>>> hi + lo == pi
True
>>> hi.hex()
'0x1.921fb58000000p+1'
>>> lo.hex()
'-0x1.dde9740000000p-26'

Note all the trailing zeros and the difference between the two exponents. Also both the lo and hi values are signed.
Read 6 tweets

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