better numerical accuracy testing

[robertwb]

If a line contains tol or tolerance, numerical results are only
verified to the given tolerance. This may be prefixed by abs[olute] or rel[ative] to specify whether to measure absolute or relative error; defaults to relative error except when the expected value is exactly zero:

        sage: RDF(pi)                               # abs tol 1e-5 
        3.14159 
        sage: [10^n for n in [0.0 .. 4]]            # rel tol 2e-4 
        [0.9999, 10.001, 100.01, 999.9, 10001] 

This can be useful when the exact output is subject to rounding error and/or processor floating point arithmetic variation.

---

Related:

• .zero_at(epsilon) methods, to fix noisy (and signed) zeroes; see for example zero_at() method for RDF/CDF vectors #11848.

---

Apply

1. attachment: 10952-tol-bin.2.patch
2. attachment: trac_10952-ref.patch
   to the Sage scripts repository.

Apply

1. attachment: 10952-tol-doc.2.patch
2. attachment: trac_10952-reviewer-docs-v3.patch
   to the Sage library repository.

CC: @jasongrout @kcrisman

Component: doctest coverage

Keywords: sd32 noise noisy doctest failure error tolerance

Author: Robert Bradshaw, Rob Beezer

Reviewer: Jason Grout, Mariah Lenox, William Stein, John Palmieri

Merged: sage-4.7.2.alpha3

Issue created by migration from https://trac.sagemath.org/ticket/10952

[robertwb]

Attachment: noise.patch.gz

[robertwb]

Description changed:

--- 
+++ 
@@ -1,5 +1,6 @@
 
 ```
-sage: 1.5 # tol 1e-3
-1.499999
+sage: print "The answers are", 1.5, 2, 1e-12 # tol 1e-3
+The answers are 1.499999 2.0001 0
 ```
+

[jasongrout]

comment:2

Amazing speed getting this up! Shouldn't "((?:abs(?:solute)?)" be "((?:abs(?:olute)?)" ?

[jasongrout]

comment:3

Nitpicky: This comment is now incorrect:

# following three: used only for parsing only_optional; list of comments 

Also, something should be added to the docs about this new option. Perhaps here: http://sagemath.org/doc/developer/conventions.html#further-conventions-for-automated-testing-of-examples

[robertwb]

apply only this patch to the bin repo

[robertwb]

Attachment: 10952-tol-bin.patch.gz

[robertwb]

comment:4

Attachment: 10952-tol-doc.patch.gz

Thanks for the comments, new patches attached.

[kiwifb]

comment:6

Would this works with test of that kind as well:

File "/usr/share/sage/devel/sage-main/sage/stats/hmm/chmm.pyx", line 579:
    sage: m.viterbi([-2,-1,.1,0.3])
Expected:
    ([1, 1, 1, 0], -9.5660236533785135)
Got:
    ([1, 1, 1, 0], -9.566023653378513)

or even

File "/usr/share/sage/devel/sage-main/sage/categories/examples/semigroups.py", line 32:
    sage: S.some_elements()
Expected:
    [3, 42, 'a', 3.3999999999999999, 'raton laveur']
Got:
    [3, 42, 'a', 3.4, 'raton laveur']

(test failures courtesy of my work on python-2.7)

[robertwb]

comment:7

Yes, it would (though you do have to explicitly annotate the test with a tolerance, and if Python 2.7 consistently gives, e.g., 3.4 then we should change the doctest rather than make it fuzzy.)

[sagetrac-mariah]

comment:8

In trying to review this ticket, I applied 10952-tol-bin.patch in the local/bin directory of a skynet/taurus (x86_64-Linux-nehalem) build of sage-4.7.rc2. I next did 'sage -b'. Yet I get

sage: print "The answers are", 1.5, 2, 1e-12 # tol 1e-3
The answers are 1.50000000000000 2 1.00000000000000e-12

Am I missing something?