53
votes
Accepted
Why does Mathematica results differ from C++ results within machine precision?
Something important to keep in mind is that Mathematica parses x / y as
Times[x, Power[y, -1]]
For actual floating point ...
- 25.1k
26
votes
Accepted
Plot of $\sin(x^x)$ is missing from $x=143$
Normally Plot uses machine precision numbers; your $x^x$ expression is hitting the limit of the numbers that can be represented in machine precision right about $x&...
- 58.1k
22
votes
Accepted
Does Mathematica have an equivalent of Python's float.as_integer_ratio?
SetPrecision[] does this:
SetPrecision[0.1, ∞]
3602879701896397/36028797018963968
22
votes
Accepted
What explains the inaccuracies in machine-precision 'integers' with trigonometry or powers?
The result given by python is completely wrong, as are the results given by Mathematica for machine numbers. To get a correct result, you need to use extended precision numbers:
...
- 122k
18
votes
Accepted
Why is Mathematica destroying this graph?
There are several important things about the way computer systems represent real numbers, which most of the time can be blithely ignored, just like the safety of bridges in the United States.
One ...
- 212k
17
votes
Accepted
Why does 1 - Exp[-10.0^12] cause an out-of-memory error?
Actually this is not a duplicate. The prior question is about underflows that require massive bignums to represent at machine precision, and that much is present here as well. So what @J.M notes is ...
- 55k
16
votes
Compilation of Total with compensated summation
Compilation of Total
As pointed out in a comment, it seems that the compensated summation form of Total can't be compiled. You can check this using ...
- 8,578
16
votes
Why does Mathematica results differ from C++ results within machine precision?
Without code and your actual results, this question cannot be answered. Here is one thing that might help: We have a compiler that can compile to C it can show you the code it creates. So why don't ...
- 111k
15
votes
Accepted
Different floating-point numbers equal?
It seems I found my answer in OleksandrR's comment to this question. He says,
Bear in mind Equal applies an extra tolerance in Mathematica. The proper ...
- 7,624
14
votes
Different floating-point numbers equal?
You can define your own "precise equal" using Congruent (≡) (entered as Esc===Esc or ...
- 27.1k
14
votes
Accepted
Really understanding precision
Here are my thoughts:
Q1
Machine numbers: For machine numbers, what you describe is correct. I would just add that you can use InputForm or FullForm to see all the digits if desired:
...
- 122k
13
votes
Accepted
How to calculate accurate answer in Mathematica?
Numerics in Mathematica can be as precise as you like. However, precision comes at price; you pay for it in computation time and in additional coding effort.
In Mathematica there are several ...
- 106k
13
votes
How to detect underflow/overflow (post 11.3)?
Update
I think your updated question shows exactly why the check for machine underflow was removed. In M11.1 we get:
...
- 122k
11
votes
Does Mathematica have an equivalent of Python's float.as_integer_ratio?
Not as clean as J.M.'s method but this seems to give the same result:
0.1 ~RealDigits~ 2 ~FromDigits~ 2
...
- 264k
11
votes
Machine-Precision and Arbitrary Precision
This is not an answer. But I don't believe we should close this question as "easily found in the documentation".
Numerics in Mathematica is an extremely complicated and mostly undocumented subject, ...
10
votes
Accepted
Machine Epsilon
There is a tolerance Internal`$EqualTolerance that is applied when testing Real numbers. If the numbers agree up to the last <...
- 212k
10
votes
Accepted
Performance Tuning - How can I make Mathematica to use less than certain digits number?
I would strongly discourage to do that globally (e.g. by setting $MaxPrecision) because this will -- ironically -- enforce calculations in arbitrary extended ...
- 94.6k
10
votes
Terrible accuracy of DawsonF
Before DawsonF[] became built-in in Mathematica, I used the following method for (small to moderately-sized) real arguments:
...
10
votes
Accepted
Inexact numbers as keys in Association?
Lookups with inexact numbers behave like other lookups: they use hashing. We can check that the results are consistent with what Hash does.
...
- 55k
10
votes
Accepted
Why is Mathematica's default precision only 16 digits?
To give a simple answer to the question in the title:
There is dedicated hardware in your CPU for machine precision computations. In contrast, arbitrary precision computations have to be emulated in ...
- 94.6k
10
votes
Accepted
Why would Evaluate[] in Plot give me warning about precision?
When you use Evaluate, the PDF evaluates to an exponential:
...
- 122k
10
votes
Accepted
FixedPoint not working, despite quick convergence of sequence
$Version
(* "13.0.1 for Mac OS X x86 (64-bit) (January 28, 2022)" *)
The Possible Issues section of the ...
- 119k
9
votes
Numerical underflow for a scaled error function
You need to avoid the underflow or overflow inside the formula $f[x\_]$. And that can be done simply by:
$$ g[x\_] := \frac{2}{\sqrt{\pi}}\ \text{HermiteH}[-1, x].$$
Use this $g[x]$ in place of the ...
- 584
9
votes
Accepted
Unexpected behavior from CoordinateBoundingBoxArray with Into[1] and 0.41
The problem arises from rounding in machine precision. For instance the second-coordinate interval {-0.41, 0.41} leads to this edge case, a one ulp error:
...
- 212k
8
votes
Elegant high precision `log1p`?
Problem
The problem with Log[1. + 1.*^-15] not yielding 1.*^-15 is not due to Log, but to <...
- 212k
8
votes
Elegant high precision `log1p`?
LogLogPlot[{Internal`Log1p[x], Log[1 + x]}, {x, 1*^-17, 1*^-14}]
ps：Of course,maybe you need Internal`Expm1,too.
- 23k
8
votes
Accepted
Can Someone Please Explain Internal`$SameQTolerance?
As the old documentation states:
$EqualTolerance gives the number of decimal digits by which two numbers can disagree and still be considered equal according to Equal.
The default setting is ...
- 264k
8
votes
Accepted
Why does this rounding happen at machine precision?
I think we can find out what's happening by comparing (real digits of) exact values with (real digits of) rounded machine precision values:
...
- 35.4k
8
votes
Accepted
MemberQ[{0.01, 0.05}, (0.01*9*2)/9/2] returns False
Contrary to MemberQ, the function ContainsAny has an option SameTest which can be adjusted. ...
- 94.6k
8
votes
Accepted
How do I convert an inexact number smaller than $MinMachineNumber to machine-precision?
This situation is comparable to
$MinMachineNumber / 2
automatically giving an arbitrary precision result.
...
- 224k
Only top scored, non community-wiki answers of a minimum length are eligible