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### What are the differences between RealDoubleField() and RealField(53) ?

Hi,

This question is related to question 2402 (still open!) and tries to collect differences between RDF=RealDoubleField() and RR=RealField(53). These are two floating point real number fields with both 53 bits of precision. The first one comes from the processor floating-point arithmetic, the second one is "emulated" by mpfr. They are assumed to follow the same rounding standards (to the nearest, according to the sagebook, but i may be wrong).

However, we can see some differences between them:

sage: RDF(1/10)*10 == RDF(1)
False
sage: RDF(1/10)*10 - RDF(1)
-1.11022302463e-16

sage: RR(1/10)*10 == RR(1)
True
sage: sage: RR(1/10)*10 - RR(1)
0.000000000000000


Could you explain that ?

There are also some specificities on which field should be used for some methods.

• For example, it seems that the eignevalues are not well computed on RR, but are correctly computed on RDF (see trac #13660). What is the reason for that ?

• Also, it seems that when dealing with huge matrices, the fast atlas library in only used when entries are in RDF, not in RR (see trac #10815).

Are there other difference that should be known between both implementations of floating point numbers in Sage ?

Thierry

### What are the differences between RealDoubleField() and RealField(53) ?

Hi,

This question is related to question 2402 (still open!) and tries to collect differences between RDF=RealDoubleField() and RR=RealField(53). These are two floating point real number fields with both 53 bits of precision. The first one comes from the processor floating-point arithmetic, the second one is "emulated" by mpfr. They are assumed to follow the same rounding standards (to the nearest, according to the sagebook, but i may be wrong).

However, we can see some differences between them:

sage: RDF(1/10)*10 == RDF(1)
False
sage: RDF(1/10)*10 - RDF(1)
-1.11022302463e-16

sage: RR(1/10)*10 == RR(1)
True
sage: sage: RR(1/10)*10 - RR(1)
0.000000000000000


Could you explain that ?? EDIT: this was a bug and it is now fixed, see trac ticket 14416.

There are also some specificities on which field should be used for some methods.

• For example, it seems that the eignevalues are not well computed on RR, but are correctly computed on RDF (see trac #13660). What is the reason for that ?

• Also, it seems that when dealing with huge matrices, the fast atlas library in only used when entries are in RDF, not in RR (see trac #10815).

Are there other difference that should be known between both implementations of floating point numbers in Sage ?

Thierry