This change adds Int.ModSqrt to compute modular square-roots via the
standard Tonelli-Shanks algorithm, and the Jacobi function that this and
many other modular-arithmetic algorithms depend on.
This is needed by change 1883 (https://golang.org/cl/1883), to add
support for ANSI-standard compressed encoding of elliptic curve points.
Change-Id: Icc4805001bba0b3cb7200e0b0a7f87b14a9e9439
Reviewed-on: https://go-review.googlesource.com/1886
Reviewed-by: Adam Langley <agl@golang.org>
There was no way to get to the error message before.
Change-Id: I4aa9d3d9f468c33f9996295bafcbed097de0389f
Reviewed-on: https://go-review.googlesource.com/8660
Reviewed-by: Alan Donovan <adonovan@google.com>
Fixed bug that caused Exp(x, y, m) ( i.e. x**y (mod m) ) to return x
instead of x (mod m) when y == 1. See issue page on github for more
details.
Added test case
Fixes#9826
Change-Id: Ibabb58275a20c4231c9474199b7f1c10e54241ce
Reviewed-on: https://go-review.googlesource.com/8409
Reviewed-by: Robert Griesemer <gri@golang.org>
I first prototyped this change in Sept 2011, and I discarded it
because it made no difference in the obvious benchmark loop.
It still makes no difference in the obvious benchmark loop,
but in a less obvious one, doing some extra computation
around the calls to Sqrt, not making the call does have a
significant effect.
benchmark old ns/op new ns/op delta
BenchmarkSqrt 4.56 4.57 +0.22%
BenchmarkSqrtIndirect 4.56 4.56 +0.00%
BenchmarkSqrtGo 69.4 69.4 +0.00%
BenchmarkSqrtPrime 4417 3647 -17.43%
This is a warmup for using hardware expansions for some
calls to 1-line assembly routines in the runtime (for example getg).
Change-Id: Ie66be23f8c09d0f7dc4ddd7ca8a93cfce28f55a4
Reviewed-on: https://go-review.googlesource.com/8356
Reviewed-by: Rob Pike <r@golang.org>
Reviewed-by: Ian Lance Taylor <iant@golang.org>
NaNs make the API more complicated for no real good reasons.
There are few operations that produce NaNs with IEEE arithmetic,
there's no need to copy the behavior. It's easy to test for these
scenarios and avoid them (on the other hand, it's not easy to test
for overflow or underflow, so we want to keep +/-Inf).
Also:
- renamed IsNeg -> Signbit (clearer, especially for x == -0)
- removed IsZero (Sign() == 0 is sufficient and efficient)
- removed IsFinite (now same as !IsInf)
Change-Id: I3f3b4445c325d9bbb1bf46ce2e298a6aeb498e07
Reviewed-on: https://go-review.googlesource.com/8280
Reviewed-by: Alan Donovan <adonovan@google.com>
- fix bounds checks for exponent range of denormalized numbers
- use correct rounding precision for denormalized numbers
- added extra tests
Change-Id: I6be56399afd0d9a603300a2e44b5539e08d6f592
Reviewed-on: https://go-review.googlesource.com/8096
Reviewed-by: Alan Donovan <adonovan@google.com>
To use a pure Go implementation of the low-level arithmetic
functions (when no platform-specific assembly implementations
are available), set the build tag math_big_pure_go.
This will make it easy to vendor the math/big package where no
assembly is available (for instance for use with gc which relies
on 1.4 functionality for now).
Change-Id: I91e17c0fdc568a20ec1512d7c64621241dc60c17
Reviewed-on: https://go-review.googlesource.com/7856
Reviewed-by: Brad Fitzpatrick <bradfitz@golang.org>
Float.Cmp used to return a value < 0, 0, or > 0 depending on how
arguments x, y compared against each other. With the possibility
of NaNs, the result was changed into an Accuracy (to include Undef).
Consequently, Float.Cmp results could still be compared for (in-)
equality with 0, but comparing if < 0 or > 0 would provide the
wrong answer w/o any obvious notice by the compiler.
This change wraps Float.Cmp results into a struct and accessors
are used to access the desired result. This prevents incorrect
use.
Change-Id: I34e6a6c1859251ec99b5cf953e82542025ace56f
Reviewed-on: https://go-review.googlesource.com/7526
Reviewed-by: Rob Pike <r@golang.org>
Also:
- Implemented NewFloat convenience factory function (analogous to
NewInt and NewRat).
- Implemented convenience accessors for Accuracy values returned
from Float.Cmp.
- Added test and example.
Change-Id: I985bb4f86e6def222d4b2505417250d29a39c60e
Reviewed-on: https://go-review.googlesource.com/6970
Reviewed-by: Alan Donovan <adonovan@google.com>
This is a fairly significant _internal_ representation change. Instead
of encoding 0, finite, infinite, and NaN values with special mantissa
and exponent values, a new (1 byte) 'form' field is used (without making
the Float struct bigger). The form field permits simpler and faster
case distinctions. As a side benefit, for zero and non-finite floats,
fewer fields need to be set. Also, the exponent range is not the full
int32 range (in the old format, infExp and nanExp were used to represent
Inf and NaN values and tests for those values sometimes didn't test
for the empty mantissa, so the range was reduced by 2 values).
The correspondence between the old and new fields is as follows.
Old representation:
x neg mant exp
---------------------------------------------------------------
+/-0 sign empty 0
0 < |x| < +Inf sign mantissa exponent
+/-Inf sign empty infExp
NaN false empty nanExp
New representation (- stands for ignored fields):
x neg mant exp form
---------------------------------------------------------------
+/-0 sign - - zero
0 < |x| < +Inf sign mantissa exponent finite
+/-Inf sign - - inf
NaN - - - nan
Client should not be affected by this change.
Change-Id: I7e355894d602ceb23f9ec01da755fe6e0386b101
Reviewed-on: https://go-review.googlesource.com/6870
Reviewed-by: Alan Donovan <adonovan@google.com>
This is a pure code move without any semantic change.
Change-Id: I2c18efc858955d07949b1241e793232f2cf1deb9
Reviewed-on: https://go-review.googlesource.com/6821
Reviewed-by: Alan Donovan <adonovan@google.com>
Shifts are trivially implemented by combining
Float.MantExp and Float.SetMantExp.
Change-Id: Ia2fb49297d8ea7aa7d64c8b1318dc3dc7c8af2f7
Reviewed-on: https://go-review.googlesource.com/6671
Reviewed-by: Alan Donovan <adonovan@google.com>
This change represents Accuracy as a bit pattern rather than
an ordered value; with a new value Undef which is both Below
and Above.
Change-Id: Ibb96294c1417fb3cf2c3cf2374c993b0a4e106b3
Reviewed-on: https://go-review.googlesource.com/6650
Reviewed-by: Alan Donovan <adonovan@google.com>
This change introduces NaNs (for situations like Inf-Inf, etc.).
The implementation is incomplete (the four basic operations produce
a NaN if any of the operands is an Inf or a NaN); and some operations
produce incorrect accuracy for NaN arguments. These are known bugs
which are documented.
Change-Id: Ia88841209e47930681cef19f113e178f92ceeb33
Reviewed-on: https://go-review.googlesource.com/6540
Reviewed-by: Alan Donovan <adonovan@google.com>
MinPrec returns the minimum precision required to represent a Float
without loss of precision. Added test.
Change-Id: I466c8e492dcdd59fae854fc4e71ef9b1add7d817
Reviewed-on: https://go-review.googlesource.com/6010
Reviewed-by: Alan Donovan <adonovan@google.com>
