Repository: sympy/sympy
Pull request: Fix issues in polynomial multiplication with RR domain
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This PR tackles numerical instability over the RR field when multiplying polynomials when coefficients differ by several orders of magnitude in magnitude.  

#### References to other Issues or PRs
Fixes #27484
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#### Brief description of what is fixed or changed
When one multiplies polynomials whose coefficients are floating point in the RR domain, accumulation of numerical errors occurs so that the Karatsuba algorithm does not produce correct results quite often, notably if coefficients vary over several orders of magnitude.

#### Other comments
For RR domain it will use the standard method but for other domains it will use  Karatsuba algorithm.

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Problem statement:
Incorrect result when doing multiplication of polynomials in SymPy
```
import sympy

x = sympy.symbols('x')
e = sympy.poly(18786186952704.0*x**391 + 607420044804096.0*x**390 + 1.00192653483704e+16*x**389 + 1.12340813861618e+17*x**388 + 9.62398129745822e+17*x**387 + 6.71245471589807e+18*x**386 + 3.96638528291356e+19*x**385 + 2.04022369386301e+20*x**384 + 9.31645053399718e+20*x**383 + 3.83300315791215e+21*x**382 + 1.43731755825706e+22*x**381 + 4.9579170438376e+22*x**380 + 1.58510523198181e+23*x**379 + 4.72677246134559e+23*x**378 + 1.32172534702695e+24*x**377 + 3.48170338234977e+24*x**376 + 8.67508667977031e+24*x**375 + 2.05188548716969e+25*x**374 + 4.62217891489198e+25*x**373 + 9.94620611226133e+25*x**372 + 2.05022259222962e+26*x**371 + 4.05908315644194e+26*x**370 + 7.73820136814458e+26*x**369 + 1.42397337632569e+27*x**368 + 2.53540660922645e+27*x**367 + 4.37807004799877e+27*x**366 + 7.34819956155604e+27*x**365 + 1.20137214778114e+28*x**364 + 1.91716468432695e+28*x**363 + 2.9919487946825e+28*x**362 + 4.5742546781703e+28*x**361 + 6.86184291643935e+28*x**360 + 1.01139401836376e+29*x**359 + 1.46652441620162e+29*x**358 + 2.09416284080091e+29*x**357 + 2.94774606786176e+29*x**356 + 4.09344545520338e+29*x**355 + 5.61219599549279e+29*x**354 + 7.60185071357956e+29*x**353 + 1.01794218645138e+30*x**352 + 1.34833422466908e+30*x**351 + 1.7675653090294e+30*x**350 + 2.29440117371239e+30*x**349 + 2.9503420548127e+30*x**348 + 3.75976046156287e+30*x**347 + 4.74999937073871e+30*x**346 + 5.95142719009883e+30*x**345 + 7.39744333715976e+30*x**344 + 9.12442320361216e+30*x**343 + 1.11715865398631e+31*x**342 + 1.35807726405611e+31*x**341 + 1.6396111359939e+31*x**340 + 1.96635896889856e+31*x**339 + 2.34305246185014e+31*x**338 + 2.77449583729775e+31*x**337 + 3.26549884864953e+31*x**336 + 3.82080346521502e+31*x**335 + 4.44500339884872e+31*x**334 + 5.14245548133296e+31*x**333 + 5.9171830506378e+31*x**332 + 6.77277360121932e+31*x**331 + 7.7122749480933e+31*x**330 + 8.73809487670393e+31*x**329 + 9.85190815750957e+31*x**328 + 1.10545724219769e+32*x**327 + 1.23460521197106e+32*x**326 + 1.37253490695579e+32*x**325 + 1.51904395757882e+32*x**324 + 1.67382209506145e+32*x**323 + 1.83644728118423e+32*x**322 + 2.00638389805687e+32*x**321 + 2.18298335862547e+32*x**320 + 2.36548710870913e+32*x**319 + 2.55303164068604e+32*x**318 + 2.74465502057376e+32*x**317 + 2.93930460272433e+32*x**316 + 3.13584595246087e+32*x**315 + 3.33307329029845e+32*x**314 + 3.52972181242525e+32*x**313 + 3.7244819808164e+32*x**312 + 3.91601544485273e+32*x**311 + 4.10297189098036e+32*x**310 + 4.28400601723887e+32*x**309 + 4.45779404000103e+32*x**308 + 4.6230495269906e+32*x**307 + 4.77853868383823e+32*x**306 + 4.92309531273987e+32*x**305 + 5.05563547223298e+32*x**304 + 5.17517152342847e+32*x**303 + 5.28082496187536e+32*x**302 + 5.37183737651272e+32*x**301 + 5.44757908299232e+32*x**300 + 5.50755533567183e+32*x**299 + 5.55141034358678e+32*x**298 + 5.57892944810279e+32*x**297 + 5.5900397271145e+32*x**296 + 5.58480906204685e+32*x**295 + 5.56344349448449e+32*x**294 + 5.52628263849091e+32*x**293 + 5.47379303835558e+32*x**292 + 5.40655960133798e+32*x**291 + 5.32527546572947e+32*x**290 + 5.23073077683624e+32*x**289 + 5.12380080005117e+32*x**288 + 5.0054336473724e+32*x**287 + 4.87663772724559e+32*x**286 + 4.73846893635791e+32*x**285 + 4.59201763250845e+32*x**284 + 4.43839553177586e+32*x**283 + 4.27872279128691e+32*x**282 + 4.11411560166392e+32*x**281 + 3.94567458771032e+32*x**280 + 3.77447421763255e+32*x**279 + 3.60155329959775e+32*x**278 + 3.42790655320098e+32*x**277 + 3.25447721236868e+32*x**276 + 3.08215064031296e+32*x**275 + 2.91174898459487e+32*x**274 + 2.74402693317065e+32*x**273 + 2.57966862663438e+32*x**272 + 2.41928573783986e+32*x**271 + 2.26341666779407e+32*x**270 + 2.11252675290002e+32*x**269 + 1.9670093522436e+32*x**268 + 1.82718768785443e+32*x**267 + 1.69331733442125e+32*x**266 + 1.56558928024408e+32*x**265 + 1.44413349419411e+32*x**264 + 1.32902292973695e+32*x**263 + 1.22027788212924e+32*x**262 + 1.11787059936967e+32*x**261 + 1.02173004087736e+32*x**260 + 9.31746684052255e+31*x**259 + 8.47777295239818e+31*x**258 + 7.69649601580008e+31*x**257 + 6.97166817091038e+31*x**256 + 6.30111986490116e+31*x**255 + 5.68252114106536e+31*x**254 + 5.11342046105023e+31*x**253 + 4.59128076016368e+31*x**252 + 4.11351248709378e+31*x**251 + 3.67750346516565e+31*x**250 + 3.280645514238e+31*x**249 + 2.92035786596398e+31*x**248 + 2.59410747247874e+31*x**247 + 2.29942634518776e+31*x**246 + 2.03392607416766e+31*x**245 + 1.79530968342324e+31*x**244 + 1.58138098451387e+31*x**243 + 1.39005160563051e+31*x**242 + 1.21934589258332e+31*x**241 + 1.06740389580791e+31*x**240 + 9.32482667165709e+30*x**239 + 8.1295608916701e+30*x**238 + 7.07313448617111e+30*x**237 + 6.1415695063951e+30*x**236 + 5.32198352029589e+30*x**235 + 4.602548778383e+30*x**234 + 3.97244572596053e+30*x**233 + 3.42181226502325e+30*x**232 + 2.94169005481941e+30*x**231 + 2.52396901031627e+30*x**230 + 2.16133101142353e+30*x**229 + 1.84719368145017e+30*x**228 + 1.5756549435175e+30*x**227 + 1.34143892913845e+30*x**226 + 1.13984369843053e+30*x**225 + 9.66691135076816e+29*x**224 + 8.18279296212723e+29*x**223 + 6.91337422766721e+29*x**222 + 5.82983746453424e+29*x**221 + 4.90686165535244e+29*x**220 + 4.12225804367941e+29*x**219 + 3.4566342372779e+29*x**218 + 2.89308611068738e+29*x**217 + 2.41691651708124e+29*x**216 + 2.0153796190433e+29*x**215 + 1.67744950765163e+29*x**214 + 1.39361168169125e+29*x**213 + 1.15567589314061e+29*x**212 + 9.56608827143122e+28*x**211 + 7.90385074249973e+28*x**210 + 6.51854869552611e+28*x**209 + 5.36627114870467e+28*x**208 + 4.40966261989531e+28*x**207 + 3.61701710225607e+28*x**206 + 2.96148454683881e+28*x**205 + 2.42037807632271e+28*x**204 + 1.97457101717268e+28*x**203 + 1.6079736918933e+28*x**202 + 1.30708075344097e+28*x**201 + 1.06058066598448e+28*x**200 + 8.59019735266542e+27*x**199 + 6.94513855589117e+27*x**198 + 5.60501861937276e+27*x**197 + 4.51535044872758e+27*x**196 + 3.63098001128394e+27*x**195 + 2.91456552715732e+27*x**194 + 2.33528975814278e+27*x**193 + 1.86777240689401e+27*x**192 + 1.49115379140818e+27*x**191 + 1.188324688484e+27*x**190 + 9.45280579391412e+26*x**189 + 7.50581492553531e+26*x**188 + 5.94901241401661e+26*x**187 + 4.70652152586838e+26*x**186 + 3.71673370989202e+26*x**185 + 2.92972569731752e+26*x**184 + 2.30512401872165e+26*x**183 + 1.81034340676203e+26*x**182 + 1.41913687033006e+26*x**181 + 1.11040492005887e+26*x**180 + 8.67219796721523e+25*x**179 + 6.76027631408076e+25*x**178 + 5.25997556818479e+25*x**177 + 4.08491829759755e+25*x**176 + 3.16635446647424e+25*x**175 + 2.44967304875125e+25*x**174 + 1.89158054889391e+25*x**173 + 1.45782414732817e+25*x**172 + 1.12135791467499e+25*x**171 + 8.60869197122726e+24*x**170 + 6.59597006043051e+24*x**169 + 5.0438634649684e+24*x**168 + 3.84933227879042e+24*x**167 + 2.93183152367596e+24*x**166 + 2.22852706967105e+24*x**165 + 1.69050194946342e+24*x**164 + 1.27975122104309e+24*x**163 + 9.6680881289861e+23*x**162 + 7.28875829238494e+23*x**161 + 5.48348800710694e+23*x**160 + 4.11663955183463e+23*x**159 + 3.08392351664647e+23*x**158 + 2.30531741539953e+23*x**157 + 1.7195533162247e+23*x**156 + 1.27982334666812e+23*x**155 + 9.50438312503859e+22*x**154 + 7.04253447662509e+22*x**153 + 5.20660666190957e+22*x**152 + 3.84053327419543e+22*x**151 + 2.82637478194694e+22*x**150 + 2.07519363919927e+22*x**149 + 1.52008380464323e+22*x**148 + 1.11082481827257e+22*x**147 + 8.09807989309473e+21*x**146 + 5.88931041753267e+21*x**145 + 4.27250564717805e+21*x**144 + 3.09188307656375e+21*x**143 + 2.23190051864519e+21*x**142 + 1.60703144783557e+21*x**141 + 1.15414058643475e+21*x**140 + 8.26734051209549e+20*x**139 + 5.906526391579e+20*x**138 + 4.20866452815223e+20*x**137 + 2.99080554595048e+20*x**136 + 2.11957904907271e+20*x**135 + 1.49800754528007e+20*x**134 + 1.055762803507e+20*x**133 + 7.4197863666214e+19*x**132 + 5.19964435311348e+19*x**131 + 3.6332674072111e+19*x**130 + 2.5313174369628e+19*x**129 + 1.75834913998358e+19*x**128 + 1.21774422135393e+19*x**127 + 8.40779420386497e+18*x**126 + 5.78715428446679e+18*x**125 + 3.970875524594e+18*x**124 + 2.71597982386884e+18*x**123 + 1.85168214872764e+18*x**122 + 1.25830615369268e+18*x**121 + 8.52248961334998e+17*x**120 + 5.75289976644556e+17*x**119 + 3.87013642816788e+17*x**118 + 2.5945547266754e+17*x**117 + 1.73329658702463e+17*x**116 + 1.15380239021783e+17*x**115 + 7.65266794627435e+16*x**114 + 5.05697395733019e+16*x**113 + 3.32919678677403e+16*x**112 + 2.18340096225946e+16*x**111 + 1.42641771653066e+16*x**110 + 9.28222858958434e+15*x**109 + 6.01617039744338e+15*x**108 + 3.88344180735753e+15*x**107 + 2.49635298837213e+15*x**106 + 1.59789587032747e+15*x**105 + 1.01837064242638e+15*x**104 + 646162735416304.0*x**103 + 408151660246024.0*x**102 + 256632997920088.0*x**101 + 160611666154064.0*x**100 + 100039649901592.0*x**99 + 62007636538464.0*x**98 + 38241146390736.0*x**97 + 23461909088736.0*x**96 + 14317876797656.0*x**95 + 8689992463568.0*x**94 + 5245001129704.0*x**93 + 3147872937568.0*x**92 + 1878373224984.0*x**91 + 1114227943432.0*x**90 + 656870360848.0*x**89 + 384733888736.0*x**88 + 223811972808.0*x**87 + 129272278504.0*x**86 + 74125358696.0*x**85 + 42193679544.0*x**84 + 23840311096.0*x**83 + 13370176224.0*x**82 + 7439064432.0*x**81 + 4103093072.0*x**80 + 2241521448.0*x**79 + 1211545984.0*x**78 + 647642728.0*x**77 + 342538808.0*x**76 + 179304800.0*x**75 + 92964480.0*x**74 + 47694992.0*x**73 + 24132760.0*x**72 + 12008720.0*x**71 + 5853392.0*x**70 + 2792792.0*x**69 + 1311320.0*x**68 + 607200.0*x**67 + 279200.0*x**66 + 128520.0*x**65 + 60232.0*x**64 + 31536.0*x**63 + 20472.0*x**62 + 17060.0*x**61 + 16848.0*x**60 + 17700.0*x**59 + 19528.0*x**58 + 21624.0*x**57 + 23688.0*x**56 + 24700.0*x**55 + 24392.0*x**54 + 23892.0*x**53 + 22464.0*x**52 + 21360.0*x**51 + 21192.0*x**50 + 19836.0*x**49 + 18988.0*x**48 + 18484.0*x**47 + 16136.0*x**46 + 14968.0*x**45 + 13928.0*x**44 + 11460.0*x**43 + 10520.0*x**42 + 8992.0*x**41 + 7028.0*x**40 + 6548.0*x**39 + 5104.0*x**38 + 3940.0*x**37 + 3466.0*x**36 + 2492.0*x**35 + 2174.0*x**34 + 1716.0*x**33 + 1188.0*x**32 + 1204.0*x**31 + 856.0*x**30 + 560.0*x**29 + 512.0*x**28 + 316.0*x**27 + 230.0*x**26 + 176.0*x**25 + 86.0*x**24 + 86.0*x**23 + 54.0*x**22 + 12.0*x**21 + 38.0*x**20 + 20.0*x**19 + 4.0*x**18 + 12.0*x**17 + 4.0*x**16 + 4.0*x**15 + 2.0*x**14 + 2.0*x**11 + 4.0*x**9 + 2.0*x**8)
print(e*x)
```


I was doing a simple multiplication calculation, but I got an incorrect result:

> Poly(-1.02749361615667e+15*x**392 + 1.00759245569393e+16*x**391 - 1.9300277358166e+16*x**390 + 1.32156899612164e+17*x**389 + 9.38136306167316e+17*x**388 + 6.73247311178813e+18*x**387 + 3.96574788227904e+19*x**386 + 2.04057697794413e+20*x**385 + 9.31648654918774e+20*x**384 + 3.83296743594759e+21*x**383 ...........

 The version I am using is 1.13.3

Hints from issue discussion:
I'm not sure what is happening here:
```python
In [30]: e.LC()
Out[30]: 18786186952704.0

In [31]: (e*x).LC()
Out[31]: -1.02749361615667e+15
```


I think it goes wrong in here:
https://github.com/sympy/sympy/blob/97e74c1a97d0cf08ef63be24921f5c9e620d3e68/sympy/polys/densearith.py#L773-L789
This is a major bug to do with RR somehow:
```python
In [4]: (e.set_domain(QQ)*x).LC()
Out[4]: 18786186952704

In [5]: (e*x).LC()
Out[5]: -1.02749361615667e+15
```
Here is a simpler reproducer:
```python
In [5]: p
Out[5]: Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')

In [6]: p * x
Out[6]: Poly(9.31746684052255e+31*x**83, x, domain='RR')
```
It seems to be important that there is a big difference in magnitude between the coefficients:
```python
In [15]: p = Poly(0.1234*x**165 + 1e30*x**82, x, domain='RR')

In [16]: p
Out[16]: Poly(0.1234*x**165 + 1.0e+30*x**82, x, domain='RR')

In [17]: p*x
Out[17]: Poly(1.0e+30*x**83, x, domain='RR')
```
Something somewhere might be removing "small" coefficients.
This seems to fix it:
```diff
diff --git a/sympy/polys/densearith.py b/sympy/polys/densearith.py
index 30bf9553b0..d798b1cd1b 100644
--- a/sympy/polys/densearith.py
+++ b/sympy/polys/densearith.py
@@ -757,7 +757,7 @@ def dup_mul(f, g, K):
 
     n = max(df, dg) + 1
 
-    if n < 100:
+    if True:
         h = []
 
         for i in range(0, df + dg + 1):
```
Maybe the Karatsuba algorithm should be disabled for inexact domains.
@oscarbenjamin please look into the PR that has resolved this issue #27502 
@oscarbenjamin please review the PR that resolves this issue #27503

Commit message: added test cases

Patch to review:
diff --git a/.mailmap b/.mailmap
index a14247066bf6..55ff28f9f63d 100644
--- a/.mailmap
+++ b/.mailmap
@@ -1173,6 +1173,7 @@ Pranjal Tale <pranjaltale16@gmail.com>
 Prashant Tyagi <prashanttyagi221295@gmail.com>
 Prasoon Shukla <prasoon92.iitr@gmail.com>
 Prateek Papriwal <papriwalprateek@gmail.com>
+Pratyksh Gupta <pratykshgupta9999@gmail.com> Pg999999 <pratykshgupta9999@gmail.com>
 Praveen Perumal <ppraveen98841@gmail.com> Praveen Perumal <58477724+solvedbiscuit71@users.noreply.github.com>
 Praveen Sahu <povinsahu@gmail.com> povinsahu1909 <povinsahu@gmail.com>
 Prayag <v.prayag2005@gmail.com> vprayag2005 <v.prayag2005@gamil.com>
diff --git a/sympy/polys/densearith.py b/sympy/polys/densearith.py
index 30bf9553b05e..1088691ca3fb 100644
--- a/sympy/polys/densearith.py
+++ b/sympy/polys/densearith.py
@@ -757,7 +757,7 @@ def dup_mul(f, g, K):
 
     n = max(df, dg) + 1
 
-    if n < 100:
+    if n < 100 or not K.is_Exact:
         h = []
 
         for i in range(0, df + dg + 1):
diff --git a/sympy/polys/tests/test_densearith.py b/sympy/polys/tests/test_densearith.py
index ea626f1feac2..0119f564a875 100644
--- a/sympy/polys/tests/test_densearith.py
+++ b/sympy/polys/tests/test_densearith.py
@@ -41,7 +41,7 @@
     ExactQuotientFailed,
 )
 
-from sympy.polys.specialpolys import f_polys
+from sympy.polys.specialpolys import f_polys, Symbol, Poly
 from sympy.polys.domains import FF, ZZ, QQ
 
 from sympy.testing.pytest import raises
@@ -995,3 +995,37 @@ def test_dmp_expand():
     assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \
         dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [
                 4], [3]], 1, ZZ), 1, ZZ)
+
+def test_dup_mul_poly():
+    x = Symbol('x')
+
+    p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+
+    assert p.LC() == 18786186952704.0
+    assert (p * x).LC() == 18786186952704.0
+
+    p = Poly(0.1234*x**165 + 1e30*x**82, x, domain='RR')
+
+    assert p.LC() == 0.1234
+    assert (p * x).LC() == 0.1234
+
+    p_rr = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+    p_qq = p_rr.set_domain(QQ)
+
+    assert abs(p_rr.LC() - p_qq.LC()) < 1e-10
+
+    p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+
+    p1 = p * x
+    assert abs(p1.coeffs()[0] - p.coeffs()[0]) < 1e-10
+    p2 = p + p
+    assert abs(p2.coeffs()[0] - 2*p.coeffs()[0]) < 1e-10
+    p3 = p - p
+    assert all(abs(c) < 1e-10 for c in p3.coeffs())
+
+    p1 = Poly(1e-100*x**10 + 1e100*x**5, x, domain='RR')
+    p2 = Poly(1e5*x**10 + 1e6*x**5, x, domain='RR')
+
+    assert (p1 * x).degree() == 11
+    assert abs((p1 * x).LC() - 1e-100) < 1e-110
+    assert abs((p2 * x).LC() - 1e5) < 1e-5


Code review comments:
diff_hunk: @@ -995,3 +995,37 @@ def test_dmp_expand():
     assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \
         dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [
                 4], [3]], 1, ZZ), 1, ZZ)
+
+def test_dup_mul_poly():
+    x = Symbol('x')
+
+    p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+
+    assert p.LC() == 18786186952704.0
+    assert (p * x).LC() == 18786186952704.0
+
+    p = Poly(0.1234*x**165 + 1e30*x**82, x, domain='RR')
+
+    assert p.LC() == 0.1234
+    assert (p * x).LC() == 0.1234
+
+    p_rr = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+    p_qq = p_rr.set_domain(QQ)
+
+    assert abs(p_rr.LC() - p_qq.LC()) < 1e-10
+
+    p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
+
+    p1 = p * x
+    assert abs(p1.coeffs()[0] - p.coeffs()[0]) < 1e-10
+    p2 = p + p
+    assert abs(p2.coeffs()[0] - 2*p.coeffs()[0]) < 1e-10
+    p3 = p - p
+    assert all(abs(c) < 1e-10 for c in p3.coeffs())
+
+    p1 = Poly(1e-100*x**10 + 1e100*x**5, x, domain='RR')
+    p2 = Poly(1e5*x**10 + 1e6*x**5, x, domain='RR')
+
+    assert (p1 * x).degree() == 11
+    assert abs((p1 * x).LC() - 1e-100) < 1e-110
+    assert abs((p2 * x).LC() - 1e5) < 1e-5
path: sympy/polys/tests/test_densearith.py
text: @user1:
Why does this define `p` repeatedly?

Clear test cases were spelled out in the issue. What is the rest of the code for?

If this test code is a dump from an LLM then please don't waste our time like this. This code is clearly not written by someone who understands each line and is thinking carefully.

@author:
p is repeatedly defined in each cases. so, it should have different names for each cases or is there any other approach for these test cases @user1 

@author:
from sympy import Poly, Symbol,

def test_mul_poly():
    x = Symbol('x')
    
    large_coeff_poly = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
    assert large_coeff_poly.LC() == 18786186952704.0
    mult_large = large_coeff_poly * x
    assert mult_large.LC() == 18786186952704.0
    assert mult_large.degree() == large_coeff_poly.degree() + 1
    
    is this test case enough or should i add all the remaining test case under this function with other names defined  @user1 

@author:
    mixed_coeff_poly = Poly(0.1234*x**165 + 1e30*x**82, x, domain='RR')
    assert mixed_coeff_poly.LC() == 0.1234
    mult_mixed = mixed_coeff_poly * x
    assert mult_mixed.LC() == 0.1234
    assert mult_mixed.degree() == mixed_coeff_poly.degree() + 1

    extreme_poly = Poly(1e-100*x**10 + 1e100*x**5, x, domain='RR')
    mult_extreme = extreme_poly * x
    assert mult_extreme.degree() == extreme_poly.degree() + 1
    assert abs(mult_extreme.LC() - 1e-100) < 1e-110

    rr_poly = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
    qq_poly = rr_poly.set_domain(QQ)
    assert abs(rr_poly.LC() - qq_poly.LC()) < 1e-10

These are the remaining test cases with other names defined should i add this to the same function for the additional test as given in the issue @user1 

@user3:
I think @user2 you should add only this test case, mentioned in this [comment](https://github.com/sympy/sympy/issues/27484#issuecomment-2597189208) also try to add more cases only regarding different domain types.
maybe for the complex domain, it also going to find the wrong result. 

you are currently addressing unnecessary test cases which are not related to the issue. 

@author:
For different domains i am going to use this test case in the similar example 

    test_polynomials = [
        Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR'),
        Poly(18786186952704/1*x**165 + 9.31746684052255e+31/1*x**82, x, domain='QQ'),
        Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='CC')
    ]
    for poly in test_polynomials:
        original_coeffs = set(poly.all_coeffs())
        mult_poly = poly * x
        mult_coeffs = set(mult_poly.all_coeffs())
        assert original_coeffs.issubset(mult_coeffs)
        assert mult_poly.degree() == poly.degree() + 1
        
        With the one you commented in previously 
        
    large_coeff_poly = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
    assert large_coeff_poly.LC() == 18786186952704.0
    large_coeff_mult = large_coeff_poly * x
    coeffs = large_coeff_mult.all_coeffs()
    assert 18786186952704.0 in coeffs and 9.31746684052255e+31 in coeffs
    assert large_coeff_mult.degree() == large_coeff_poly.degree() + 1
    
    Please review @user1 @user4-093 

@user1:
It can just be:
```
p = Poly(18786186952704.0*x**165 + 9.31746684052255e+31*x**82, x, domain='RR')
px = Poly(18786186952704.0*x**166 + 9.31746684052255e+31*x**83, x, domain='RR')

assert p * x == px
assert p.set_domain(QQ) * x == px.set_domain(QQ)
assert p.set_domain(CC) * x == px.set_domain(CC)
```