実行例（３）
CGS-EQ の深作亮也先生（東京理科大）のサイト
http://www.rs.tus.ac.jp/fukasaku/software/CGSQE-20160509/benchmark/computation-time/
http://www.rs.tus.ac.jp/fukasaku/software/CGSQE-20160509/benchmark/input/04/log/
出典
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.642.8545&rep=rep1&type=pdf Example 9
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([ex,ex],[x,y,z],0,0,r11,0 ),
qe( bfpcad(ext( '( x^2+y^2*z+z^3=0 and 3*x^2+3*y^2+z^2-1=0 and x^2+z^2-y^3*(y-1)^3<0 ) ))) );
Evaluation took 34.9400 seconds (34.9500 elapsed) using 4777.359 MB.
(%o1) [[root(387420489*x^36+473513931*x^34+1615049199*x^32-5422961745*x^30
+2179233963*x^28-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8-446456947*x^6
+43657673*x^4-1631864*x^2-40328,1)
< x
and x < root(387420489*x^36+473513931*x^34+1615049199*x^32
-5422961745*x^30+2179233963*x^28
-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8
-446456947*x^6+43657673*x^4-1631864*x^2
-40328,2),true,true]]
(%i2) (qvpeds ([ex,ex],[x,y,z],1,0,r11,0 ),
qe( bfpcad(ext( '( x^2+y^2*z+z^3=0 and 3*x^2+3*y^2+z^2-1=0 and x^2+z^2-y^3*(y-1)^3<0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [16,30,1.0b-3,1.0b-16]
equal(z^3+y^2*z+x^2,0) and equal(z^2+3*y^2+3*x^2-1,0)
and z^2-y^6+3*y^5-3*y^4+y^3+x^2 < 0
["and",z^2-y^6+3*y^5-3*y^4+y^3+x^2 < 0,equal(z^2+3*y^2+3*x^2-1,0),
equal(z^3+y^2*z+x^2,0)]
[[z^2+3*y^2+3*x^2-1],[z^2-y^6+3*y^5-3*y^4+y^3+x^2],[z^3+y^2*z+x^2]]
[[3*y^2+3*x^2-1],[y^6-3*y^5+3*y^4-y^3-x^2],
[y^6-3*y^5+3*y^4-y^3+3*y^2+2*x^2-1],[4*y^6+27*x^4],
[12*y^6+48*x^2*y^4-16*y^4+63*x^4*y^2-42*x^2*y^2+7*y^2+27*x^6-26*x^4+9*x^2-1],
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13-3*x^2*y^12+114*y^12
+18*x^2*y^11-76*y^11-45*x^2*y^10+40*y^10+60*x^2*y^9-16*y^9-49*x^2*y^8
+5*y^8+30*x^2*y^7-y^7+3*x^4*y^6-15*x^2*y^6-9*x^4*y^5+4*x^2*y^5+9*x^4*y^4
-x^2*y^4-3*x^4*y^3+2*x^4*y^2-x^6-x^4]]
[[x,x^2-2,2*x^2-1,3*x^2-1,27*x^6-155*x^4+9*x^2-1,27*x^6-26*x^4+9*x^2-1,
54*x^6-297*x^4+18*x^2-2,108*x^6-837*x^4+36*x^2-4,
729*x^12+729*x^10+1215*x^8-6129*x^6+4185*x^4-504*x^2-8,
387420489*x^36+473513931*x^34+926301663*x^32-4590725139*x^30+3855398877*x^28
-5886383967*x^26+14051402938*x^24-13565732679*x^22
+9151246767*x^20-15472018962*x^18+21618673717*x^16
-15614937665*x^14+6066412333*x^12-1269760998*x^10
+127675066*x^8-3130295*x^6-196075*x^4-2296*x^2-8,
387420489*x^36+473513931*x^34+1615049199*x^32-5422961745*x^30
+2179233963*x^28-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20-126478283472*x^18
+164389796305*x^16-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8-446456947*x^6+43657673*x^4
-1631864*x^2-40328,
387420489*x^36+1162261467*x^34+4649045868*x^32-9125904852*x^30
-4864279473*x^28-75245668308*x^26+228853906389*x^24
-233182493334*x^22+526458209184*x^20-1632367306233*x^18
+2577322981359*x^16-2213846330418*x^14+1103473933785*x^12
-328088633355*x^10+61540990272*x^8-7555532337*x^6
+598204197*x^4-19509336*x^2-512072],
[3*y^2+3*x^2-1,y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3+3*y^2+2*x^2-1,
4*y^6+27*x^4,
12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4],
[z^2+3*y^2+3*x^2-1,z^2-y^6+3*y^5-3*y^4+y^3+x^2,z^3+y^2*z+x^2]]
1 multi-roots: 0.0b0 0.0b0 [3*y^2+3*x^2-1,3*y^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-5.72536737891363b0*y^4+8.963235209558525b-1*y^2]
dist: 0.0b0
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
common-roots: -3.731302026563126b-1 -3.731302026563127b-1
dist: 4.163336342344337b-17
common-roots: -4.492182869390443b-1 -4.492182869390443b-1
dist: 6.938893903907228b-17
common-roots: -4.293461173741275b-1 -4.293461173741276b-1
dist: 4.163336342344337b-17
common-roots: -4.293461173741276b-1 -4.293461173741276b-1
dist: 2.081668171172169b-17
common-roots: 1.242338414099984b0 1.241578377687144b0
dist: 7.600364128400594b-4
1 multi-roots: 0.0b0 0.0b0 [y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 0.0b0
3 multi-roots: 9.999999999999998b-1-1.908195823574488b-16*%i
1.006139616066548b-16*%i+1.0b0 [y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 4.425687238657765b-16
4 multi-roots: 1.006139616066548b-16*%i+1.0b0 9.020562075079397b-17*%i+1.0b0
[y^6-3*y^5+3*y^4-y^3-x^2,y^6-3*y^5+3*y^4-y^3]
dist: 1.040834085586084b-17
1 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
4 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
5 multi-roots: 0.0b0 0.0b0 [4*y^6+27*x^4,4*y^6]
dist: 0.0b0
1 multi-roots: (-3.118675378203367b-11*%i)-7.071067811740978b-1
3.11867695771263b-11*%i-7.071067811989973b-1
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-16*y^4+7*y^2-1]
dist: 6.715982037737266b-11
2 multi-roots: 7.071067811770959b-1-1.273326148703733b-11*%i
1.273326308308289b-11*%i+7.071067811959992b-1
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-16*y^4+7*y^2-1]
dist: 3.171562834653794b-11
1 multi-roots: (-6.025654200204613b-1*%i)-1.891279438990948b-2
(-6.025654199767368b-1*%i)-1.891279438040217b-2
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 4.474615144603693b-11
2 multi-roots: 6.025654209207471b-1*%i-1.891280116685924b-2
6.025654190764511b-1*%i-1.89127876034522b-2
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 1.368822261249517b-8
3 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
4 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
5 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
6 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
7 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
8 multi-roots: 0.0b0 0.0b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 0.0b0
9 multi-roots: 2.717041990286581b-6*%i+9.99993414966278b-1
1.00000093787557b0-7.060909170695758b-6*%i
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 1.233703745313714b-5
10 multi-roots: 1.00000093787557b0-7.060909170695758b-6*%i
4.343867253830464b-6*%i+1.000005647158198b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 1.233881145655348b-5
11 multi-roots: 1.518912661150419b0-6.666097968941921b-1*%i
1.518912927619844b0-6.666098929698847b-1*%i
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 2.832604685158154b-7
12 multi-roots: 6.666098479459857b-1*%i+1.51891279382378b0
6.666098419180176b-1*%i+1.518912794946535b0
[y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+((-3*x^2)+114)*y^12
+(18*x^2-76)*y^11+((-45*x^2)+40)*y^10+(60*x^2-16)*y^9+((-49*x^2)+5)*y^8
+(30*x^2-1)*y^7+(3*x^4-15*x^2)*y^6+((-9*x^4)+4*x^2)*y^5+(9*x^4-x^2)*y^4
-3*x^4*y^3+2*x^4*y^2-x^6-x^4,
y^18-9*y^17+36*y^16-84*y^15+128*y^14-138*y^13+114*y^12-76*y^11+40*y^10
-16*y^9+5*y^8-y^7]
dist: 6.131637614094183b-9
common-roots: 1.000000000000015b0 1.0b0
dist: 1.515454428613339b-14
common-roots: 5.773502691896258b-1 5.773502691896257b-1
dist: 5.551115123125783b-17
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -5.773502691896258b-1 -5.773502691896258b-1
dist: 5.551115123125783b-17
common-roots: 1.242338414100015b0 1.241578377687144b0
dist: 7.600364128707848b-4
common-roots: -4.293461173741275b-1 -4.293461173741275b-1
dist: 1.387778780781446b-17
common-roots: -4.293461173741275b-1 -4.293461173741276b-1
dist: 3.469446951953614b-17
common-roots: -4.492182869390442b-1 -4.492182869390443b-1
dist: 9.020562075079397b-17
common-roots: -3.731302026563125b-1 -3.731302026563127b-1
dist: 2.012279232133096b-16
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
common-roots: -3.990820394214746b-1 -3.990820394214746b-1
dist: 6.938893903907228b-18
1 multi-roots: 0.0b0 0.0b0
[12*y^6+(48*x^2-16)*y^4+(63*x^4-42*x^2+7)*y^2+27*x^6-26*x^4+9*x^2-1,
12*y^6-5.72536737891363b0*y^4+8.963235209558528b-1*y^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [3*y^2+3*x^2-1,3*y^2]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 1.9369471621906b-1 1.9369471621906b-1
dist: 0.0b0
common-roots: -1.9369471621906b-1 -1.9369471621906b-1
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.704199308450861b-1 2.704199308450861b-1
dist: 0.0b0
common-roots: -2.704199308450861b-1 -2.704199308450861b-1
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.755098124531884b-1 2.755098124531884b-1
dist: 0.0b0
common-roots: -2.755098124531884b-1 -2.755098124531884b-1
dist: 0.0b0
common-roots: -4.190704888041301b-1 -4.190704888041305b-1
dist: 3.747002708109903b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.799706718183064b-1 -3.799706718183065b-1
dist: 4.85722573273506b-17
common-roots: 2.804840673490828b-1 2.804840673490828b-1
dist: 0.0b0
common-roots: -2.804840673490828b-1 -2.804840673490828b-1
dist: 0.0b0
common-roots: -4.152590546912595b-1 -4.152590546912597b-1
dist: 1.179611963664229b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.655627214498231b-1 -3.655627214498232b-1
dist: 5.551115123125783b-17
common-roots: 3.184417034569177b-1 3.184417034569177b-1
dist: 0.0b0
common-roots: -3.184417034569177b-1 -3.184417034569177b-1
dist: 0.0b0
common-roots: -3.831359850498015b-1 -3.831359850498017b-1
dist: 1.52655665885959b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 3.509771861657056b-1 3.509771861657056b-1
dist: 0.0b0
common-roots: -3.509771861657056b-1 -3.509771861657056b-1
dist: 0.0b0
common-roots: -3.509771861657056b-1 -3.509771861657057b-1
dist: 1.110223024625157b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -8.666384296627297b-2 -8.666384296627914b-2
dist: 6.168676680573526b-15
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -8.666384296627296b-2 -8.666384296627922b-2
dist: 6.257147577848343b-15
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 3.509771861657057b-1 3.509771861657057b-1
dist: 0.0b0
common-roots: -3.509771861657057b-1 -3.509771861657057b-1
dist: 0.0b0
common-roots: -3.509771861657057b-1 -3.509771861657057b-1
dist: 2.081668171172169b-17
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.655627214498231b-1 -3.655627214498232b-1
dist: 5.551115123125783b-17
common-roots: 3.184417034569177b-1 3.184417034569177b-1
dist: 0.0b0
common-roots: -3.184417034569177b-1 -3.184417034569177b-1
dist: 0.0b0
common-roots: -3.831359850498015b-1 -3.831359850498017b-1
dist: 1.52655665885959b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: -3.799706718183065b-1 -3.799706718183065b-1
dist: 2.081668171172169b-17
common-roots: 2.80484067349083b-1 2.80484067349083b-1
dist: 0.0b0
common-roots: -2.80484067349083b-1 -2.80484067349083b-1
dist: 0.0b0
common-roots: -4.152590546912595b-1 -4.152590546912595b-1
dist: 6.245004513516506b-17
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.755098124531883b-1 2.755098124531883b-1
dist: 0.0b0
common-roots: -2.755098124531883b-1 -2.755098124531883b-1
dist: 0.0b0
common-roots: -4.190704888041303b-1 -4.190704888041304b-1
dist: 1.110223024625157b-16
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 2.704199308450858b-1 2.704199308450858b-1
dist: 0.0b0
common-roots: -2.704199308450858b-1 -2.704199308450858b-1
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [z^2+3*y^2+3*x^2-1,z^2]
dist: 0.0b0
common-roots: 1.936947162190598b-1 1.936947162190598b-1
dist: 0.0b0
common-roots: -1.936947162190598b-1 -1.936947162190598b-1
dist: 0.0b0
[T,F]: [3,1064]
Evaluation took 32.3300 seconds (32.4100 elapsed) using 4531.459 MB.
(%o2) [[root(387420489*x^36+473513931*x^34+1615049199*x^32-5422961745*x^30
+2179233963*x^28-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8-446456947*x^6
+43657673*x^4-1631864*x^2-40328,1)
< x
and x < root(387420489*x^36+473513931*x^34+1615049199*x^32
-5422961745*x^30+2179233963*x^28
-14860773459*x^26+43317737551*x^24
-45925857657*x^22+60356422059*x^20
-126478283472*x^18+164389796305*x^16
-121571730573*x^14+54842719755*x^12
-16059214980*x^10+3210573925*x^8
-446456947*x^6+43657673*x^4-1631864*x^2
-40328,2),true,true]]
実行例（２）
SyNRAC の岩根秀直さん（富士通研究所）のサイト
https://github.com/hiwane/qe_problems/blob/master/problems/exam/manual-fof/tsukuba2010-Ri-1-m.mpl
他の QE ツールの出力との比較
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2019-13.pdf Appendix
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([ex],[a,b,x],0,0,r11,0 ),
qe( bfpcad(ext( '( -1 <= x and x <= 3 and b = x^3/3 - a*x^2/2 and a > 0 ) ))) );
Evaluation took 1.7400 seconds (1.7400 elapsed) using 188.173 MB.
(%o1) [[root(a,1) < a and a < root(a-2,1),
root(6*b+3*a+2,1) <= b and b <= root(2*b+9*a-18,1),true],
[a = root(a-2,1),root(6*b+a^3,1) <= b and b <= root(2*b+9*a-18,1),
true],
[root(a-2,1) < a and a <= root(a-3,1),
root(6*b+a^3,1) <= b and b <= root(b,1),true],
[root(a-3,1) < a,root(2*b+9*a-18,1) <= b and b <= root(b,1),true]]
(%i2) (qvpeds ([ex],[a,b,x],1,0,r11,0 ),
qe( bfpcad(ext( '( -1 <= x and x <= 3 and b = x^3/3 - a*x^2/2 and a > 0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [16,30,1.0b-3,1.0b-16]
x+1 >= 0 and x-3 <= 0 and equal((2*x^3-3*a*x^2-6*b)/6,0) and a > 0
["and",equal((2*x^3-3*a*x^2-6*b)/6,0),x-3 <= 0,x+1 >= 0,a > 0]
[[x-3],[x+1],[2*x^3-3*a*x^2-6*b]]
[[b],[2*b+9*a-18],[6*b+3*a+2],[6*b+a^3]]
[[a-3,a-2,a,a+1,a+6,3*a-7,3*a+2],[b,2*b+9*a-18,6*b+3*a+2,6*b+a^3],
[x-3,x+1,2*x^3-3*a*x^2-6*b]]
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -1.333333333333333b0 -1.333333333333333b0
dist: 0.0b0
common-roots: -1.5b0 -1.5b0
dist: 0.0b0
common-roots: -4.5b0 -4.5b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 9.999999999999999b-1-3.469517903370656b-17*%i
3.469446951953614b-17*%i+1.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-3.0b0*x^2+1.0b0]
dist: 1.309231643767475b-16
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-3.0b0*x^2]
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 6.938893903907228b-17*%i+2.0b0 2.0b0-6.939035806741312b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-6*x^2+8]
dist: 2.61846328753495b-16
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-6*x^2]
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 2.166666666666667b0-4.163490115220211b-17*%i
4.163336342344337b-17*%i+2.166666666666667b0
[2*x^3-3*a*x^2-6*b,2*x^3-6.5b0*x^2+1.01712962962963b1]
dist: 1.861907491918325b-16
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-6.5b0*x^2]
dist: 0.0b0
1 multi-roots: 2.333333333333333b0-5.551280766043446b-17*%i
5.551115123125783b-17*%i+2.333333333333334b0
[2*x^3-3*a*x^2-6*b,2*x^3-7*x^2+1.27037037037037b1]
dist: 1.171009175116753b-15
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-7*x^2]
dist: 0.0b0
1 multi-roots: 1.387778780781446b-17*%i+2.666666666666667b0
2.666666666666667b0-1.387968122423659b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-8.0b0*x^2+1.896296296296296b1]
dist: 1.144396291987333b-16
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-8.0b0*x^2]
dist: 0.0b0
1 multi-roots: 3.0b0-4.163336342344337b-17*%i 4.163123343336605b-17*%i+3.0b0
[2*x^3-3*a*x^2-6*b,2*x^3-9*x^2+27]
dist: 2.371429941323172b-16
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-9*x^2]
dist: 0.0b0
1 multi-roots: 5.550866638069772b-17*%i+3.5b0 3.5b0-5.551115123125783b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-1.05b1*x^2+4.2875b1]
dist: 5.661043993863231b-16
common-roots: 3.0b0 3.0b0
dist: 5.551115123125783b-17
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-1.05b1*x^2]
dist: 0.0b0
[T,F]: [39,253]
Evaluation took 2.5500 seconds (2.5600 elapsed) using 169.662 MB.
(%o2) [[root(a,1) < a and a < root(a-2,1),
root(6*b+3*a+2,1) <= b and b <= root(2*b+9*a-18,1),true],
[a = root(a-2,1),root(6*b+a^3,1) <= b and b <= root(2*b+9*a-18,1),
true],
[root(a-2,1) < a and a <= root(a-3,1),
root(6*b+a^3,1) <= b and b <= root(b,1),true],
[root(a-3,1) < a,root(2*b+9*a-18,1) <= b and b <= root(b,1),true]]
projection set に対する数値解の誤差（その２）
．．．とは言え，sep の評価なしでは，いくら R を小さくしても，前回の冒頭で述べたように，近接根に対応した数値解をカウントしてしまう可能性を排除出来ないかのように見えます．
しかし，我々が扱うのは一般の系ではなく，CAD の projection set であり，特に pscs を含む Collins-Hong タイプの projection（定義は，例えば，D.Wilson. Advances in Cylindrical Algebraic Decomposition の 2.3.1 Collins’Algorithm，2.4.1 Alternative Projection Operators をご参照ください）を利用すれば，sep の情報なしに，重根に対応した数値解全体のみを正確にカウントすることが出来ます．以下，そのことをお話しましょう．
まず，pscs（principal subresultant coefficient sequence）とは，多項式 f(x), g(x) に対して，それらの Sylvester 行列
syl(f1,f2,v):=block([d1:hipow(f1,v),d2:hipow(f2,v),c1,c2,z0,i,k],
c1:makelist(coeff(f1,v,i),i,d1,0,-1),
c2:makelist(coeff(f2,v,i),i,d2,0,-1),z0:makelist(0,d1+d2),
[makelist(firstn(append(firstn(z0,k-1),c1,z0),d1+d2),k,d2),
makelist(firstn(append(firstn(z0,k-1),c2,z0),d1+d2),k,d1)])$
の一部を除いた行列の行列式の列（普通の resultant はその初項です）
pscs(f01,f02,v):=block(
[uratmx:ratmx,s1,s2,d,d1,d2,f1:expand(f01),f2:expand(f02),j],
ratmx:on,[s1,s2]:syl(f1,f2,v),d1:hipow(f1,v),d2:hipow(f2,v),
d:makelist(determinant(apply(matrix,
append(
map(lambda([e],firstn(e,d1+d2-2*j)),
firstn(s1,d2-j)),
map(lambda([e],firstn(e,d1+d2-2*j)),
firstn(s2,d1-j))))),j,0,
min(d1,d2)-1),ratmx:uratmx,d)$
のことで
( pscs の初項から続く 0 の個数 ) = ( gcd( f(x), g(x) ) の次数 )
という性質（Polynomial greatest common divisor - Wikipedia）を持ちます．例えば，上の（素朴な）コードを読み込んで，実行すると
(%i24) pscs( (x-1) * (x-2) , (x-3) * (x-4), x ); (%o24) [12,-4] (%i25) pscs( (x-1) * (x-2)^3 * (x-3)^2 , (x-2)^2 * (x-3)^3 * (x-4), x ); (%o25) [0,0,0,0,12,-4] (%i26) factor(pscs( (x-1) * (x-a)^3 * (x-b)^2 , (x-a)^2 * (x-b)^3 * (x-4), x )); (%o26) [0,0,0,0,-3*(a-4)*(b-1)*(b-a),-(b-a+3)]
といった具合になります．
さて，上の性質を g(x) = f'(x) の場合に用いると
gcd( f(x), f'(x) ) の次数
の値が pscs から特定できる訳ですが，この値が f(x) の
(重根の個数) - (相異なる重根の個数)
であることは，積の微分公式により簡単に判ります．例で確認してみると
(%i31) pscs( f : (x-1) * (x-2)^3 * (x-3)^5 , diff(f,x) , x ); (%o31) [0,0,0,0,0,0,-60,-38]
のように 8 - 2 = 6 となっています．
ところが，この差は前回の冒頭で述べた方法における，重根に対応する数値解の検出カウントに他なりません．例えば，上の重根 2, 2, 2, 3, 3, 3, 3, 3 に対応した数値解 s1, s2, s3, t1, t2, t3, t4, t5 は
2*R >= | s1 - s2 |, 2*R >= | s2 - s3 |,
2*R >= | t1 - t2 |, 2*R >= | t2 - t3 |, 2*R >= | t3 - t4 |, 2*R >= | t4 - t5 |
として，合計 (3-1) + (5-1) = 6 回カウントされ，もし，検出カウントがこれを超えるなら，そこには近接根に対応する数値解のカウントが含まれているので，R を小さくして retry すれば，有限回で近接根に対応する数値解を排除でき，同時に，カウントの長さを見れば，各重根の重複度（各クラスタに属する数値解の個数）も判る，と言う仕組みです．
近接根の識別
さて，r, R を先の条件を満たすようにとるとして，そも，R をどのように定めるのか？という問題があります．
確かに，数値解の誤差 R の系において，互いの距離が 2*R 以下である数値解を全て集めれば，重根に対する数値解は必ずそれに属します．しかし，互いの距離が 4*R 以下である厳密相異根（近接根）が存在すると，それらに対する数値解もメンバーになるかも知れません．
ここでもし，系における厳密相異根間の距離の最小値（根差）sep の値が判るならば，sep > 4*R を満たすように R をとっておけば，相異根 r1, r2 に対する数値解 s1, s2 間の距離は
| s2 - s1 | = | (r2 - r1) - (s1 - r1) + (s2 - r2) | >= sep - R - R > 2*R
のように 2*R より大きくなるので，上記のように集めた数値解全体が，重根に対する数値解全体になります．
では如何にして sep の値を得るかですが，それには resultant を繰り返し用いて１変数化する方法があります．maxima では
(%i7) factor(eliminate([2*c+1,4096*c^3+27*b^4,8*a*c-9*b^2-2*a^3,x^4+a*x^2+b*x+c],[c,b,a]));
Evaluation took 0.0300 seconds (0.0400 elapsed) using 2.309 MB.
(%o7) [33554432*x^8*(2*x^4-27)*(6*x^4-1)^3
*(186624*x^32+2985984*x^28+40621824*x^24+12082176*x^20
+41597280*x^16+64694016*x^12+262151536*x^8
-644544*x^4+729)]
のように必要条件を定める多項式が得られ，この実根
-1.91683..., -0.63894..., 0, 0.63894..., 1.91683...
から，sep >= 0.63894...（実際には c は 0 でないので，1.91683...-0.63894...）と判るので，
R < 0.63894.../4 = 0.15973...
のようにとっておけば，上記の系の近接根に対する数値解間の距離はカウントされず，原理的には，これをすべての多項式の組合せに対して実行し，最小の sep に対して， R を定めればよいことになります．
．．．が，一般に，resultant の係数は爆発し，モニター画面が埋まります．
projection set に対する数値解の誤差（その１）
誤差 r の数値解を係数に代入した方程式系の数値解の誤差 R との関係を考えると
・数値解の代入による係数の摂動は Taylor の定理により r 程度．
・係数の摂動 r に対する厳密 n 重根の摂動は同じく r^{1/n} 程度．
なので，例えば，前回の
4096*c^3+27*b^4（4次）, 8*a*c-9*b^2-2*a^3（3次）, x^4+a*x^2+b*x+c（4次）
の場合，次数の最大値 4 により，r を最悪でも R^{4} 程度まで小さくすればよく，先の例のように %ez : 1.0b-5 とするならば，fpprec は 25 程度にすればよいことになります．
重根に対する数値解の誤差
先の出力の dist は，同じ値に対すると見做した数値解間の複素平面上での距離ですが，単根の場合の（絶対）誤差 10^{-30} に対して，例えば，下の部分では 10^{-11} まで膨らんでいます．
1 multi-roots:
(-6.2414826969557199979633340867b-13*%i)-6.38943104254668599248510111018b-1
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.45826385597869537841530414623b-11
2 multi-roots:
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
7.58333029391685279498189770221b-12*%i-6.38943104242614942415515511763b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.4582638571553690760678343437b-11
これは，一般に n 重根に対する数値解の誤差は，単根に対する数値解の誤差の 1/n 乗程度となる性質によるもので，上の場合は，3 重根なので指数が -30 の 1/3 程度となっている訳です．
この例の出所をもう少し詳しく述べると，上記は，下位の lifting で projection factor から得た
c = root(2*c+1, 1), b = root(4096*c^3+27*b^4, 1), a = root(8*a*c-9*b^2-2*a^3, 1)
の数値解を x^4+a*x^2+b*x+c = 0 に代入した
x^4-2.44948974278317809819728407471b0*x^2-2.08677944009771642211264733221b0*x-5.0b-1 = 0
を解いたもので，厳密解
c = - 1/2 = - 0.5, b = - (4*2^(1/4))/(3^(3/4)) = 2.08677944009771642211264733221..., a = - 6^(1/2) = 2.449489742783178098197284074705...
に対して，数値解は
c = - 5.0b-1, b = - 2.08677944009771642211264733221b0, a = - 2.44948974278317809819728407471b0
なので，c,b,a までの精度は保たれており，最後の
x^4+a*x^2+b*x+c = (x-3^(3/4)/2^(1/4))*(x+1/6^(1/4))^3 = 0
の 3 重根
-1/6^(1/4) = - 0.6389431042462724758553493051605564336...
に対してのみ，対応する数値解が
(-6.2414826969557199979633340867b-13*%i)-6.38943104254668599248510111018b-1, (-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1, 7.58333029391685279498189770221b-12*%i-6.38943104242614942415515511763b-1
のようにばらけますが，相加平均をとると
(-3.21582999003662290927010077574b-32*%i)-6.38943104246272475855349305161b-1
となり，この実部として，30 桁精度のサンプルが得られます．
QE on maxima
以下は，現在作成中の QE ツールの出力例です．
・有効桁数が fpprec の多倍長浮動小数点数を使用．
・２つの数値解間の距離が %ez より小ならばそれらは同じ値の根（重根や共通根）に対応し，%ez 以上ならばそれらは異なる値の根に対応する数値解と見做しています．
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([all], [c,b,a,x],0,h1,r11,0 ),qe( bfpcad(ext( '( x^4+a*x^2+b*x+c>=0 ) ))) );
Evaluation took 11.3900 seconds (11.4100 elapsed) using 1307.972 MB.
(%o1) [[c = root(c,1),b = root(126976*c^3-135*b^4,1),
root(8*a*c-9*b^2-2*a^3,1) <= a,true],
[root(c,1) < c,b < root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true],
[root(c,1) < c,b = root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,1)
<= a,true],
[root(c,1) < c,root(b,1) < b,
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true]]
(%i2) (qvpeds ([all], [c,b,a,x],1,h1,r11,0 ),qe( bfpcad(ext( '( x^4+a*x^2+b*x+c>=0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [30,30,1.0b-5,1.0b-30]
x^4+a*x^2+b*x+c >= 0
x^4+a*x^2+b*x+c >= 0
[[c],
[b,256*c^3-27*b^4,1024*c^3-2187*b^4,1024*c^3+3*b^4,4096*c^3+27*b^4,
126976*c^3-135*b^4],
[a,8*a*c-9*b^2-2*a^3,
256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2],
[x^4+a*x^2+b*x+c]]
1 multi-roots: 0.0b0 0.0b0 [256*c^3-27*b^4,-27*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [256*c^3-27*b^4,-27*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [256*c^3-27*b^4,-27*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [1024*c^3-2187*b^4,-2187*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [1024*c^3-2187*b^4,-2187*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [1024*c^3-2187*b^4,-2187*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [1024*c^3+3*b^4,3*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [1024*c^3+3*b^4,3*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [1024*c^3+3*b^4,3*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [4096*c^3+27*b^4,27*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [4096*c^3+27*b^4,27*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [4096*c^3+27*b^4,27*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [126976*c^3-135*b^4,-135*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [126976*c^3-135*b^4,-135*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [126976*c^3-135*b^4,-135*b^4]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
4.52295331325015833831979417428b-16*%i-2.44948974278317847258028706083b0
(-4.52295331325015213319900815541b-16*%i)-2.44948974278317772381428108859b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-5.0b-1*(16*a^4+6.27069374152493593138504723125b2*a))
-1.74185937264581553649584645312b1*a^3-3.2b1*a^2-5.44b2]
dist: 1.17428054512251951790867706837b-15
common-roots: -2.44948974278317809819728407471b0
-2.44948974278317809819728407471b0
dist: 1.57772181044202361082345713057b-30
1 multi-roots: -1.41421356237309504880168872421b0*%i
(-1.41421356237309504880168872421b0*%i)-3.86941038573562221171322357299b-43
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-8.0b0*a^4)-3.2b1*a^2-3.2b1]
dist: 7.88860905221011805411728660181b-31
2 multi-roots:
1.41421356237309504880261489839b0*%i-2.95418118567534149019656617711b-22
1.41421356237309504880076255003b0*%i+2.9541811856753414902004355875b-22
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-8.0b0*a^4)-3.2b1*a^2-3.2b1]
dist: 1.94429469761416284867540568328b-21
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
4.52295330147550776509999452971b-16*%i-2.44948974278317847258028887801b0
(-4.52295330147551542569579253843b-16*%i)-2.44948974278317772381427927141b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-5.0b-1*(16*a^4+6.27069374152493593138504723125b2*a))
-1.74185937264581553649584645313b1*a^3-3.2b1*a^2
-5.44000000000000000000000000001b2]
dist: 1.17428054562583903487602624948b-15
common-roots: -2.44948974278317809819728407471b0
-2.44948974278317809819728407471b0
dist: 1.57772181044202361082345713057b-30
1 multi-roots: 0.0b0 0.0b0 [8*a*c-9*b^2-2*a^3,-2*a^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [8*a*c-9*b^2-2*a^3,-2*a^3]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
8.16496580927726032731894282802b-1-1.6183536594179216924381257046b-22*%i
1.61835365941792169243742042834b-22*%i+8.16496580927726032732961767002b-1
[8*a*c-9*b^2-2*a^3,(-2*a^3)+4.0b0*a-2.17732421580726942061980806641b0]
dist: 1.11547535241290012202233114443b-21
1 multi-roots:
(-8.47286141428164487440940737737b-20*%i)-1.41421356237309504909493338483b0
8.47286141428164487616706655174b-20*%i-1.41421356237309504850844406359b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
8.0b0*a^4-3.2b1*a^2+3.2b1]
dist: 6.10479709847422546500953724134b-19
2 multi-roots:
1.41421356237309504879618188426b0-1.96859282073234240509586747592b-21*%i
1.96859282073234238751927573217b-21*%i+1.41421356237309504880719556416b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
8.0b0*a^4-3.2b1*a^2+3.2b1]
dist: 1.16962633227785224686827731017b-20
common-roots: 1.41421356237309504880168872421b0
1.41421356237309504880168872421b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -1.41421356237309504880168872421b0
-1.41421356237309504880168872421b0
dist: 0.0b0
1 multi-roots:
8.1649658092772603273200008634b-1-1.30109252983440372998384918995b-22*%i
1.30109252983440372998339457472b-22*%i+8.16496580927726032732855963464b-1
[8*a*c-9*b^2-2*a^3,(-2*a^3)+4.0b0*a-2.17732421580726942061980806641b0]
dist: 8.94560966410596198552565674457b-22
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
(-1.16997742006955532323735088816b-17*%i)-3.35546475144558657978639517931b-1
1.16997742006955532102907767889b-17*%i-3.35546475144558652208708812765b-1
[x^4+a*x^2+b*x+c,
x^4-4.77860936379455500840332276085b0*x^2
-3.05577241698508990342139722064b0*x-5.0b-1]
dist: 2.41004349697564216587243138313b-17
1 multi-roots:
(-2.24047760152893241737421755167b-20*%i)-1.0109852688012821415609910299b0
2.24047760152893269859199915596b-20*%i-1.01098526880128214151321562239b0
[x^4+a*x^2+b*x+c,
x^4-3.55546677120777806279111679242b0*x^2
-3.05577241698508990342139722064b0*x-5.0b-1]
dist: 6.55010344684744407831148213933b-20
1 multi-roots:
2.22935715805258081302014670225b-18*%i-4.14333797761439630281210468432b-1
(-2.22935715805258088784912140497b-18*%i)-4.14333797761439627962993519833b-1
[x^4+a*x^2+b*x+c,
x^4-3.42753994633320075795235578053b0*x^2
-2.55577241698508990342139722064b0*x-5.0b-1]
dist: 5.02536199426577147169718392348b-18
1 multi-roots:
(-7.63751536911083876261312227633b-19*%i)-8.96295167765296091409417104731b-1
7.63751536911083876548501856105b-19*%i-8.96295167765296090286367856344b-1
[x^4+a*x^2+b*x+c,
x^4-3.03243266180629550631765220844b0*x^2
-2.55577241698508990342139722064b0*x-5.0b-1]
dist: 1.89591805066530344113644713169b-18
1 multi-roots:
4.44329956060120776495651206012b-21*%i-4.7445898461131539492366468499b-1
(-4.44329956060120776491378978904b-21*%i)-4.74458984611315394915833267933b-1
[x^4+a*x^2+b*x+c,
x^4-2.89645721365288302103527609247b0*x^2
-2.32127592854140316276702227643b0*x-5.0b-1]
dist: 1.18449456333131606287348159343b-20
1 multi-roots:
(-5.13518395612155029572934523107b-18*%i)-8.19617912427072101369893178291b-1
5.13518395612155030519077584595b-18*%i-8.19617912427072096157128200993b-1
[x^4+a*x^2+b*x+c,
x^4-2.75961902983845013509216731558b0*x^2
-2.32127592854140316276702227643b0*x-5.0b-1]
dist: 1.15175247237145207854523711501b-17
1 multi-roots:
(-6.2414826969557199979633340867b-13*%i)-6.38943104254668599248510111018b-1
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.45826385597869537841530414623b-11
2 multi-roots:
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
7.58333029391685279498189770221b-12*%i-6.38943104242614942415515511763b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.4582638571553690760678343437b-11
1 multi-roots:
6.38943104246144034405208996102b-1-4.422502465774425554602186706b-14*%i
1.33349305972666630723236835579b-13*%i+6.38943104246298402619456223129b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
+2.08677944009771642211264733221b0*x-5.0b-1]
dist: 2.3529170930722626582908906827b-13
2 multi-roots:
1.33349305972666630723236835579b-13*%i+6.38943104246298402619456223129b-1
6.38943104246374990541382696251b-1-8.91242813149223756054934246976b-14*%i
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
+2.08677944009771642211264733221b0*x-5.0b-1]
dist: 2.35287498234870156742355194154b-13
1 multi-roots:
3.72610157565100454685245607331b-20*%i+4.74458984611315394913552119431b-1
4.74458984611315394925945833492b-1-3.72610157565100454690915365651b-20*%i
[x^4+a*x^2+b*x+c,
x^4-2.89645721365288302103527609247b0*x^2
+2.32127592854140316276702227643b0*x-5.0b-1]
dist: 7.55455976814101740503286439967b-20
1 multi-roots:
8.19617912427072098761958869152b-1-3.13943116569295322775812726995b-21*%i
3.13943116569295322775468371035b-21*%i+8.19617912427072098765062510133b-1
[x^4+a*x^2+b*x+c,
x^4-2.75961902983845013509216731558b0*x^2
+2.32127592854140316276702227643b0*x-5.0b-1]
dist: 7.00404879414403669951788837319b-21
1 multi-roots:
2.46936845160507392345254636383b-17*%i+4.14333797761439612007308924107b-1
4.14333797761439646236895064158b-1-2.46936845160507398188288178659b-17*%i
[x^4+a*x^2+b*x+c,
x^4-3.42753994633320075795235578053b0*x^2
+2.55577241698508990342139722064b0*x-5.0b-1]
dist: 6.00897394505265450935583153395b-17
1 multi-roots:
8.96295167765296090779340009951b-1-8.69963119046618331638821004739b-20*%i
8.69963119046618331598771326227b-20*%i+8.96295167765296090916444951125b-1
[x^4+a*x^2+b*x+c,
x^4-3.03243266180629550631765220844b0*x^2
+2.55577241698508990342139722064b0*x-5.0b-1]
dist: 2.21520197802208732404660033187b-19
1 multi-roots:
3.35546475144558654573716194067b-1-3.2321873145433364152600510801b-19*%i
3.23218731454333641635946277158b-19*%i+3.35546475144558655613632136628b-1
[x^4+a*x^2+b*x+c,
x^4-4.77860936379455500840332276085b0*x^2
+3.05577241698508990342139722064b0*x-5.0b-1]
dist: 1.2244617433979626836819761986b-18
1 multi-roots:
1.01098526880128213782781893505b0-2.07015345747187669505738755424b-18*%i
2.07015345747187669030859350914b-18*%i+1.01098526880128214524638771724b0
[x^4+a*x^2+b*x+c,
x^4-3.55546677120777806279111679242b0*x^2
+3.05577241698508990342139722064b0*x-5.0b-1]
dist: 8.49572269592501254899699170041b-18
1 multi-roots:
1.09731695855008129432850901121b-16*%i-6.29960524947436974960114115801b-1
(-1.09731695855008175020983658478b-16*%i)-6.29960524947436189807096491478b-1
[x^4+a*x^2+b*x+c,x^4-1.19055078897614960606377922946b0*x^2-5.0b-1*x]
dist: 8.15248085790726489126442096385b-16
1 multi-roots: 0.0b0 0.0b0 [x^4+a*x^2+b*x+c,x^4-5.0b-1*x^2]
dist: 0.0b0
1 multi-roots:
6.29960524947436582381894689913b-1-5.1915207100556332839659272546b-22*%i
5.19152071005563328395652911378b-22*%i+6.29960524947436582385315917366b-1
[x^4+a*x^2+b*x+c,x^4-1.19055078897614960606377922946b0*x^2+5.0b-1*x]
dist: 3.57531436008650390541071687971b-21
1 multi-roots:
(-8.67770249132643708238919796611b-23*%i)-1.31549862931202979917891670099b0
8.67770249132643708239032323801b-23*%i-1.31549862931202979917841983362b0
[x^4+a*x^2+b*x+c,
x^4-4.90268221512206394838567282523b0*x^2
-3.79286913672985525962281904563b0*x+5.0b-1]
dist: 5.26306169342786570228478616546b-22
1 multi-roots:
2.61198261984724332021478496099b-1-3.73328701456410231125396194552b-23*%i
3.73328701456410231125394316086b-23*%i+2.61198261984724332021514776701b-1
[x^4+a*x^2+b*x+c,
x^4+7.12406849810570318368614385351b0*x^2
-3.79286913672985525962281904563b0*x+5.0b-1]
dist: 8.30135823846948556696946435113b-23
1 multi-roots:
(-8.35793034501348062623930605099b-22*%i)-1.26839633131681513709955065996b0
8.35793034501348062625014797815b-22*%i-1.26839633131681513709478200116b0
[x^4+a*x^2+b*x+c,
x^4-4.51570275957450820467053128978b0*x^2
-3.29286913672985525962281904563b0*x+5.0b-1]
dist: 5.05314820570176695705561472995b-21
1 multi-roots:
3.49188104507532910404015377995b-23*%i+2.98842309414307218065285325202b-1
2.98842309414307218065613316054b-1-3.49188104507532910403992032144b-23*%i
[x^4+a*x^2+b*x+c,
x^4+5.33076228416614959738464811917b0*x^2
-3.29286913672985525962281904563b0*x+5.0b-1]
dist: 3.35343542885119089772093274405b-22
1 multi-roots:
(-5.10955124220135577792387388171b-22*%i)-1.1495332358829680004419098722b0
5.10955124220135577792837906846b-22*%i-1.14953323588296800043902649925b0
[x^4+a*x^2+b*x+c,
x^4-3.58590105561473348013717487389b0*x^2
-2.16812942838935673533957135587b0*x+5.0b-1]
dist: 3.05910773620696643491835445611b-21
1 multi-roots:
4.29760429000264513227806165013b-1-4.9825698827127380563723750383b-17*%i
4.9825698827127377393878597465b-17*%i+4.29760429000264633921325792835b-1
[x^4+a*x^2+b*x+c,
x^4+2.15309806062091933677807113863b0*x^2
-2.16812942838935673533957135587b0*x+5.0b-1]
dist: 1.56516218758945019086233255227b-16
1 multi-roots:
(-1.09877730976256428518192469349b-21*%i)-1.00617122801969079931775349078b0
1.09877730976254185519229728272b-21*%i-1.00617122801969079931174334307b0
[x^4+a*x^2+b*x+c,
x^4-2.54325618867219212419273854599b0*x^2
-1.04338972004885821105632366611b0*x+5.0b-1]
dist: 6.39930635240535273049586094033b-21
1 multi-roots:
6.38943104246272475524586123887b-1-7.84439082707517235056253391037b-20*%i
7.84439082708190432497441062349b-20*%i+6.38943104246272476186112486434b-1
[x^4+a*x^2+b*x+c,x^4-1.04338972004885821105632366611b0*x+5.0b-1]
dist: 6.79875661664383746823339492112b-19
1 multi-roots:
(-4.8300774831537619911512151596b-22*%i)-9.65994523816328465899519279333b-1
4.8300774707834158989233615011b-22*%i-9.65994523816328465899094849168b-1
[x^4+a*x^2+b*x+c,
x^4-2.26361409430139643717899746966b0*x^2
-7.67624175513389459396781893817b-1*x+5.0b-1]
dist: 1.05514307182643029247458493579b-21
1 multi-roots:
1.119137819361247989040796719b-15*%i+6.94963770046457076218883279372b-1
6.94963770046457864444214852857b-1-1.11913781936124711403020335568b-15*%i
[x^4+a*x^2+b*x+c,
x^4-4.13672885217763849890412496005b-1*x^2
-7.67624175513389459396781893817b-1*x+5.0b-1]
dist: 2.37301011549313127333803802423b-15
1 multi-roots:
6.33988972265475161381443097681b-22*%i-9.23416931585132456001273979169b-1
(-6.33988972265475161353799507713b-22*%i)-9.23416931585132455992195460933b-1
[x^4+a*x^2+b*x+c,
x^4-1.9717229852393250817227538327b0*x^2
-4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 9.16663850192252560176095861665b-21
1 multi-roots:
7.49558352273654755133389531399b-1-9.2980021082756567072476690926b-18*%i
9.29800210827546330169426966815b-18*%i+7.49558352273654772974086850138b-1
[x^4+a*x^2+b*x+c,
x^4-7.95576487395121227355033572426b-1*x^2
-4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 2.57701737215882295851533416416b-17
1 multi-roots:
1.03401156272799309188393045594b-19*%i-8.83251837810700396804617510575b-1
(-1.03401156272799304752829508618b-19*%i)-8.83251837810700395251773718947b-1
[x^4+a*x^2+b*x+c,
x^4-1.69948573504594766924564690096b0*x^2
-2.45929315488960353868620060764b-1*x+5.0b-1]
dist: 1.56655387320991285239749179312b-18
1 multi-roots:
7.96303936790265302555727618361b-1-3.40419552049028707429902475843b-21*%i
3.40419552049944803968351960399b-21*%i+7.96303936790265302568553128937b-1
[x^4+a*x^2+b*x+c,
x^4-1.1137806681726386454165880796b0*x^2
-2.45929315488960353868620060764b-1*x+5.0b-1]
dist: 1.45206029523254927419705052618b-20
1 multi-roots:
(-5.70609945371498130990553057283b-21*%i)-8.40896415253714543050874201664b-1
5.70609945371498131018153821252b-21*%i-8.40896415253714543011376750803b-1
[x^4+a*x^2+b*x+c,x^4-1.41421356237309504880168872421b0*x^2+5.0b-1]
dist: 4.11130989883287052898186383085b-20
2 multi-roots:
8.40896415253714543012849282445b-1-6.53339775168399828589095914175b-21*%i
6.53339775168399828561495150206b-21*%i+8.40896415253714543049401670022b-1
[x^4+a*x^2+b*x+c,x^4-1.41421356237309504880168872421b0*x^2+5.0b-1]
dist: 3.88177560186052570543478329165b-20
1 multi-roots:
2.15893555484106906165912452485b-16*%i+8.83251837810700183846822208909b-1
8.83251837810700608209569020613b-1-2.15893555484106797595047500567b-16*%i
[x^4+a*x^2+b*x+c,
x^4-1.69948573504594766924564690097b0*x^2
+2.45929315488960353868620060765b-1*x+5.0b-1]
dist: 6.05412132418777109501344790091b-16
1 multi-roots:
6.37787066397529175282449379547b-20*%i-7.96303936790265302727748521798b-1
(-6.37787066396800919273345251121b-20*%i)-7.963039367902653023965322255b-1
[x^4+a*x^2+b*x+c,
x^4-1.1137806681726386454165880796b0*x^2
+2.45929315488960353868620060765b-1*x+5.0b-1]
dist: 3.54929751662912295409283425277b-19
1 multi-roots:
9.23416931585132422243653899598b-1-6.27472747421841058511382108183b-18*%i
6.2747274742183885879114162131b-18*%i+9.23416931585132489749815540504b-1
[x^4+a*x^2+b*x+c,
x^4-1.9717229852393250817227538327b0*x^2
+4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 6.86627313685598614587092899877b-17
1 multi-roots:
(-1.99636848170413194768633600558b-15*%i)-7.49558352273655235457557488862b-1
1.99636848170413258293302923552b-15*%i-7.49558352273654292649918892678b-1
[x^4+a*x^2+b*x+c,
x^4-7.95576487395121227355033572426b-1*x^2
+4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 4.10254003056179335802065997111b-15
1 multi-roots:
9.65994523816328465899290503781b-1-2.66934379792368063978571497834b-22*%i
2.66934379792368063978568047301b-22*%i+9.65994523816328465899323624718b-1
[x^4+a*x^2+b*x+c,
x^4-2.26361409430139643717899746966b0*x^2
+7.67624175513389459396781893818b-1*x+5.0b-1]
dist: 5.34895175649003922093343053329b-22
1 multi-roots:
(-4.29381502396318990315032909949b-19*%i)-6.9496377004645747104791949945b-1
4.29381502396139055973603979073b-19*%i-6.94963770046457469615178632778b-1
[x^4+a*x^2+b*x+c,
x^4-4.13672885217763849890412496004b-1*x^2
+7.67624175513389459396781893818b-1*x+5.0b-1]
dist: 1.67039524946424430892380307845b-18
1 multi-roots:
1.00617122801969079926553973356b0-1.60075239406241646868279412713b-19*%i
1.60075239406241646862513736191b-19*%i+1.00617122801969079936395710029b0
[x^4+a*x^2+b*x+c,
x^4-2.54325618867219212419273854599b0*x^2
+1.04338972004885821105632366611b0*x+5.0b-1]
dist: 3.34936273279166823298162816124b-19
1 multi-roots:
(-3.77185753862812953984534199095b-24*%i)-6.38943104246272475855369706517b-1
3.77185746499376146262928840184b-24*%i-6.38943104246272475855328903805b-1
[x^4+a*x^2+b*x+c,x^4+1.04338972004885821105632366611b0*x+5.0b-1]
dist: 4.14942038334135943047182928317b-23
1 multi-roots:
1.14953323588296799935017434052b0-1.25168566885221419821833160929b-18*%i
1.25168566885221419738370885239b-18*%i+1.14953323588296800153076203092b0
[x^4+a*x^2+b*x+c,
x^4-3.58590105561473348013717487389b0*x^2
+2.16812942838935673533957135587b0*x+5.0b-1]
dist: 3.31991426545575543023864496728b-18
1 multi-roots:
(-2.58295399767902797007565021885b-19*%i)-4.29760429000264573649957309613b-1
2.58295399767902797030577470367b-19*%i-4.29760429000264573499174648236b-1
[x^4+a*x^2+b*x+c,
x^4+2.15309806062091933677807113863b0*x^2
+2.16812942838935673533957135587b0*x+5.0b-1]
dist: 5.38146323165938018060210022577b-19
1 multi-roots:
1.2683963313168151249994797521b0-1.02128852191271037829509986173b-17*%i
1.02128852191271037157319651058b-17*%i+1.26839633131681514919485290902b0
[x^4+a*x^2+b*x+c,
x^4-4.51570275957450820467053128978b0*x^2
+3.29286913672985525962281904563b0*x+5.0b-1]
dist: 3.16643045115257576716717302982b-17
1 multi-roots:
8.98733613243950412511068357208b-21*%i-2.98842309414307218067753665713b-1
(-8.98733613243950412511912321165b-21*%i)-2.98842309414307218063144975543b-1
[x^4+a*x^2+b*x+c,
x^4+5.33076228416614959738464811916b0*x^2
+3.29286913672985525962281904563b0*x+5.0b-1]
dist: 1.85561005632311052538934339445b-20
1 multi-roots:
1.315498629312029794561387534b0-3.5935851244110502148689984252b-18*%i
3.5935851244110502062082304748b-18*%i+1.31549862931202980379594900061b0
[x^4+a*x^2+b*x+c,
x^4-4.90268221512206394838567282523b0*x^2
+3.79286913672985525962281904563b0*x+5.0b-1]
dist: 1.1701817878693290753327731022b-17
1 multi-roots:
2.54178527994425949722680256301b-23*%i-2.61198261984724332021510649724b-1
(-2.54178527994425949722681244282b-23*%i)-2.61198261984724332021482623075b-1
[x^4+a*x^2+b*x+c,
x^4+7.12406849810570318368614385351b0*x^2
+3.79286913672985525962281904563b0*x+5.0b-1]
dist: 5.80496512960753920779203206589b-23
[T,F]: [92,175]
Evaluation took 13.7400 seconds (13.7800 elapsed) using 1299.854 MB.
(%o2) [[c = root(c,1),b = root(126976*c^3-135*b^4,1),
root(8*a*c-9*b^2-2*a^3,1) <= a,true],
[root(c,1) < c,b < root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true],
[root(c,1) < c,b = root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,1)
<= a,true],
[root(c,1) < c,root(b,1) < b,
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true]]
