Question 3875: Exact Moment Integral Minus a Finite Left Sum

Original question and givens

On all R, f(x)=(25)x+(12). On a=-2≤b=-1, use N=50 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, f(x)=(25)x+(12). On a=-2≤b=-1, use N=50 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-definite-integral

Step-by-step derivation

explicit real polynomial moment or constant integrand model

oriented I(a,b)=H(b)−H(a);do not sort endpoints

interval width := sub(-1, -2)

interval width = 1

partition step := div(1, 50)

partition step = 1/50

left sample0 offset := mul(0, 1/50)

left sample0 offset = 0

left sample0 x := sum(-2, 0)

left sample0 x = -2

left sample0 mx := mul(25, -2)

left sample0 mx = -50

left sample0 value := sum(-50, 12)

left sample0 value = -38

left sample0 rectangle := mul(-38, 1/50)

left sample0 rectangle = -19/25

[-2, -38, -19/25]

finite left-sum term index=0

left sample1 offset := mul(1, 1/50)

left sample1 offset = 1/50

left sample1 x := sum(-2, 1/50)

left sample1 x = -99/50

left sample1 mx := mul(25, -99/50)

left sample1 mx = -99/2

left sample1 value := sum(-99/2, 12)

left sample1 value = -75/2

left sample1 rectangle := mul(-75/2, 1/50)

left sample1 rectangle = -3/4

[-99/50, -75/2, -3/4]

finite left-sum term index=1

left sample2 offset := mul(2, 1/50)

left sample2 offset = 1/25

left sample2 x := sum(-2, 1/25)

left sample2 x = -49/25

left sample2 mx := mul(25, -49/25)

left sample2 mx = -49

left sample2 value := sum(-49, 12)

left sample2 value = -37

left sample2 rectangle := mul(-37, 1/50)

left sample2 rectangle = -37/50

[-49/25, -37, -37/50]

finite left-sum term index=2

left sample3 offset := mul(3, 1/50)

left sample3 offset = 3/50

left sample3 x := sum(-2, 3/50)

left sample3 x = -97/50

left sample3 mx := mul(25, -97/50)

left sample3 mx = -97/2

left sample3 value := sum(-97/2, 12)

left sample3 value = -73/2

left sample3 rectangle := mul(-73/2, 1/50)

left sample3 rectangle = -73/100

[-97/50, -73/2, -73/100]

finite left-sum term index=3

left sample4 offset := mul(4, 1/50)

left sample4 offset = 2/25

left sample4 x := sum(-2, 2/25)

left sample4 x = -48/25

left sample4 mx := mul(25, -48/25)

left sample4 mx = -48

left sample4 value := sum(-48, 12)

left sample4 value = -36

left sample4 rectangle := mul(-36, 1/50)

left sample4 rectangle = -18/25

[-48/25, -36, -18/25]

finite left-sum term index=4

left sample5 offset := mul(5, 1/50)

left sample5 offset = 1/10

left sample5 x := sum(-2, 1/10)

left sample5 x = -19/10

left sample5 mx := mul(25, -19/10)

left sample5 mx = -95/2

left sample5 value := sum(-95/2, 12)

left sample5 value = -71/2

left sample5 rectangle := mul(-71/2, 1/50)

left sample5 rectangle = -71/100

[-19/10, -71/2, -71/100]

finite left-sum term index=5

left sample6 offset := mul(6, 1/50)

left sample6 offset = 3/25

left sample6 x := sum(-2, 3/25)

left sample6 x = -47/25

left sample6 mx := mul(25, -47/25)

left sample6 mx = -47

left sample6 value := sum(-47, 12)

left sample6 value = -35

left sample6 rectangle := mul(-35, 1/50)

left sample6 rectangle = -7/10

[-47/25, -35, -7/10]

finite left-sum term index=6

left sample7 offset := mul(7, 1/50)

left sample7 offset = 7/50

left sample7 x := sum(-2, 7/50)

left sample7 x = -93/50

left sample7 mx := mul(25, -93/50)

left sample7 mx = -93/2

left sample7 value := sum(-93/2, 12)

left sample7 value = -69/2

left sample7 rectangle := mul(-69/2, 1/50)

left sample7 rectangle = -69/100

[-93/50, -69/2, -69/100]

finite left-sum term index=7

left sample8 offset := mul(8, 1/50)

left sample8 offset = 4/25

left sample8 x := sum(-2, 4/25)

left sample8 x = -46/25

left sample8 mx := mul(25, -46/25)

left sample8 mx = -46

left sample8 value := sum(-46, 12)

left sample8 value = -34

left sample8 rectangle := mul(-34, 1/50)

left sample8 rectangle = -17/25

[-46/25, -34, -17/25]

finite left-sum term index=8

left sample9 offset := mul(9, 1/50)

left sample9 offset = 9/50

left sample9 x := sum(-2, 9/50)

left sample9 x = -91/50

left sample9 mx := mul(25, -91/50)

left sample9 mx = -91/2

left sample9 value := sum(-91/2, 12)

left sample9 value = -67/2

left sample9 rectangle := mul(-67/2, 1/50)

left sample9 rectangle = -67/100

[-91/50, -67/2, -67/100]

finite left-sum term index=9

left sample10 offset := mul(10, 1/50)

left sample10 offset = 1/5

left sample10 x := sum(-2, 1/5)

left sample10 x = -9/5

left sample10 mx := mul(25, -9/5)

left sample10 mx = -45

left sample10 value := sum(-45, 12)

left sample10 value = -33

left sample10 rectangle := mul(-33, 1/50)

left sample10 rectangle = -33/50

[-9/5, -33, -33/50]

finite left-sum term index=10

left sample11 offset := mul(11, 1/50)

left sample11 offset = 11/50

left sample11 x := sum(-2, 11/50)

left sample11 x = -89/50

left sample11 mx := mul(25, -89/50)

left sample11 mx = -89/2

left sample11 value := sum(-89/2, 12)

left sample11 value = -65/2

left sample11 rectangle := mul(-65/2, 1/50)

left sample11 rectangle = -13/20

[-89/50, -65/2, -13/20]

finite left-sum term index=11

left sample12 offset := mul(12, 1/50)

left sample12 offset = 6/25

left sample12 x := sum(-2, 6/25)

left sample12 x = -44/25

left sample12 mx := mul(25, -44/25)

left sample12 mx = -44

left sample12 value := sum(-44, 12)

left sample12 value = -32

left sample12 rectangle := mul(-32, 1/50)

left sample12 rectangle = -16/25

[-44/25, -32, -16/25]

finite left-sum term index=12

left sample13 offset := mul(13, 1/50)

left sample13 offset = 13/50

left sample13 x := sum(-2, 13/50)

left sample13 x = -87/50

left sample13 mx := mul(25, -87/50)

left sample13 mx = -87/2

left sample13 value := sum(-87/2, 12)

left sample13 value = -63/2

left sample13 rectangle := mul(-63/2, 1/50)

left sample13 rectangle = -63/100

[-87/50, -63/2, -63/100]

finite left-sum term index=13

left sample14 offset := mul(14, 1/50)

left sample14 offset = 7/25

left sample14 x := sum(-2, 7/25)

left sample14 x = -43/25

left sample14 mx := mul(25, -43/25)

left sample14 mx = -43

left sample14 value := sum(-43, 12)

left sample14 value = -31

left sample14 rectangle := mul(-31, 1/50)

left sample14 rectangle = -31/50

[-43/25, -31, -31/50]

finite left-sum term index=14

left sample15 offset := mul(15, 1/50)

left sample15 offset = 3/10

left sample15 x := sum(-2, 3/10)

left sample15 x = -17/10

left sample15 mx := mul(25, -17/10)

left sample15 mx = -85/2

left sample15 value := sum(-85/2, 12)

left sample15 value = -61/2

left sample15 rectangle := mul(-61/2, 1/50)

left sample15 rectangle = -61/100

[-17/10, -61/2, -61/100]

finite left-sum term index=15

left sample16 offset := mul(16, 1/50)

left sample16 offset = 8/25

left sample16 x := sum(-2, 8/25)

left sample16 x = -42/25

left sample16 mx := mul(25, -42/25)

left sample16 mx = -42

left sample16 value := sum(-42, 12)

left sample16 value = -30

left sample16 rectangle := mul(-30, 1/50)

left sample16 rectangle = -3/5

[-42/25, -30, -3/5]

finite left-sum term index=16

left sample17 offset := mul(17, 1/50)

left sample17 offset = 17/50

left sample17 x := sum(-2, 17/50)

left sample17 x = -83/50

left sample17 mx := mul(25, -83/50)

left sample17 mx = -83/2

left sample17 value := sum(-83/2, 12)

left sample17 value = -59/2

left sample17 rectangle := mul(-59/2, 1/50)

left sample17 rectangle = -59/100

[-83/50, -59/2, -59/100]

finite left-sum term index=17

left sample18 offset := mul(18, 1/50)

left sample18 offset = 9/25

left sample18 x := sum(-2, 9/25)

left sample18 x = -41/25

left sample18 mx := mul(25, -41/25)

left sample18 mx = -41

left sample18 value := sum(-41, 12)

left sample18 value = -29

left sample18 rectangle := mul(-29, 1/50)

left sample18 rectangle = -29/50

[-41/25, -29, -29/50]

finite left-sum term index=18

left sample19 offset := mul(19, 1/50)

left sample19 offset = 19/50

left sample19 x := sum(-2, 19/50)

left sample19 x = -81/50

left sample19 mx := mul(25, -81/50)

left sample19 mx = -81/2

left sample19 value := sum(-81/2, 12)

left sample19 value = -57/2

left sample19 rectangle := mul(-57/2, 1/50)

left sample19 rectangle = -57/100

[-81/50, -57/2, -57/100]

finite left-sum term index=19

left sample20 offset := mul(20, 1/50)

left sample20 offset = 2/5

left sample20 x := sum(-2, 2/5)

left sample20 x = -8/5

left sample20 mx := mul(25, -8/5)

left sample20 mx = -40

left sample20 value := sum(-40, 12)

left sample20 value = -28

left sample20 rectangle := mul(-28, 1/50)

left sample20 rectangle = -14/25

[-8/5, -28, -14/25]

finite left-sum term index=20

left sample21 offset := mul(21, 1/50)

left sample21 offset = 21/50

left sample21 x := sum(-2, 21/50)

left sample21 x = -79/50

left sample21 mx := mul(25, -79/50)

left sample21 mx = -79/2

left sample21 value := sum(-79/2, 12)

left sample21 value = -55/2

left sample21 rectangle := mul(-55/2, 1/50)

left sample21 rectangle = -11/20

[-79/50, -55/2, -11/20]

finite left-sum term index=21

left sample22 offset := mul(22, 1/50)

left sample22 offset = 11/25

left sample22 x := sum(-2, 11/25)

left sample22 x = -39/25

left sample22 mx := mul(25, -39/25)

left sample22 mx = -39

left sample22 value := sum(-39, 12)

left sample22 value = -27

left sample22 rectangle := mul(-27, 1/50)

left sample22 rectangle = -27/50

[-39/25, -27, -27/50]

finite left-sum term index=22

left sample23 offset := mul(23, 1/50)

left sample23 offset = 23/50

left sample23 x := sum(-2, 23/50)

left sample23 x = -77/50

left sample23 mx := mul(25, -77/50)

left sample23 mx = -77/2

left sample23 value := sum(-77/2, 12)

left sample23 value = -53/2

left sample23 rectangle := mul(-53/2, 1/50)

left sample23 rectangle = -53/100

[-77/50, -53/2, -53/100]

finite left-sum term index=23

left sample24 offset := mul(24, 1/50)

left sample24 offset = 12/25

left sample24 x := sum(-2, 12/25)

left sample24 x = -38/25

left sample24 mx := mul(25, -38/25)

left sample24 mx = -38

left sample24 value := sum(-38, 12)

left sample24 value = -26

left sample24 rectangle := mul(-26, 1/50)

left sample24 rectangle = -13/25

[-38/25, -26, -13/25]

finite left-sum term index=24

left sample25 offset := mul(25, 1/50)

left sample25 offset = 1/2

left sample25 x := sum(-2, 1/2)

left sample25 x = -3/2

left sample25 mx := mul(25, -3/2)

left sample25 mx = -75/2

left sample25 value := sum(-75/2, 12)

left sample25 value = -51/2

left sample25 rectangle := mul(-51/2, 1/50)

left sample25 rectangle = -51/100

[-3/2, -51/2, -51/100]

finite left-sum term index=25

left sample26 offset := mul(26, 1/50)

left sample26 offset = 13/25

left sample26 x := sum(-2, 13/25)

left sample26 x = -37/25

left sample26 mx := mul(25, -37/25)

left sample26 mx = -37

left sample26 value := sum(-37, 12)

left sample26 value = -25

left sample26 rectangle := mul(-25, 1/50)

left sample26 rectangle = -1/2

[-37/25, -25, -1/2]

finite left-sum term index=26

left sample27 offset := mul(27, 1/50)

left sample27 offset = 27/50

left sample27 x := sum(-2, 27/50)

left sample27 x = -73/50

left sample27 mx := mul(25, -73/50)

left sample27 mx = -73/2

left sample27 value := sum(-73/2, 12)

left sample27 value = -49/2

left sample27 rectangle := mul(-49/2, 1/50)

left sample27 rectangle = -49/100

[-73/50, -49/2, -49/100]

finite left-sum term index=27

left sample28 offset := mul(28, 1/50)

left sample28 offset = 14/25

left sample28 x := sum(-2, 14/25)

left sample28 x = -36/25

left sample28 mx := mul(25, -36/25)

left sample28 mx = -36

left sample28 value := sum(-36, 12)

left sample28 value = -24

left sample28 rectangle := mul(-24, 1/50)

left sample28 rectangle = -12/25

[-36/25, -24, -12/25]

finite left-sum term index=28

left sample29 offset := mul(29, 1/50)

left sample29 offset = 29/50

left sample29 x := sum(-2, 29/50)

left sample29 x = -71/50

left sample29 mx := mul(25, -71/50)

left sample29 mx = -71/2

left sample29 value := sum(-71/2, 12)

left sample29 value = -47/2

left sample29 rectangle := mul(-47/2, 1/50)

left sample29 rectangle = -47/100

[-71/50, -47/2, -47/100]

finite left-sum term index=29

left sample30 offset := mul(30, 1/50)

left sample30 offset = 3/5

left sample30 x := sum(-2, 3/5)

left sample30 x = -7/5

left sample30 mx := mul(25, -7/5)

left sample30 mx = -35

left sample30 value := sum(-35, 12)

left sample30 value = -23

left sample30 rectangle := mul(-23, 1/50)

left sample30 rectangle = -23/50

[-7/5, -23, -23/50]

finite left-sum term index=30

left sample31 offset := mul(31, 1/50)

left sample31 offset = 31/50

left sample31 x := sum(-2, 31/50)

left sample31 x = -69/50

left sample31 mx := mul(25, -69/50)

left sample31 mx = -69/2

left sample31 value := sum(-69/2, 12)

left sample31 value = -45/2

left sample31 rectangle := mul(-45/2, 1/50)

left sample31 rectangle = -9/20

[-69/50, -45/2, -9/20]

finite left-sum term index=31

left sample32 offset := mul(32, 1/50)

left sample32 offset = 16/25

left sample32 x := sum(-2, 16/25)

left sample32 x = -34/25

left sample32 mx := mul(25, -34/25)

left sample32 mx = -34

left sample32 value := sum(-34, 12)

left sample32 value = -22

left sample32 rectangle := mul(-22, 1/50)

left sample32 rectangle = -11/25

[-34/25, -22, -11/25]

finite left-sum term index=32

left sample33 offset := mul(33, 1/50)

left sample33 offset = 33/50

left sample33 x := sum(-2, 33/50)

left sample33 x = -67/50

left sample33 mx := mul(25, -67/50)

left sample33 mx = -67/2

left sample33 value := sum(-67/2, 12)

left sample33 value = -43/2

left sample33 rectangle := mul(-43/2, 1/50)

left sample33 rectangle = -43/100

[-67/50, -43/2, -43/100]

finite left-sum term index=33

left sample34 offset := mul(34, 1/50)

left sample34 offset = 17/25

left sample34 x := sum(-2, 17/25)

left sample34 x = -33/25

left sample34 mx := mul(25, -33/25)

left sample34 mx = -33

left sample34 value := sum(-33, 12)

left sample34 value = -21

left sample34 rectangle := mul(-21, 1/50)

left sample34 rectangle = -21/50

[-33/25, -21, -21/50]

finite left-sum term index=34

left sample35 offset := mul(35, 1/50)

left sample35 offset = 7/10

left sample35 x := sum(-2, 7/10)

left sample35 x = -13/10

left sample35 mx := mul(25, -13/10)

left sample35 mx = -65/2

left sample35 value := sum(-65/2, 12)

left sample35 value = -41/2

left sample35 rectangle := mul(-41/2, 1/50)

left sample35 rectangle = -41/100

[-13/10, -41/2, -41/100]

finite left-sum term index=35

left sample36 offset := mul(36, 1/50)

left sample36 offset = 18/25

left sample36 x := sum(-2, 18/25)

left sample36 x = -32/25

left sample36 mx := mul(25, -32/25)

left sample36 mx = -32

left sample36 value := sum(-32, 12)

left sample36 value = -20

left sample36 rectangle := mul(-20, 1/50)

left sample36 rectangle = -2/5

[-32/25, -20, -2/5]

finite left-sum term index=36

left sample37 offset := mul(37, 1/50)

left sample37 offset = 37/50

left sample37 x := sum(-2, 37/50)

left sample37 x = -63/50

left sample37 mx := mul(25, -63/50)

left sample37 mx = -63/2

left sample37 value := sum(-63/2, 12)

left sample37 value = -39/2

left sample37 rectangle := mul(-39/2, 1/50)

left sample37 rectangle = -39/100

[-63/50, -39/2, -39/100]

finite left-sum term index=37

left sample38 offset := mul(38, 1/50)

left sample38 offset = 19/25

left sample38 x := sum(-2, 19/25)

left sample38 x = -31/25

left sample38 mx := mul(25, -31/25)

left sample38 mx = -31

left sample38 value := sum(-31, 12)

left sample38 value = -19

left sample38 rectangle := mul(-19, 1/50)

left sample38 rectangle = -19/50

[-31/25, -19, -19/50]

finite left-sum term index=38

left sample39 offset := mul(39, 1/50)

left sample39 offset = 39/50

left sample39 x := sum(-2, 39/50)

left sample39 x = -61/50

left sample39 mx := mul(25, -61/50)

left sample39 mx = -61/2

left sample39 value := sum(-61/2, 12)

left sample39 value = -37/2

left sample39 rectangle := mul(-37/2, 1/50)

left sample39 rectangle = -37/100

[-61/50, -37/2, -37/100]

finite left-sum term index=39

left sample40 offset := mul(40, 1/50)

left sample40 offset = 4/5

left sample40 x := sum(-2, 4/5)

left sample40 x = -6/5

left sample40 mx := mul(25, -6/5)

left sample40 mx = -30

left sample40 value := sum(-30, 12)

left sample40 value = -18

left sample40 rectangle := mul(-18, 1/50)

left sample40 rectangle = -9/25

[-6/5, -18, -9/25]

finite left-sum term index=40

left sample41 offset := mul(41, 1/50)

left sample41 offset = 41/50

left sample41 x := sum(-2, 41/50)

left sample41 x = -59/50

left sample41 mx := mul(25, -59/50)

left sample41 mx = -59/2

left sample41 value := sum(-59/2, 12)

left sample41 value = -35/2

left sample41 rectangle := mul(-35/2, 1/50)

left sample41 rectangle = -7/20

[-59/50, -35/2, -7/20]

finite left-sum term index=41

left sample42 offset := mul(42, 1/50)

left sample42 offset = 21/25

left sample42 x := sum(-2, 21/25)

left sample42 x = -29/25

left sample42 mx := mul(25, -29/25)

left sample42 mx = -29

left sample42 value := sum(-29, 12)

left sample42 value = -17

left sample42 rectangle := mul(-17, 1/50)

left sample42 rectangle = -17/50

[-29/25, -17, -17/50]

finite left-sum term index=42

left sample43 offset := mul(43, 1/50)

left sample43 offset = 43/50

left sample43 x := sum(-2, 43/50)

left sample43 x = -57/50

left sample43 mx := mul(25, -57/50)

left sample43 mx = -57/2

left sample43 value := sum(-57/2, 12)

left sample43 value = -33/2

left sample43 rectangle := mul(-33/2, 1/50)

left sample43 rectangle = -33/100

[-57/50, -33/2, -33/100]

finite left-sum term index=43

left sample44 offset := mul(44, 1/50)

left sample44 offset = 22/25

left sample44 x := sum(-2, 22/25)

left sample44 x = -28/25

left sample44 mx := mul(25, -28/25)

left sample44 mx = -28

left sample44 value := sum(-28, 12)

left sample44 value = -16

left sample44 rectangle := mul(-16, 1/50)

left sample44 rectangle = -8/25

[-28/25, -16, -8/25]

finite left-sum term index=44

left sample45 offset := mul(45, 1/50)

left sample45 offset = 9/10

left sample45 x := sum(-2, 9/10)

left sample45 x = -11/10

left sample45 mx := mul(25, -11/10)

left sample45 mx = -55/2

left sample45 value := sum(-55/2, 12)

left sample45 value = -31/2

left sample45 rectangle := mul(-31/2, 1/50)

left sample45 rectangle = -31/100

[-11/10, -31/2, -31/100]

finite left-sum term index=45

left sample46 offset := mul(46, 1/50)

left sample46 offset = 23/25

left sample46 x := sum(-2, 23/25)

left sample46 x = -27/25

left sample46 mx := mul(25, -27/25)

left sample46 mx = -27

left sample46 value := sum(-27, 12)

left sample46 value = -15

left sample46 rectangle := mul(-15, 1/50)

left sample46 rectangle = -3/10

[-27/25, -15, -3/10]

finite left-sum term index=46

left sample47 offset := mul(47, 1/50)

left sample47 offset = 47/50

left sample47 x := sum(-2, 47/50)

left sample47 x = -53/50

left sample47 mx := mul(25, -53/50)

left sample47 mx = -53/2

left sample47 value := sum(-53/2, 12)

left sample47 value = -29/2

left sample47 rectangle := mul(-29/2, 1/50)

left sample47 rectangle = -29/100

[-53/50, -29/2, -29/100]

finite left-sum term index=47

left sample48 offset := mul(48, 1/50)

left sample48 offset = 24/25

left sample48 x := sum(-2, 24/25)

left sample48 x = -26/25

left sample48 mx := mul(25, -26/25)

left sample48 mx = -26

left sample48 value := sum(-26, 12)

left sample48 value = -14

left sample48 rectangle := mul(-14, 1/50)

left sample48 rectangle = -7/25

[-26/25, -14, -7/25]

finite left-sum term index=48

left sample49 offset := mul(49, 1/50)

left sample49 offset = 49/50

left sample49 x := sum(-2, 49/50)

left sample49 x = -51/50

left sample49 mx := mul(25, -51/50)

left sample49 mx = -51/2

left sample49 value := sum(-51/2, 12)

left sample49 value = -27/2

left sample49 rectangle := mul(-27/2, 1/50)

left sample49 rectangle = -27/100

[-51/50, -27/2, -27/100]

finite left-sum term index=49

finite left sum := sum(-19/25, -3/4, -37/50, -73/100, -18/25, -71/100, -7/10, -69/100, -17/25, -67/100, -33/50, -13/20, -16/25, -63/100, -31/50, -61/100, -3/5, -59/100, -29/50, -57/100, -14/25, -11/20, -27/50, -53/100, -13/25, -51/100, -1/2, -49/100, -12/25, -47/100, -23/50, -9/20, -11/25, -43/100, -21/50, -41/100, -2/5, -39/100, -19/50, -37/100, -9/25, -7/20, -17/50, -33/100, -8/25, -31/100, -3/10, -29/100, -7/25, -27/100)

finite left sum = -103/4

affine integral primitive coefficient1 := div(12, 1)

affine integral primitive coefficient1 = 12

affine integral primitive coefficient2 := div(25, 2)

affine integral primitive coefficient2 = 25/2

[0, 12, 25/2]

H′=the supplied polynomial;integrate from -2 to -1

affine integral H(b) ascending coefficients=[0, 12, 25/2]

evaluate at x=-1

affine integral H(b) Horner multiply before degree1 := mul(25/2, -1)

affine integral H(b) Horner multiply before degree1 = -25/2

affine integral H(b) Horner add degree1 := sum(-25/2, 12)

affine integral H(b) Horner add degree1 = -1/2

affine integral H(b) Horner multiply before degree0 := mul(-1/2, -1)

affine integral H(b) Horner multiply before degree0 = 1/2

affine integral H(b) Horner add degree0 := sum(1/2, 0)

affine integral H(b) Horner add degree0 = 1/2

affine integral H(a) ascending coefficients=[0, 12, 25/2]

evaluate at x=-2

affine integral H(a) Horner multiply before degree1 := mul(25/2, -2)

affine integral H(a) Horner multiply before degree1 = -25

affine integral H(a) Horner add degree1 := sum(-25, 12)

affine integral H(a) Horner add degree1 = -13

affine integral H(a) Horner multiply before degree0 := mul(-13, -2)

affine integral H(a) Horner multiply before degree0 = 26

affine integral H(a) Horner add degree0 := sum(26, 0)

affine integral H(a) Horner add degree0 = 26

affine integral oriented moment := sub(1/2, 26)

affine integral oriented moment = -51/2

integral minus left sum := sub(-51/2, -103/4)

integral minus left sum = 1/4

I=-51/2;S_N=-103/4;N=50

signed finite-sum difference=1/4

Reverse checks

• S320 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S319 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.
• S318 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.
• S317 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S316 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S315 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S314 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S313 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S312 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S311 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S310 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S309 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S308 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S307 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S306 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.
• S305 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.
• S304 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.
• S303 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S302 · premise_and_result_audit — Recompute left sample49 rectangle with exact rational arithmetic, not rounded decimals.
• S301 · premise_and_result_audit — Recompute left sample49 value with exact rational arithmetic, not rounded decimals.
• S300 · premise_and_result_audit — Recompute left sample49 mx with exact rational arithmetic, not rounded decimals.
• S299 · premise_and_result_audit — Recompute left sample49 x with exact rational arithmetic, not rounded decimals.
• S298 · premise_and_result_audit — Recompute left sample49 offset with exact rational arithmetic, not rounded decimals.
• S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S296 · premise_and_result_audit — Recompute left sample48 rectangle with exact rational arithmetic, not rounded decimals.
• S295 · premise_and_result_audit — Recompute left sample48 value with exact rational arithmetic, not rounded decimals.
• S294 · premise_and_result_audit — Recompute left sample48 mx with exact rational arithmetic, not rounded decimals.
• S293 · premise_and_result_audit — Recompute left sample48 x with exact rational arithmetic, not rounded decimals.
• S292 · premise_and_result_audit — Recompute left sample48 offset with exact rational arithmetic, not rounded decimals.
• S291 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S290 · premise_and_result_audit — Recompute left sample47 rectangle with exact rational arithmetic, not rounded decimals.
• S289 · premise_and_result_audit — Recompute left sample47 value with exact rational arithmetic, not rounded decimals.
• S288 · premise_and_result_audit — Recompute left sample47 mx with exact rational arithmetic, not rounded decimals.
• S287 · premise_and_result_audit — Recompute left sample47 x with exact rational arithmetic, not rounded decimals.
• S286 · premise_and_result_audit — Recompute left sample47 offset with exact rational arithmetic, not rounded decimals.
• S285 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S284 · premise_and_result_audit — Recompute left sample46 rectangle with exact rational arithmetic, not rounded decimals.
• S283 · premise_and_result_audit — Recompute left sample46 value with exact rational arithmetic, not rounded decimals.
• S282 · premise_and_result_audit — Recompute left sample46 mx with exact rational arithmetic, not rounded decimals.
• S281 · premise_and_result_audit — Recompute left sample46 x with exact rational arithmetic, not rounded decimals.
• S280 · premise_and_result_audit — Recompute left sample46 offset with exact rational arithmetic, not rounded decimals.
• S279 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S278 · premise_and_result_audit — Recompute left sample45 rectangle with exact rational arithmetic, not rounded decimals.
• S277 · premise_and_result_audit — Recompute left sample45 value with exact rational arithmetic, not rounded decimals.
• S276 · premise_and_result_audit — Recompute left sample45 mx with exact rational arithmetic, not rounded decimals.
• S275 · premise_and_result_audit — Recompute left sample45 x with exact rational arithmetic, not rounded decimals.
• S274 · premise_and_result_audit — Recompute left sample45 offset with exact rational arithmetic, not rounded decimals.
• S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S272 · premise_and_result_audit — Recompute left sample44 rectangle with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Recompute left sample44 value with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Recompute left sample44 mx with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Recompute left sample44 x with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Recompute left sample44 offset with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S266 · premise_and_result_audit — Recompute left sample43 rectangle with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute left sample43 value with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute left sample43 mx with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute left sample43 x with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute left sample43 offset with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S260 · premise_and_result_audit — Recompute left sample42 rectangle with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Recompute left sample42 value with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Recompute left sample42 mx with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Recompute left sample42 x with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Recompute left sample42 offset with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S254 · premise_and_result_audit — Recompute left sample41 rectangle with exact rational arithmetic, not rounded decimals.
• S253 · premise_and_result_audit — Recompute left sample41 value with exact rational arithmetic, not rounded decimals.
• S252 · premise_and_result_audit — Recompute left sample41 mx with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute left sample41 x with exact rational arithmetic, not rounded decimals.
• S250 · premise_and_result_audit — Recompute left sample41 offset with exact rational arithmetic, not rounded decimals.
• S249 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S248 · premise_and_result_audit — Recompute left sample40 rectangle with exact rational arithmetic, not rounded decimals.
• S247 · premise_and_result_audit — Recompute left sample40 value with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Recompute left sample40 mx with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute left sample40 x with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute left sample40 offset with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S242 · premise_and_result_audit — Recompute left sample39 rectangle with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute left sample39 value with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Recompute left sample39 mx with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Recompute left sample39 x with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Recompute left sample39 offset with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S236 · premise_and_result_audit — Recompute left sample38 rectangle with exact rational arithmetic, not rounded decimals.
• S235 · premise_and_result_audit — Recompute left sample38 value with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute left sample38 mx with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Recompute left sample38 x with exact rational arithmetic, not rounded decimals.
• S232 · premise_and_result_audit — Recompute left sample38 offset with exact rational arithmetic, not rounded decimals.
• S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S230 · premise_and_result_audit — Recompute left sample37 rectangle with exact rational arithmetic, not rounded decimals.
• S229 · premise_and_result_audit — Recompute left sample37 value with exact rational arithmetic, not rounded decimals.
• S228 · premise_and_result_audit — Recompute left sample37 mx with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Recompute left sample37 x with exact rational arithmetic, not rounded decimals.
• S226 · premise_and_result_audit — Recompute left sample37 offset with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S224 · premise_and_result_audit — Recompute left sample36 rectangle with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute left sample36 value with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute left sample36 mx with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute left sample36 x with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute left sample36 offset with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S218 · premise_and_result_audit — Recompute left sample35 rectangle with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute left sample35 value with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute left sample35 mx with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Recompute left sample35 x with exact rational arithmetic, not rounded decimals.
• S214 · premise_and_result_audit — Recompute left sample35 offset with exact rational arithmetic, not rounded decimals.
• S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S212 · premise_and_result_audit — Recompute left sample34 rectangle with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Recompute left sample34 value with exact rational arithmetic, not rounded decimals.
• S210 · premise_and_result_audit — Recompute left sample34 mx with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Recompute left sample34 x with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute left sample34 offset with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S206 · premise_and_result_audit — Recompute left sample33 rectangle with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Recompute left sample33 value with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute left sample33 mx with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute left sample33 x with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute left sample33 offset with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S200 · premise_and_result_audit — Recompute left sample32 rectangle with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute left sample32 value with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute left sample32 mx with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute left sample32 x with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Recompute left sample32 offset with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S194 · premise_and_result_audit — Recompute left sample31 rectangle with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute left sample31 value with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Recompute left sample31 mx with exact rational arithmetic, not rounded decimals.
• S191 · premise_and_result_audit — Recompute left sample31 x with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Recompute left sample31 offset with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S188 · premise_and_result_audit — Recompute left sample30 rectangle with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Recompute left sample30 value with exact rational arithmetic, not rounded decimals.
• S186 · premise_and_result_audit — Recompute left sample30 mx with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Recompute left sample30 x with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Recompute left sample30 offset with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S182 · premise_and_result_audit — Recompute left sample29 rectangle with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute left sample29 value with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute left sample29 mx with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute left sample29 x with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute left sample29 offset with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S176 · premise_and_result_audit — Recompute left sample28 rectangle with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute left sample28 value with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute left sample28 mx with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Recompute left sample28 x with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute left sample28 offset with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S170 · premise_and_result_audit — Recompute left sample27 rectangle with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute left sample27 value with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Recompute left sample27 mx with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Recompute left sample27 x with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute left sample27 offset with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S164 · premise_and_result_audit — Recompute left sample26 rectangle with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute left sample26 value with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute left sample26 mx with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute left sample26 x with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute left sample26 offset with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S158 · premise_and_result_audit — Recompute left sample25 rectangle with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute left sample25 value with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute left sample25 mx with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute left sample25 x with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute left sample25 offset with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Recompute left sample24 rectangle with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute left sample24 value with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute left sample24 mx with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute left sample24 x with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute left sample24 offset with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S146 · premise_and_result_audit — Recompute left sample23 rectangle with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute left sample23 value with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute left sample23 mx with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute left sample23 x with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute left sample23 offset with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S140 · premise_and_result_audit — Recompute left sample22 rectangle with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute left sample22 value with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute left sample22 mx with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute left sample22 x with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute left sample22 offset with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S134 · premise_and_result_audit — Recompute left sample21 rectangle with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute left sample21 value with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute left sample21 mx with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute left sample21 x with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute left sample21 offset with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Recompute left sample20 rectangle with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute left sample20 value with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute left sample20 mx with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute left sample20 x with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute left sample20 offset with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Recompute left sample19 rectangle with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute left sample19 value with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute left sample19 mx with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute left sample19 x with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute left sample19 offset with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Recompute left sample18 rectangle with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute left sample18 value with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute left sample18 mx with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute left sample18 x with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute left sample18 offset with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S110 · premise_and_result_audit — Recompute left sample17 rectangle with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute left sample17 value with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute left sample17 mx with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute left sample17 x with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute left sample17 offset with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S104 · premise_and_result_audit — Recompute left sample16 rectangle with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute left sample16 value with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute left sample16 mx with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute left sample16 x with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute left sample16 offset with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Recompute left sample15 rectangle with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute left sample15 value with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute left sample15 mx with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute left sample15 x with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute left sample15 offset with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Recompute left sample14 rectangle with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute left sample14 value with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute left sample14 mx with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute left sample14 x with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute left sample14 offset with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Recompute left sample13 rectangle with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute left sample13 value with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute left sample13 mx with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute left sample13 x with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute left sample13 offset with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Recompute left sample12 rectangle with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute left sample12 value with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute left sample12 mx with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute left sample12 x with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute left sample12 offset with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute left sample11 value with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute left sample11 mx with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute left sample11 x with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute left sample11 offset with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute left sample10 value with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute left sample10 mx with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute left sample10 x with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute left sample10 offset with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute left sample9 value with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute left sample9 mx with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute left sample9 x with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute left sample9 offset with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute left sample8 value with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute left sample8 mx with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute left sample8 x with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute left sample8 offset with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute left sample7 rectangle with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute left sample7 value with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute left sample7 mx with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute left sample7 x with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute left sample7 offset with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute left sample6 value with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute left sample6 mx with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute left sample6 x with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute left sample6 offset with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute left sample5 value with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute left sample5 mx with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute left sample5 x with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute left sample5 offset with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute left sample4 value with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute left sample4 mx with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute left sample4 x with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute left sample4 offset with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute left sample3 value with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute left sample3 mx with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute left sample3 x with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute left sample3 offset with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute left sample2 value with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute left sample2 mx with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute left sample2 x with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute left sample2 offset with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute left sample1 value with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute left sample1 mx with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute left sample1 x with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute left sample1 offset with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute left sample0 value with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute left sample0 mx with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute left sample0 x with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute left sample0 offset with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute partition step with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute interval width with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

1/4

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.