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# A New Abstract Domain for the Representation of Mathematically Equivalent Expressions

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## A New Abstract Domain for the Representation ofMathematically Equivalent Expressions

(2)

1,2

1,2

1

2

(3)

2.0

### × +

2.62 [3.14 , 3.15]

8.62

7.61

### Fig. 1. Syntactic tree of expression e.

2.0

× +

2.62 [3.14 , 3.15] +

8.62

+

[3.14 , 3.15]

8.62 +

2.62

7.61

### APEGs are designed to represent, in polynomial size, many expressions that are

(4)

2.0

× +

2.62 [3.14 , 3.15]

+

8.62

+

×

[3.14 , 3.15]

2.0

×

2.0

+

× ×

2.62

2.0 2.0 8.62

7.61

### Fig. 3. Example of product propagation.

2.0

× +

2.62 [3.14 , 3.15] +

8.62

7.61

+ ([3.14 , 3.15], 2.62, 8.62) [3.14 , 3.15]

+

+ (2.62, 8.62)

### all the possible propagations of the subtraction without growing exponentially.

(5)

+

+

2.0

×

— +

— × ×

+

7.61 7.61

2.0

2.0

2.62 8.62 2.62

8.62

+

— —

[3.14 , 3.15] [3.14 , 3.15]

i

(6)

1

2

1

2

1

2

1

2

1

2

?

?

i

1

2

3

4

i=4

i=1

i

1

2

1

2

3

x3∈Y

i=3

i=1

i

i=3

i=1

i

B

B

B

1

n

B

1

n

1

n

B

1

n

1

n

1

n

1

n

01

0m

1

n

01

0m

(7)

1

n

i

0≤i≤n

0

1

1

0

1

2

1

2

(8)

×

a +

d

+

×

a c

×

a b

+

× ×

+ a

b

a d

c ek

e1

+

× ×

+ a

b

a d

c ek

e1

-

×

a b

×

- b

a -

+

a b

ek e1

-

- b

a - +

a b

ek e1

ek

e1 e1 ek

-

×

a b

×

a -

b +

- b

1

k

1

k

01

k00

1

1

k

2

01

k00

k

i=1

i

k0

j=1

0j

k

i=1

i

k0 j=1

0j

### duce only O(2n) boxes, among the exponential number of possible combinations.

(9)

+

+ l1,…,lk

+ l'1,…,l'k'

+

+(l1,…,lk)

ek e1

+

+ l1,…,lk

+ l'1,…,l'k' ek

e1

+ l1,…,lk l'1,…,l'k'

+

+ l1,…,lk l'1,…,l'k'

+

ek e1

+

+(l1,…,lk)

+ l'1,…,l'k' ek

e1

1

k

1

n

01

0k

01

0k

0

0

0

0

0

00

0

00

B

(10)

ni=1

i

i

i

1

2

2

2

3

4

1

2

i

0

i

0

1

01

1

2

01

2

2

02

1

2

1

02

B

1

n

i

i

i+1

1

B

B

B

]

]

B

]

]

x∈X

]

]

B

]

B

(2)

B

c∈C

]

c∈C

B

B

B

0

n

0

n

(11)

1

n

1

n

1

n

1

2

1

2

1

2

1

2

1

2

1

2

1

2

B

0

n

0

m

0

0

1

1

n+m

n+m

i

i+1

i

i+1

i

i+1

i

i+1

1

n

i

j

i

0

j

0

n

0

00

0m

ij

k

0l

j

i

j

i

j

ij

0

k

0l

m

1

2

p∈P

1

n

1

1

n

2

0

01

0m

2

1

(12)

0

0i

01

0m

0i

j

1

n

j

0i

0i

01

0m

0i

3

00

001

00k

00

00j

001

00k

00j

00j

1

n

00j

i

1

n

i

00j

i

00

j

1

n

j

0i

j

1

1

n

2

0

01

0m

3

1

n

01

0m

1

2

3

0

0

1

2

1

2

B

1

2

1

2

0

0

1

2

1

2

0

0

0

0

0

0

1

1

n

2

01

0m

i

0j

1

1

n

2

0

i

n

0

1

1

n

2

l1

lk

l

r1

rk

r

i

1

lj

rk

i

lj

rk

1

2

1

2

B

1

2

1

2

1

2

1

2

B

α

γ

B

(13)

x

x

y

y

x

y

x

y

12

12

x

y

x

y

x

y

x

y

x

y

x

y

x

y

y

x

x

y

x

y

x

y

x

x

y

y

1

2

1

2

1

01

0n

2

001

00n

0i

00j

1

2

1

2

0i

00j

0i

1

00j

2

1

n

(14)

i

j

i

j

i

j

ij

i

j

16

−16

(15)

−12

### the roundoff error on the result of the evaluation of one summation (i.e. for a

(16)

0 100000 200000 300000 400000 500000 600000

0 20 40 60 80 100 120 140

Number of expression

Bound on the absolute error Number of Expressions (Dataset 1)

0 100000 200000 300000 400000 500000 600000

0 20 40 60 80 100 120 140

Number of expression

Bound on the absolute error Number of Optimized Expressions (Dataset 1)

0 500000 1e+06 1.5e+06 2e+06 2.5e+06

0 10 20 30 40 50 60 70

Number of expression

Bound on the absolute error Number of Expressions (Dataset 3)

0 500000 1e+06 1.5e+06 2e+06 2.5e+06

0 10 20 30 40 50 60 70

Number of expression

Bound on the absolute error Number of Optimized Expressions (Dataset 3)

### abstraction box: we perform a greedy association of terms and, in the case of

(17)

0 100000 200000 300000 400000 500000

0 10 20 30 40 50 60

Number of expression

Bound on the absolute error Number of Expressions (Dataset 5)

0 100000 200000 300000 400000 500000

0 10 20 30 40 50 60

Number of expression

Bound on the absolute error Number of Optimized Expressions (Dataset 5)

0 20000 40000 60000 80000 100000 120000

0 20 40 60 80 100 120

Number of expression

Bound on the absolute error Number of Expressions (Dataset 7)

0 20000 40000 60000 80000 100000 120000

0 20 40 60 80 100 120

Number of expression

Bound on the absolute error Number of Optimized Expressions (Dataset 7)

n

n

k=0

k

k

n

n

n

i=1

−12

5

−15

−14

5

−15

−15

### 50% improvement of the numerical accuracy.

(18)

0 2000 4000 6000 8000 10000 12000 14000 16000

2e-15 4e-15 6e-15 8e-15 1e-14 1.2e-14 1.4e-14

Number of Expression

Bound on the absolute error Polynomial (x-1)5 initial schemes

0 2000 4000 6000 8000 10000 12000 14000 16000

2e-15 4e-15 6e-15 8e-15 1e-14 1.2e-14 1.4e-14

Number of Expression

Bound on the absolute error Polynomial (x-1)5 optimized schemes

5

(19)

### Arithmetic Library, 5.0.2 edition, 2011. http://gmplib.org.

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