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Testing%20COSY

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... sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, log, ... and ASIN. 2. The COS and the SIN errors only occur at. order 17 of dimension 1 or dimension 2. ... – PowerPoint PPT presentation

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Title: Testing%20COSY

1
Testing COSYs Taylor Model Arithmetic

Jun Yu and George F. CorlissMarquette
University, Dec. 16, 2002
Supported by Martin Berz, MSU

2
What is correct?

Domain X
Expression f
COSYs Taylor model expression of f TM_F
COSYs interval expression of f INV_F
For all x ? X, f (x) ? TM_F (x).
For all x ? X, INV_F (x) intersects
TM_F(x) .

3
Goal

Find test cases when x ? X, f (x) ? TM_F (x)
Or find test cases when x ? X, INV_F (x)
and TM_F (x) are disjoint.

4
Verification Process

Process 1 Use COSYs interval arithmetic to
verify COSYs Taylor model
arithmetic.
Process 2 Use Maple as a referee.

5
Verification process 1

Evaluate the function f in Taylor model
arithmetic over a domain.E.g., f COS (3.14
1.57 X), domain -1, 1.
Construct the Taylor model expression of f
(TM_EXPR) in COSYTM_EXPR COS(-3.14 TM_ONE
(1.57 TM_ONE) TM_INDEP)
Construct the interval expression of f (INL_EXPR)
in COSY.IVL_EXPR COS(INTV(-3.14, -3.14)
INTV(1.57, 1.57) VAR1)
Pick a point , say A, inside the domain and
convert it to a tight interval A.
Evaluate the polynomial part of TM_EXPR on A
and add the remainder bound.
Evaluate IVL_EXPR on A.
Compare the results of 5 and 6. If the intervals
are disjoint, there is an error.

6
Verification process 2

Follow the first four steps of verification
process 1.
Output the information about Taylor model
expression in binary form.
In Maple, we code the function f.
Read the binary information of the Taylor model
expression.
Evaluate f at the challenge point. If the point
result of f is not in the interval result of
Taylor model expression, we have a violation of
containment.

7
Testing Scope

Test cases are designed to evaluate the COSY
operations of addition, subtraction,
multiplication, sin, cos, tan, asin, acos, atan,
sinh, cosh, tanh, log, exp, sqrt, sqr, isqrt,
unary , unary -.
We test Taylor models at
dimension 1 order 1, 7, 15, 17, and 20
dimension 2 order 1, 7, 15, 17, and 18
dimension 7 order 1, 2, 3, and 4

8
COSY Driver
9
Maple Driver
10
Error Report

Using process 1 (all in COSY)
Observations (For all the test cases we ran)
1. Errors are found in 3 functions COS, SIN,
and ASIN.
2. The COS and the SIN errors only occur at
order 17 of dimension 1 or dimension 2.
3. The ACOS errors only occur at order 1 of
dimension 1 and dimension 2, and order 1 or order
3 of dimension 7.

11
Error 1 (cos)

IMD 0
Dimension 1
Order 17
TM_EXPR COS(-3.141592653590006 TM_ONE
(1.570796326794687
TM_ONE) TM_INDEP (1))
Point value -0.9999990000000004,
-0.9999989999999995
ERROR Lower bd of expression gt upper bd of
Taylor model
Lower bound of expression -0.1570796328686757E-05
Upper bound of Taylor model -0.1570796816860323E-
05
Difference 0.4881735668679223E-12

12
Error 2 (cos)

IMD 1
Dimension 2
Order 17
TM_EXPR COS(-3.141592653590006 TM_ONE
(1.570796326794687
TM_ONE) TM_INDEP (1))
Point value 0.9999899999999996,
0.9999900000000005
Point value 0.9999899999999996,
0.9999900000000005
ERROR Lower bd of expression gt upper bd of
Taylor model
Lower bound of expression -0.1570796369223327E-04
Upper bound of Taylor model -0.1570796418440198E-
04
Difference 0.4921687142306775E-12

13
Error 3 (cos)

IMD 1
Dimension 2
Order 17
TM_EXPR COS(-3.141592653589796 TM_ONE
(0.7853981633974484
TM_ONE) TM_INDEP(1)
(0.7853981633974484 TM_ONE) TM_INDEP(2))
Point value -0.9999900000000005,
-0.9999899999999996
Point value -0.9999900000000005,
-0.9999899999999996
ERROR Lower bd of expression gt upper bd of
Taylor model
Lower bound of expression -0.1570796327002688E-04
Upper bound of Taylor model -0.1570796373588637E-
04
Difference 0.4658594880300668E-12

14
Error 4 (sin)

IMD 0
Dimension 1
Order 17
TM_EXPR SIN(-4.712388980384691 TM_ONE
(1.570796326794690
TM_ONE) TM_INDEP(1))
Point value -0.9999990000000004,
-0.9999989999999995
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression 0.1570796538419324E-05
Lower bound of Taylor model 0.1570797020611943E-0
5
Difference -0.4821926187347235E-12

15
Error 5 (sin)

IMD 1
Dimension 2
Order 17
TM_EXPR SIN(-4.712388980384691 TM_ONE
(1.570796326794690
TM_ONE) TM_INDEP(1))
Point value -0.9999900000000005,
-0.9999899999999996
Point value -0.5000000000000002,
-0.4999999999999998
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression 0.1570796347887127E-04
Lower bound of Taylor model 0.1570796396102999E-0
4
Difference -0.4821587160292514E-12

16
Error 6 (sin)

IMD 1
Dimension 2
Order 17
TM_EXPR SIN(4.712388980384691 TM_ONE
(0.7853981633974483
TM_ONE) TM_INDEP (1)
(0.7853981633974483
TM_ONE) TM_INDEP (2))
Point value 0.9999899999999996,
0.9999900000000005
Point value 0.9999899999999996,
0.9999900000000005
ERROR Lower bd of expression gt upper bd of
Taylor model
Lower bound of expression -0.1570796327370206E-04
Upper bound of Taylor model -0.1570796373932624E-
04
Difference 0.4656241856654513E-12

17
Error 7 (asin)

IMD 0
Dimension 1
Order 1
TM_EXPR ASIN((0.9999999999999983E-03
TM_ONE)
TM_INDEP(1))
Point value -0.9999990000000004,
-0.9999989999999995
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression -0.9999991666662369E-03
Lower bound of Taylor model -0.9999990000000064E-
03
Difference -0.1666662305128269E-09

18
Error 8 (asin)

IMD 0
Dimension 1
Order 1
TM_EXPR ASIN(-0.4935 TM_ONE
(0.3499999999999997E-02
TM_ONE)
TM_INDEP(1))
Point value -0.9999990000000004,
-0.9999989999999995
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression -0.5201381202432389
Lower bound of Taylor model -0.5201381145553354
Difference -0.5687903481543799E-08

19
Error 9 (asin)

IMD 0
Dimension 2
Order 1
TM_EXPR ASIN((0.4999999999999991E-03
TM_ONE)
TM_INDEP(1)
(0.4999999999999991E-03
TM_ONE) TM_INDEP
(2))
Point value -0.9999900000000005,
-0.9999899999999996
Point value -0.9999900000000005,
-0.9999899999999996
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression -0.9999901666617370E-03
Lower bound of Taylor model -0.9999900000000064E-
03
Difference -0.1666617306401302E-09

20
Error 10 (asin)

IMD 1
Dimension 7
Order 1
TM_EXPR ASIN((-0.4935 TM_ONE)
TM_INDEP(1))
Point value 0.4999999999999998,
0.5000000000000002
Point value -0.5000000000000002,
-0.4999999999999998
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
Point value -0.5000000000000002,
-0.4999999999999998
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression -0.2493251165458725
Lower bound of Taylor model -0.2467500000000038
Difference -0.2575116545868672E-02

21
Error 11 (asin)

IMD 1
Dimension 7
Order 3
TM_EXPR ASIN((-0.4935 TM_ONE)
TM_INDEP(1))
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
Point value 0.4999999999999998,
0.5000000000000002
ERROR Upper bd of expression lt lower bd of
Taylor model
Upper bound of expression -0.2493251165458725
Lower bound of Taylor model -0.2492539187578167
Difference -0.7119778805578236E-04

Using verification process 2 (Maple as referee)
Observation (For all the test cases we ran)
Verification process 2 detects the exactly the
same errors we have detected in verification
process 1.

23
Questions?

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