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Floating Point Copyright © Software Carpentry 2010 This work is licensed under the Creative Commons Attribution License See

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Presentation on theme: "Floating Point Copyright © Software Carpentry 2010 This work is licensed under the Creative Commons Attribution License See"— Presentation transcript:

1 Floating Point Copyright © Software Carpentry 2010 This work is licensed under the Creative Commons Attribution License See http://software-carpentry.org/license.html for more information. Testing

2 Floating Point Still looking at fields in Saskatchewan…

3 TestingFloating Point Still looking at fields in Saskatchewan… Fields Corner 1Corner 2Crop ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮

4 TestingFloating Point Still looking at fields in Saskatchewan… Fields Corner 1Corner 2Crop ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ Latitude/longitude

5 TestingFloating Point Still looking at fields in Saskatchewan… Fields Corner 1Corner 2Crop ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ Latitude/longitude Floating point numbers

6 TestingFloating Point Still looking at fields in Saskatchewan… Fields Corner 1Corner 2Crop ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ ⋮⋮⋮⋮⋮ Latitude/longitude Floating point numbers That's when trouble starts

7 TestingFloating Point Finding a good representation for floating-point numbers is hard

8 TestingFloating Point Finding a good representation for floating-point numbers is hard Can't actually represent an infinite number of real values with a finite set of bit patterns

9 TestingFloating Point Finding a good representation for floating-point numbers is hard Can't actually represent an infinite number of real values with a finite set of bit patterns What follows is (over-)simplified

10 TestingFloating Point Finding a good representation for floating-point numbers is hard Can't actually represent an infinite number of real values with a finite set of bit patterns What follows is (over-)simplified Goldberg (1991): "What Every Computer Scientist Should Know About Floating-Point Arithmetic"

11 TestingFloating Point Use sign, magnitude, and exponent

12 TestingFloating Point Use sign, magnitude, and exponent signmagnitudeexponent 1238

13 TestingFloating Point Use sign, magnitude, and exponent signmagnitudeexponent 1238 To illustrate problems, we'll use a simpler format

14 TestingFloating Point Use sign, magnitude, and exponent signmagnitudeexponent 1238 To illustrate problems, we'll use a simpler format And only positive values without fractions

15 TestingFloating Point Use sign, magnitude, and exponent signmagnitudeexponent 1238 To illustrate problems, we'll use a simpler format And only positive values without fractions magnitudeexponent 32

16 TestingFloating Point Possible values Exponent 00011011 0000000 0011248 01024816 011361224 100481632 1015102040 1106122448 1017142856 Mantissa

17 TestingFloating Point Possible values Exponent 00011011 0000000 0011248 01024816 011361224 100481632 1015102040 1106122448 1017142856 Mantissa 110 × 2 11

18 TestingFloating Point Possible values Exponent 00011011 0000000 0011248 01024816 011361224 100481632 1015102040 1106122448 1017142856 Mantissa 110 × 2 11 6 × 2 3

19 TestingFloating Point Possible values Exponent 00011011 0000000 0011248 01024816 011361224 100481632 1015102040 1106122448 1017142856 Mantissa 110 × 2 11 6 × 2 3 6 × 8

20 TestingFloating Point Possible values Exponent 00011011 0000000 0011248 01024816 011361224 100481632 1015102040 1106122448 1017142856 Mantissa 110 × 2 11 6 × 2 3 6 × 8 Actual representation doesn't have redundancy

21 TestingFloating Point 0 8 16 24 32 40 48 56 A clearer view of those values

22 TestingFloating Point 0 8 16 24 32 40 48 56 There are numbers we can't represent A clearer view of those values

23 TestingFloating Point 0 8 16 24 32 40 48 56 There are numbers we can't represent Just as 1/3 must be 0.3333 or 0.3334 in decimal A clearer view of those values

24 TestingFloating Point 0 8 16 24 32 40 48 56 This scheme has no representation for 9 A clearer view of those values

25 TestingFloating Point 0 8 16 24 32 40 48 56 This scheme has no representation for 9 So 8+1 must be either 8 or 10 A clearer view of those values

26 TestingFloating Point 0 8 16 24 32 40 48 56 This scheme has no representation for 9 So 8+1 must be either 8 or 10 If 8+1 = 8, what is 8+1+1? A clearer view of those values

27 TestingFloating Point 0 8 16 24 32 40 48 56 This scheme has no representation for 9 So 8+1 must be either 8 or 10 If 8+1 = 8, what is 8+1+1? (8+1)+1 = 8+1 (if we round down) = 8 again A clearer view of those values

28 TestingFloating Point 0 8 16 24 32 40 48 56 This scheme has no representation for 9 So 8+1 must be either 8 or 10 If 8+1 = 8, what is 8+1+1? (8+1)+1 = 8+1 (if we round down) = 8 again But 8+(1+1) = 8+2 = 10, which we can represent A clearer view of those values

29 TestingFloating Point 0 8 16 24 32 40 48 56 This scheme has no representation for 9 So 8+1 must be either 8 or 10 If 8+1 = 8, what is 8+1+1? (8+1)+1 = 8+1 (if we round down) = 8 again But 8+(1+1) = 8+2 = 10, which we can represent "Sort then sum" would give the same answer... A clearer view of those values

30 TestingFloating Point 0 8 16 24 32 40 48 56 Another observation

31 TestingFloating Point 0 8 16 24 32 40 48 56 Spacing is uneven Another observation

32 TestingFloating Point 0 8 16 24 32 40 48 56 Spacing is uneven But relative spacing stays the same Another observation

33 TestingFloating Point 0 8 16 24 32 40 48 56 Spacing is uneven But relative spacing stays the same Because we're multiplying the same few mantissas by ever-larger exponents Another observation

34 TestingFloating Point Absolute error = |val – approx|

35 TestingFloating Point Absolute error = |val – approx| Relative error = |val – approx| / val

36 TestingFloating Point OperationActual ResultIntended Result Absolute Error Relative Error 8+18911/9 (11.11%) 56+1565711/57 (1.75%) Absolute error = |val – approx| Relative error = |val – approx| / val

37 TestingFloating Point OperationActual ResultIntended Result Absolute Error Relative Error 8+18911/9 (11.11%) 56+1565711/57 (1.75%) Absolute error = |val – approx| Relative error = |val – approx| / val Relative error is more useful

38 TestingFloating Point OperationActual ResultIntended Result Absolute Error Relative Error 8+18911/9 (11.11%) 56+1565711/57 (1.75%) Absolute error = |val – approx| Relative error = |val – approx| / val Relative error is more useful Makes little sense to say "off by 0.01" if the value you're approximating is 0.000000001

39 TestingFloating Point What does this have to do with testing?

40 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected)

41 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) i = 1, 2, 3,..., 9

42 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) 0.9, 0.09, 0.009,...

43 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) 0.9, 0.99, 0.999,...

44 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) 0.9, 0.99, 0.999,... But is it?

45 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) 1-0.1, 1-0.01,...

46 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) 1-0.1, 1-0.01,... Should also make 0.9, 0.99,...

47 TestingFloating Point What does this have to do with testing? vals = [] for i in range(1, 10): number = 9.0 * 10.0 ** -i vals.append(number) total = sum(vals) expected = 1.0 – (1.0 * 10.0 ** i) diff = total – expected print '%2d %22.21f %22.21f' % \ (i, total, total-expected) Check and print

48 TestingFloating Point And the answer is: 1 0.900000000000000022204 0.000000000000000000000 2 0.989999999999999991118 0.000000000000000000000 3 0.998999999999999999112 0.000000000000000000000 4 0.999900000000000011013 0.000000000000000000000 5 0.999990000000000045510 0.000000000000000000000 6 0.999999000000000082267 0.000000000000000111022 7 0.999999900000000052636 0.000000000000000000000 8 0.999999990000000060775 0.000000000000000111022 9 0.999999999000000028282 0.000000000000000000000

49 TestingFloating Point And the answer is: 1 0.900000000000000022204 0.000000000000000000000 2 0.989999999999999991118 0.000000000000000000000 3 0.998999999999999999112 0.000000000000000000000 4 0.999900000000000011013 0.000000000000000000000 5 0.999990000000000045510 0.000000000000000000000 6 0.999999000000000082267 0.000000000000000111022 7 0.999999900000000052636 0.000000000000000000000 8 0.999999990000000060775 0.000000000000000111022 9 0.999999999000000028282 0.000000000000000000000 Already slightly off

50 TestingFloating Point And the answer is: 1 0.900000000000000022204 0.000000000000000000000 2 0.989999999999999991118 0.000000000000000000000 3 0.998999999999999999112 0.000000000000000000000 4 0.999900000000000011013 0.000000000000000000000 5 0.999990000000000045510 0.000000000000000000000 6 0.999999000000000082267 0.000000000000000111022 7 0.999999900000000052636 0.000000000000000000000 8 0.999999990000000060775 0.000000000000000111022 9 0.999999999000000028282 0.000000000000000000000 But at least they're consistent

51 TestingFloating Point And the answer is: 1 0.900000000000000022204 0.000000000000000000000 2 0.989999999999999991118 0.000000000000000000000 3 0.998999999999999999112 0.000000000000000000000 4 0.999900000000000011013 0.000000000000000000000 5 0.999990000000000045510 0.000000000000000000000 6 0.999999000000000082267 0.000000000000000111022 7 0.999999900000000052636 0.000000000000000000000 8 0.999999990000000060775 0.000000000000000111022 9 0.999999999000000028282 0.000000000000000000000 Sometimes, they're not

52 TestingFloating Point And the answer is: 1 0.900000000000000022204 0.000000000000000000000 2 0.989999999999999991118 0.000000000000000000000 3 0.998999999999999999112 0.000000000000000000000 4 0.999900000000000011013 0.000000000000000000000 5 0.999990000000000045510 0.000000000000000000000 6 0.999999000000000082267 0.000000000000000111022 7 0.999999900000000052636 0.000000000000000000000 8 0.999999990000000060775 0.000000000000000111022 9 0.999999999000000028282 0.000000000000000000000 Sometimes errors cancel out later

53 TestingFloating Point So what does this have to do with testing?

54 TestingFloating Point So what does this have to do with testing? What do you compare the actual test result to?

55 TestingFloating Point So what does this have to do with testing? What do you compare the actual test result to? sum(vals[0:7]) = 0.9999999 could well be False

56 TestingFloating Point So what does this have to do with testing? What do you compare the actual test result to? sum(vals[0:7]) = 0.9999999 could well be False Even if vals = [0.9, 0.09,...]

57 TestingFloating Point So what does this have to do with testing? What do you compare the actual test result to? sum(vals[0:7]) = 0.9999999 could well be False Even if vals = [0.9, 0.09,...] And there are no bugs in our code

58 TestingFloating Point So what does this have to do with testing? What do you compare the actual test result to? sum(vals[0:7]) = 0.9999999 could well be False Even if vals = [0.9, 0.09,...] And there are no bugs in our code This is hard

59 TestingFloating Point So what does this have to do with testing? What do you compare the actual test result to? sum(vals[0:7]) = 0.9999999 could well be False Even if vals = [0.9, 0.09,...] And there are no bugs in our code This is hard No one has a good answer

60 TestingFloating Point What can you do?

61 TestingFloating Point What can you do? 1.Compare to analytic solutions (when you can)

62 TestingFloating Point What can you do? 1.Compare to analytic solutions (when you can) 2.Compare complex to simple

63 TestingFloating Point What can you do? 1.Compare to analytic solutions (when you can) 2.Compare complex to simple 3.Never use == or != with floating-point numbers

64 TestingFloating Point What can you do? 1.Compare to analytic solutions (when you can) 2.Compare complex to simple 3.Never use == or != with floating-point numbers (It's OK to trust =, etc.)

65 TestingFloating Point What can you do? 1.Compare to analytic solutions (when you can) 2.Compare complex to simple 3.Never use == or != with floating-point numbers (It's OK to trust =, etc.) Use nose.assert_almost_equals(expected, actual)

66 TestingFloating Point What can you do? 1. Compare to analytic solutions (when you can) 2.Compare complex to simple 3.Never use == or != with floating-point numbers (It's OK to trust =, etc.) Use nose.assert_almost_equals(expected, actual) Fail if the two objects are unequal as determined by their difference rounded to the given number of decimal places (default 7) and comparing to zero.

67 TestingFloating Point Is that absolute or relative? What can you do? 1.Compare to analytic solutions (when you can) 2.Compare complex to simple 3.Never use == or != with floating-point numbers (It's OK to trust =, etc.) Use nose.assert_almost_equals(expected, actual) Fail if the two objects are unequal as determined by their difference rounded to the given number of decimal places (default 7) and comparing to zero.

68 August 2010 created by Greg Wilson Copyright © Software Carpentry 2010 This work is licensed under the Creative Commons Attribution License See http://software-carpentry.org/license.html for more information.

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