Floating Point 101

A very very basic introduction to FP.
With some inaccuracies.

Transcript

1. FLOATING 101 POINT
2. FLOATING 100.999998 POINT
3. engineers we are
4. researchers we are
5. 3.14159265358979 3238462643383279 5028841971693993 7510582097494459 2307816406286208 NUMBERS WE PLAY WITH ALL DAY LONG
6. well, sometimes even at night. (yawn).
7. So, what is a floating point?
8. A floating point is ± D 1 .D 2 D 3 ···D n x Be
9. A floating point is sign ± D 1 .D 2 D 3 ···D n x Be
10. A floating point is significand ± D 1 .D 2 D 3 ···D n x Be
11. A floating point is base ± D 1 .D 2 D 3 ···D n x Be
12. A floating point is exponent ± D 1 .D 2 D 3 ···D n x Be
13. A floating point represents ± (D 1 + D 2 * B-1 + D 3 * B-2 + … + D n * B(n-1)) * Be
14. For example + 3.14 x 100 = (3 + 1*0.1 + 4*0.01)*1 = 3.14
15. The point can float ! + 3.14 x 10-1 = 0.314
16. The point can float ! + 3.14 x 10+1 = 31.4
17. What if B = 2 ? + 1.00 x 2+2 = 4.0
18. Like machines do. http://grouper.ieee.org/groups/754/
19. Normalization of floating point
20. Multiple representations + 0.01 x 22 = 1.0 + 0.10 x 21 = 1.0 + 1.00 x 20 = 1.0
21. Normalized representation + 0.01 x 22 = 1.0 + 0.10 x 21 = 1.0 + 1.00 x 20 = 1.0
22. Normalized representation + (1.)000 x 20 1 is omitted
23. Normalized representation + (1.)000 x 20 there's room for an extra digit!
24. Excess-127 representation -127 → 0 -126 → +1 … -1 → +126 0 → +127
25. #include <float.h> FLT_MIN, FLT_MAX, ... #include <math.h> M_PI, M_E, NAN, INFINITY, ...
26. Why no exact representation for 0.1?
27. FLOATING POINT REAL NUMBERS is used to represent
28. FLOATING POINT RATIONAL NUMBERS denotes a (finite) subset of
29. 0.1 cannot be expressed as a power of 2 + ??? x 2??
30. + 00 x 20 1 It's also a matter of precision
31. + 01 x 20 1 1.25 It's also a matter of precision
32. + 10 x 20 1 1.25 1.5 It's also a matter of precision
33. + 11 x 20 1 1.25 1.5 1.75 It's also a matter of precision
34. + 11 x 20 π/2 It's also a matter of precision
35. + 11 x 20 π/2 It's also a matter of precision
36. + 00 x 21 1 1.25 1.5 1.75 2.0 Not just a matter of precision or basis...
37. + 01 x 21 1 1.25 1.5 1.75 2.0 2.5 Not just a matter of precision or basis...
38. + 10 x 21 1 1.25 1.5 1.75 2.0 2.5 3.0 Not just a matter of precision or basis...
39. Like death and taxes rounding errors are a fact of life. http://wiki.octave.org/FAQ
40. + 110 x 21 Operands that differ greatly + 100 x 2-2
41. + 110000 x 21 Operands that differ greatly + 000101 x 21
42. + 110000 x 21 Operands that differ greatly + 000101 x 21 = 110
44. Operands that are really close + 111 x 21 - 110 x 21 = 001 x 21
45. Operands that are really close + 111 x 21 - 110 x 21 = 100 x 2-2
47. Fixed point representation + 100.001010 = 22 + 2-3+ 2-5 = 4.15625
48. POINT WHAT'S THE WITH FLOATING
49. FP ARITHMETIC IS FAST Embedded in HW.
50. Single precision up to ~10+38. FP REPRESENTS A WIDE RANGE
51. HE APPROVES FP
52. Anyway, errors still there.
53. Okay, what about increasing the number of digits use decimal representations estimating errors think before you type
54. More digits, please! double (52 significant bits) long double (112 significant bits) arbitrary precision * * language support needed
55. Use decimal representations! decimal (C# only) BigDecimal (Java only) std::decimal (C++, coming soon)* * after IEEE-754 2008
56. Estimate the error of your algo rel_err = fabs(f – fp) / f
57. Use float to represent time float time; while (true) time += 0.20;
58. Use float to represent time float time; while (true) time += 0.20; This is BAD. And you should feel BAD.
59. Compare float numbers (a == b)
60. Compare float numbers (a == b) fabs(a -b) <= FLT_EPSILON
61. Compare float numbers (a == b) fabs(a -b) <= FLT_EPSILON fabs(a - b) <= max(fabs(a),fabs(b)) * pc
62. There is no silver bullet.
63. Use libraries (when available).
64. Vector addition (naive) float t[SIZE]; float result; for (i = 0; i < SIZE; ++i) result += t[i];
65. RESCUE GNU GSL TO THE
67. that's all folks! @lorisfichera – https://kid-a.github.com References and source code available at https://github.com/kid-a/floating-point-seminar Credits Font: Yanone Kaffeesatz (http://www.yanone.de/typedesign/kaffeesatz/)