Floating Point 101 - Speaker Deck

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FLOATING 101 POINT

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FLOATING 100.999998 POINT

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engineers we are

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researchers we are

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3.14159265358979 3238462643383279 5028841971693993 7510582097494459 2307816406286208 NUMBERS WE PLAY WITH ALL DAY LONG

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well, sometimes even at night. (yawn).

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So, what is a floating point?

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A floating point is ± D 1 .D 2 D 3 ···D n x Be

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A floating point is sign ± D 1 .D 2 D 3 ···D n x Be

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A floating point is significand ± D 1 .D 2 D 3 ···D n x Be

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A floating point is base ± D 1 .D 2 D 3 ···D n x Be

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A floating point is exponent ± D 1 .D 2 D 3 ···D n x Be

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A floating point represents ± (D 1 + D 2 * B-1 + D 3 * B-2 + … + D n * B(n-1)) * Be

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For example + 3.14 x 100 = (3 + 1*0.1 + 4*0.01)*1 = 3.14

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The point can float ! + 3.14 x 10-1 = 0.314

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The point can float ! + 3.14 x 10+1 = 31.4

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What if B = 2 ? + 1.00 x 2+2 = 4.0

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Like machines do. http://grouper.ieee.org/groups/754/

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Normalization of floating point

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Multiple representations + 0.01 x 22 = 1.0 + 0.10 x 21 = 1.0 + 1.00 x 20 = 1.0

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Normalized representation + 0.01 x 22 = 1.0 + 0.10 x 21 = 1.0 + 1.00 x 20 = 1.0

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Normalized representation + (1.)000 x 20 1 is omitted

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Normalized representation + (1.)000 x 20 there's room for an extra digit!

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Excess-127 representation -127 → 0 -126 → +1 … -1 → +126 0 → +127

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#include FLT_MIN, FLT_MAX, ... #include M_PI, M_E, NAN, INFINITY, ...

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Why no exact representation for 0.1?

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FLOATING POINT REAL NUMBERS is used to represent

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FLOATING POINT RATIONAL NUMBERS denotes a (finite) subset of

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0.1 cannot be expressed as a power of 2 + ??? x 2??

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+ 00 x 20 1 It's also a matter of precision

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+ 01 x 20 1 1.25 It's also a matter of precision

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+ 10 x 20 1 1.25 1.5 It's also a matter of precision

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+ 11 x 20 1 1.25 1.5 1.75 It's also a matter of precision

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+ 11 x 20 π/2 It's also a matter of precision

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+ 11 x 20 π/2 It's also a matter of precision

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+ 00 x 21 1 1.25 1.5 1.75 2.0 Not just a matter of precision or basis...

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+ 01 x 21 1 1.25 1.5 1.75 2.0 2.5 Not just a matter of precision or basis...

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+ 10 x 21 1 1.25 1.5 1.75 2.0 2.5 3.0 Not just a matter of precision or basis...

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Like death and taxes rounding errors are a fact of life. http://wiki.octave.org/FAQ

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+ 110 x 21 Operands that differ greatly + 100 x 2-2

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+ 110000 x 21 Operands that differ greatly + 000101 x 21

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+ 110000 x 21 Operands that differ greatly + 000101 x 21 = 110

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Operands that are really close + 111 x 21 - 110 x 21 = 001 x 21

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Operands that are really close + 111 x 21 - 110 x 21 = 100 x 2-2

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Fixed point representation + 100.001010 = 22 + 2-3+ 2-5 = 4.15625

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POINT WHAT'S THE WITH FLOATING

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FP ARITHMETIC IS FAST Embedded in HW.

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Single precision up to ~10+38. FP REPRESENTS A WIDE RANGE

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HE APPROVES FP

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Anyway, errors still there.

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Okay, what about increasing the number of digits use decimal representations estimating errors think before you type

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More digits, please! double (52 significant bits) long double (112 significant bits) arbitrary precision * * language support needed

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Use decimal representations! decimal (C# only) BigDecimal (Java only) std::decimal (C++, coming soon)* * after IEEE-754 2008

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Estimate the error of your algo rel_err = fabs(f – fp) / f

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Use float to represent time float time; while (true) time += 0.20;

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Use float to represent time float time; while (true) time += 0.20; This is BAD. And you should feel BAD.

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Compare float numbers (a == b)

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Compare float numbers (a == b) fabs(a -b) <= FLT_EPSILON

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Compare float numbers (a == b) fabs(a -b) <= FLT_EPSILON fabs(a - b) <= max(fabs(a),fabs(b)) * pc

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There is no silver bullet.

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Use libraries (when available).

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Vector addition (naive) float t[SIZE]; float result; for (i = 0; i < SIZE; ++i) result += t[i];

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RESCUE GNU GSL TO THE

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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/)