Numerical Precision. Or, why my numbers aren t numbering right. 1 of 15

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1 Numerical Precision Or, why my numbers aren t numbering right 1 of 15

2 What s the deal? Maybe you ve seen this #include <stdio.h> int main() { float val = 3.6f; printf( %.20f \n, val); Print a float with value 3.6 to 20 decimal places } cslab6c 123sh>./myAmazingProgram cslab6c 123sh> of 15

3 What s the deal? We trust computers to do complicated arithmetic, using both integers and floating point numbers Like 6 41 or Sometimes, computers let us down :( Numerical overflow (integers above max wrap around to min and vice versa) Numerical underflow (very small floats become 0) Just plain incorrect results (like the example of 3.6 we just saw) Why do errors happen? 3 of 15

4 Some intuition Conceptually, in mathematics / the universe / life, there are Infinitely many whole numbers Infinitely many fractional numbers What if we constrain the range? For example, what happens when we only count values from 1-5? Finite amount of whole numbers: 1, 2, 3, 4, and 5 Still infinitely many fractional numbers 4 of 15

5 Some background All numbers are stored on the computer in binary Binary is a base 2 number system: each digit can be one of two values, 0 or 1 You probably write numbers in base 10, where digits go from 0-9 Ex. you type 552 computer stores Both binary and decimal can represent any number, given enough digits If theoretically this should work, then the problem must lie in implementation! Take CS33 or CS22 to learn more about number representations and binary! On CS department machines, data types have these specific, finite sizes* int 32 bits float 32 bits double 64 bits * Not guaranteed on all platforms! You can check a type s size in bytes using sizeof(float), etc. 5 of 15

6 Computers have finite resources Computers have limited hard drive space, and data points stored on a computer are size-limited by their type (as seen on previous slide) We simply cannot store infinite data Thankfully, within these limits integer representations (and arithmetic) work perfectly But floating point numbers run into issues... Big problem: If we want to compute precisely with floats, how can we represent infinitely many floating point numbers in limited space? We can t :( We have to cheat 6 of 15

7 Computers have finite resources IEEE standard for floating point numbers you do not need to memorize this 32 bits: (1) (8) (23) Sign bit: is it positive or negative? Fraction bits: what is the actual value? Exponent bits: what order of magnitude is it? What does this mean? A float stores numbers a lot like scientific notation: Fraction (base num) exponent Ex. decimal: Ex. floating point: Note about fraction bits in binary: since we omit leading zeros in the fraction (part of conversion to normalized form), every fraction will start with a 1, which means we can omit this bit and gain an extra bit of precision for free! 7 of 15

8 Computers have finite resources IEEE floats are actually pretty great we can represent a lot of numbers with just 32 bits! When we can t represent a number, we take the closest number that we can represent This leads to unexpected results sometimes Let s look at a (simplified) example of this format to see what s going on We ll use base 10 (because only Cylons think in binary) We ll constrain the size of our data 3 possible values for the exponent: {-1, 0, 1} 1 decimal digit for the fraction 8 of 15

9 Example: floating point imprecision Fraction Exponent = = = = = = = = = = = = = = = 40 What s this? In scientific / floating point notation, we re trying every fraction and every exponent = = = = = = = = = = = = 80 This will compute every value we can represent in our example scheme. Next, let s plot a picture! = = = 90 9 of 15

10 Example: floating point imprecision (A) Exponents = {-1} (B) Exponents = {-1, 0} (C) Exponents = {-1, 0, 1} In this example, we can already see 2 big issues 1) We can only represent samples of the entire continuum 2) The samples are not distributed uniformly! The closer we are to zero, the more numbers we can represent With an adequate amount of fraction and exponent bits, the distribution is less extreme, but still not uniform 10 of 15

11 Example: floating point imprecision Yikes. I don t want to compute with that system. Can we make our numbers more precise? Yes. We can get higher precision by adding more digits to the fraction We can get a wider range by adding more digits to the exponent In fact, plain floats are often big enough to store anything you d need in CS123 Our output is graphical and will be viewed by the human eye, which cannot detect these small errors Little value in using double-precision data: more work, small ROI Precision will always be an issue to some degree: with any kind of similar representation, you will always have some margin of error 11 of 15

12 How do we counter imprecision? In cases of comparison, allow some margin of acceptable error Never ever do something like this: if (mybigcomputation == 0.123)... BAD! May fail unexpectedly. Instead, do something like this: if (abs(mybigcomputation - targetvalue) < epsilon)... GOOD! Will catch computer error. abs() is absolute value; try picking epsilon to be near 1e-4 (0.0001) *but your mileage may vary!* 12 of 15

13 Imprecision in the wild Example from your Ray project: floating point imprecision can lead to speckling on objects. 13 of 15

14 Warning: error propagation Be careful of accumulators and repeated error! Small errors can build up if used over and over Ex. plotting points around a circle using a for loop float angle = 0; float inc = 1.0/20 * 360; for (int i=0; i<20; i++) { plotpointat(angle); angle += inc; } Will accumulate error over time. for (int i=0; i<20; i++) { float angle = i/20.0 * 360; plotpointat(angle); } Will not drift from true values. Yikes! On the left, an error of could become 0.02, 2, 200, or worse It can be difficult to debug where large errors come from, as they may seem unrelated to the minor errors ignored earlier! Try to avoid accumulating error by computing absolute position each time OR testing values for error before accumulating them 14 of 15

15 Takeaway Computers are less precise than true mathematics We encourage you to use floats in your code But be cautious! Testing for equality between floats is dangerous, because there is no guarantee that either value can be represented by the computer. Be careful of propagating errors in your code! Don t let floats accumulate too much error! 15 of 15

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