Essential Mathematics for Games Programmers (Fixed/Float Tutorial)

1. Essential Mathematics for Games Programmers(Fixed/Float Tutorial) Lars Bishop (lmb@ndl.com)

2. Number Spaces • Cardinal – Positive numbers, no fractions • Integer – Pos., neg., zero, no fractions • Rational – Fractions • Irrational – Non-repeating decimals (,e) • Real – Rationals+irrationals • Complex – Real + multiple of -1 a+bi Essential Math for Games

3. Numerical Representations • In graphics, we often deal with Real numbers (3.14159, 1.5, etc) • Unlike integers, Real numbers have fractional components • There are several common ways of representing these on a computer Essential Math for Games

4. Approximating Reals • When we write a Real number on paper, we generally write only a few digits past the decimal point: 1.5, 3.45 • Most reals cannot be represented exactly by a few fractional digits • Any written number has inherent precision Essential Math for Games

5. Finite Representations • Any number in a computer has finite representation • As a result, any such representation cannot represent every number exactly • When representing numbers, we will always be coping with these limitations • We need to understand and limit error Essential Math for Games

6. Fixed Point Numbers • Use integer-like representation • Assume that the least-significant bit is some negative power of 2 (⅛, ¼, etc) • The “binary point” is in the middle of the number, not after the least-significant bit Essential Math for Games

7. Fixed-Point Nomenclature • Applications can adjust their precision and range by moving the “binary point” • If a fixed point number has • M bits of integral precision • N bits of fractional precision, • It is called an M.N number • This is pronounced “M dot N” • A 32-bit integer would be “32 dot 0” Essential Math for Games

8. Fixed Point Benefits • Can represent fractional values with only integer arithmetic • Simple to use and understand • Addition and Subtraction are the same as for integers • Multiplication and Division are only slightly different from their integer siblings Essential Math for Games

9. Fixed-point vs. Floating-point • A 32-bit fixed-point actually has more inherent precision than a 32-bit Floating-point number! • Floating-point trades accuracy for range by using some bits for the exponent • “Floating point is for the lazy” - John von Neumann (paraphrased) Essential Math for Games

10. Floating-point ↔ Fixed-point • Assuming an M.N fixed-point system: • IntToFix(i) = i << N • FloatToFix(f) = (int)(f * 2.0N) • Look out for overflow on these! • FixToInt(i) = i >> N • FixToFloat(i) = ((float)i) * 2.0-N • Conversion to int can lose precision Essential Math for Games

11. Basic Fixed-Point Math • For A and B, both M.N fixed point values, • A+B = (int)A + (int)B • A-B = (int)A – (int)B • where (int)A is simply the fixed point treated bitwise as an integer • This works because the binary points line up when A and B are both M.N Essential Math for Games

12. Multiplication – Basic Idea • Multiplication is a bit more complex, but it is analogous to the grade-school base 10 trick: We multiply the numbers as integers, and then slide the decimal point to the left by 1+1=2 Essential Math for Games

13. Multiplication We want to compute 0.5f x 1.0f = 0.5f. In 4.4 fixed-point, this is: X After the integer multiply, we get (8x16=128): Then, we need to shift right by 4 (not 8 – we don’t want an integer result, we want a 4.4 result, not 8.0) Essential Math for Games

14. Challenges with Fixed Point • Range (Overflow) • In the previous multiplication example, if we compute 1.0x1.0=1.0 using the given method, the intermediate value overflows • Precision (Underflow) • If we pre-shift numbers down to avoid overflow, we can end up shifting to 0 • Extra shifting required for mul/div Essential Math for Games

15. Fixed Point Help • Some CPUs have special instructions • ARM9 (common in handhelds) • Has a 32-bit X 32-bit → 64-bit multiply • Also has similar 64 + 32 x 32 → 64 • Can avoid overflow or underflow in intermediate values • Allows a free shift in every ALU operation Essential Math for Games

16. Fixed Point Summary • Allows fractional math to be done quickly with integer hardware • Requires careful range and precision analysis • Is the only option on many embedded and handheld devices, which don’t have FPU hardware Essential Math for Games

17. Floating Point Numbers • Used to represent general non-integers • Often thought of as the set of Reals • This is far from the truth, as you’ll see • Most of this discussion will be about • IEEE 754 32-bit single-precision • Known in C/C++ as float • Mention of doubles later Essential Math for Games

18. FP and Scientific Notation • FP is analogous to scientific notation • Scientific notation is: • ± D.DDDD x 10E, where • D.DDDD has a nonzero leading digit • D.DDDD has fixed # of fractional digits • E is a signed integer value Essential Math for Games

19. FP/Sci Notation Range/Precision • Precision is not fixed: • Precision of 1.000x10-2 is 100 times more fine-grained than that of 1.000x100 • Precision and range are related • The larger the number, the less precise • 32-bit int is not a subset of float Essential Math for Games

20. FP/Sci Notation Components • Sign • Mantissa • Normalized – has a nonzero integer digit • Limited precision – fixed number of digits • Exponent • Chosen so that the mantissa is normalized Essential Math for Games

21. Sign • FP numbers have an explicit sign bit • Float has both 0.0f and -0.0f • By the standard, 0.0f = -0.0f • But the bits are different • Do notmemcmp floats! • This is only one of many reasons Essential Math for Games

22. Exponent • Like the exponent in scientific notation • But, the exponent’s base is 2, not 10 • Stored as a biased number: • ExponentBits = Exponent – Bias • For float, Bias = 127 • For float, -125 ≤ Exponent ≤ 128 • Exponent term is Essential Math for Games

23. Mantissa • Represented as 1-dot-23 fixed point • Store the 23 fractional bits explicitly • Integer bit is implied (“hidden bit”) • Generally, mantissa is normalized • In other words, mantissa is 1.MantissaBits • Similar to the scientific notation standard • In the smallest numbers, the integer bit is assumed to be 0 Essential Math for Games

24. Binary Representation • Put together, the representation is: • S=Sign bit • E=Exponent bits (8) • M=Fractional mantissa bits (23) • We will write as A = (SA,EA,MA) Essential Math for Games

25. Special Values • 0: S=0, E=All 0s, M=All 0s • -0: S=1, E=All 0s, M=All 0s • +∞: S=0, E=All 1s, M=All 0s • Ex: 1.0f / 0.0f = +∞ • −∞: S=1, E=All 1s, M=All 0s • Ex: -1.0f / 0.0f = -∞ Essential Math for Games

26. Not a Number • Represents undefined results • 0.0f / 0.0f = NaN • ACOS(2.0f) = Nan • Two kinds – Quiet and Signaling • Quiet can be passed on to other ops • Signaling traps the code • NaN: E=All 1s, M=Not all 0s Essential Math for Games

27. Very Small Numbers • What happens when we run out of smaller and smaller exponents? • Could flush to zero • But, this can lead to the following problem • X-Y=0 does not imply X=Y! • Need to gradually underflow to zero • FP does this by allowing denormals Essential Math for Games

28. Denormals • A denormal is an FP number whose hidden mantissa bit is 0, not 1 • Indicated by E=0, M=Not all 0s • A denormal is equal to • (-1)S x 2-126 x 0.MantissaBits • This allows precision to gradually roll off to zero. Essential Math for Games

29. Floating-point Add Basics • To add two positive floating point numbers A (EA, MA) and B (EB, MB): • Swap as needed so A has the greater exponent, i.e. EB≤ EA • Shift MB to the right by EA-EB bits • Add MA+MB and use as the new mantissa • Adjust the new exponent up or down to re-normalize the result. Essential Math for Games

30. FP Add Notes • Not a simple process (even for pos #’s) • If A>>B, then B can be shifted to 0 • Repeatedly adding small numbers to an accumulator (i.e. A+=B) can gradually lead to huge error • At some point, A stops growing, no matter how many times B is added! Essential Math for Games

31. Fun with Floats – The Real World • Can’t discuss every FP issue here: this is an example of why you should care • 3D engine saw a spike in basic FP code • Code (SLERP) was +,-,* only • No loops, no complex functions • What could be the problem? Essential Math for Games

32. Breaking Down the Problem • Input values looked valid (no NaN) • After a “while”, in a demo, the spike hit • We saved the values, along with timing • In a small app, ran the slow and fast cases in tight loops • The slow cases all had some tiny numbers (~1.0x10-43) Essential Math for Games

33. Tiny ≠ 0.0f • Slow cases were denormals • We assumed these numbers to be 0 • But FPU was taking care to be accurate • Denormal ops seemed to be slow Essential Math for Games

34. Denormal Performance • Did some more timing tests • Even loading a denormal was slow! • Pentium takes a big hit on denormals • True even with exceptions masked! • FPU pipeline gets flushed on denormals • Little things matter Essential Math for Games

35. “Don’t Doubles Solve this?” • Doubles do help with most range and many precision problems. But: • Need twice the memory of floats (duh) • Frequently, significantly slower than floats • Some platforms don’t support them • Avoid switching to them without tracking down the problem first Essential Math for Games

36. Floating Point Wrap-up • Floats ≠ Reals • Understand the limits of FP • Analyze your FP issues • Don’t just jump to doubles at the first sign of trouble • Be willing to rework your math functions to be FP-friendly Essential Math for Games