class: big, middle # Engineering 1020: Introduction to Programming .title[ .lecture[Lecture 25:] .title[Real numbers] ] .footer[[/lecture/25/](/lecture/25/)] --- # Last time ### Computer memory stores _bits_ ### Can use bits for _binary_ integer representation ### Yet to come: * floating-point representation -- #### Today: -- floating-point representation -- (surprise!) --- # Review: ## Integer representation --- # Real numbers ### So how do we get decimal places? -- ### _Fixed-point_ representation -- ### _Floating-point_ representation --- # Fixed-point representation -- ### _Integer bits_ and _fractional bits_ ??? This is just an extension of what we've already been doing with the place values of binary numbers: counting down from $2^2$ to $2^1$ to $2^0$, we can keep going with $2^{-1}$, $2^{-2}$, etc. A concrete example: binary 110.11 -- ### Example: dollars and cents! --- # Floating-point representation -- ### Like scientific notation (e.g., 8.9e5) -- $$ b^e \times f $$ $b$: _base_ (usually 2 or 10) $e$: _exponent_ (can be represented as an integer) $f$: _fraction_ or _mantissa_ (can also be represented as an integer) -- ### Scientific notation: $b = 10$ --- # Floating-point in practice ### Widely-implemented standard: [IEEE 754](https://en.wikipedia.org/wiki/IEEE_754) -- .center[ <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/IEEE_754_Single_Floating_Point_Format.svg/1024px-IEEE_754_Single_Floating_Point_Format.svg.png" width="900"/> ] Sign bit, _biased_ exponent (integer), fraction (fixed-point) -- Special values: -- infinity (`math.inf`) -- , not-a-number (`math.nan`) --- # Single-precision representation (32b) .floatleft[ $$ 2^{e-127} \cdot f $$ ] .center[ <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/IEEE_754_Single_Floating_Point_Format.svg/1024px-IEEE_754_Single_Floating_Point_Format.svg.png" width="700"/> ] -- #### Eight-bit _biased exponent_ -- * range of 8b integer: -- 0 to 255 -- * range of $e-127$: -- -127 to 128 ??? With this range of $e-127$, the exponential part of this number can range from $2^{-127}$ up to $2^{128}$... which is pretty big! --- # Single-precision fractional part .floatleft[ $$ 2^{e-127} \cdot f $$ ] .center[ <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/IEEE_754_Single_Floating_Point_Format.svg/1024px-IEEE_754_Single_Floating_Point_Format.svg.png" width="700"/> ] #### 23b fractional part -- Uses a **fixed-point** representation -- (assume prefix of 1) -- .floatleft[ $$ 1 + \frac{b_0}{2} + \frac{b_1}{4} + \frac{b_2}{8} + \frac{b_3}{16} + \dots $$ ] -- .floatright[ $$ = 1 + \sum_{i=0}^{22} \frac{b_i}{2^{i+1}} $$ ] --- # Double-precision representation (64b) .floatleft[ $$ 2^{e-1023} \cdot f $$ ] .center[ <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/IEEE_754_Double_Floating_Point_Format.svg/1024px-IEEE_754_Double_Floating_Point_Format.svg.png" width="700"/> ] #### Eleven-bit biased exponent * range of 11b integer: 0 to 2048 * range of $e-1023$: -1023 to 1024 --- # Floating-point precision .floatright[ <a href="https://commons.wikimedia.org/wiki/File:IEEE754.svg"> <img src="https://upload.wikimedia.org/wikipedia/commons/1/18/IEEE754.svg" height="400" alt="Precision of IEEE 754 single- and double-precision"/> </a> <div class="source"> Source: <a href="https://commons.wikimedia.org/wiki/File:IEEE754.svg"> Wikimedia commons</a> </div> ] ### Hence "single-precision" and "double-precision" -- For numbers close to 0: $\sim 10^{-7}$ or $\sim 10^{-15}$ --- # Floating-point visualization ### A neat visualization tool: .center[ https://www.h-schmidt.net/FloatConverter/IEEE754.html ] --- # Summary ### Real numbers * _fixed-point_ representation * _floating-point_ representation --- class: big, middle (here endeth the lesson)