Assignment 6: Floating Point

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CSE 30: Computer Organization and Systems, Fall 2023

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Instructors: Paul Cao and Bryan Chin

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Due: Tuesday Nov 14, 2023

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Please read over the entire assignment before starting to get a sense of what you will need to

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get done in the next week. REMEMBER: Everyone procrastinates but it is important to know

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that you are procrastinating and still leave yourself enough time to finish. Start early, start often.

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You MUST run the assignment on the pi-cluster. You HAVE to SSH: You will not be able to

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compile or run the assignment otherwise.

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ACADEMIC INTEGRITY REMINDER: you should do this assignment on your own.

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If you work with others, be sure to cite them in your submission. Never copy from

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others.

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Please read the FAQ and search the existing questions on Edstem before asking for help.

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This reduces the load on the teaching staff, and clutter/duplicate questions on Edstem.

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Version updates:

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● 1.0 [Nov 8] Final Draft

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● 1.1 [Nov 8] Fix midpoint due date Sunday -> Friday

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● 1.2 [Nov 9] Fix: somehexnums.txt 0x8000 to 0x4000 since it is 15 bit representation.

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● 1.3 [Nov 9] Clarify: style won’t be regraded during resubmission

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● 1.4 [Nov 10] Prerelease: Midpoint answers are visible before due date.

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● 1.5 [Nov 12] Fix git clone link, fix # of bits in mantissas in page 4 table to be 8 bits

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Table of Contents

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1. Learning Goals

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2. Assignment Overview

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3. Getting Started

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4. How the Program Works

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5. Program Implementation

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a. Functions to Implement

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b. Developing Your Code

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c. Testing Your Code

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6. Submission and Grading

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a. Submitting

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b. Grading Breakdown [50 pts]

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Learning Goals

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● Programming in ARM assembly

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○ Bit masking

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○ Function call

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○ Branching

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● Working with floating point numbers

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● Coordinating a program with both ARM assembly and C code (you aren’t writing in C)

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Assignment Overview

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At the peak of time where pirates and bounty hunters are in the air. Porco Rosso makes his

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rounds in the vast ocean to capture any air pirates that disturb the peace near Adriano hotel.

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On the radio, Porco Rosso tunes in to listen to his next job, however he discovers an issue. The

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coordinates given out are in 15-bit floating-point format. He doesn’t know how to convert from

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this format and only knows the standardized IEEE 754 format. Gina only has devices that are

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written in ARM so Porco plans to rely on your assembly skills to create the conversion function.

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Help him write and test code to convert the coordinates into IEEE format!

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A note about representing number literals

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In the number 8’b1101_0011:

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● 8 is the number of binary bits in this number, in base-10.

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● b means binary. Other formats are d for decimal, o for octal, and h for hexadecimal.

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● To conserve space, we may also write the bits in hexadecimal, 0xd3 is equivalent to

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8’b1101_0011.

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● _ is a spacer character that is only there to make it easier to read. It has no numerical

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meaning. A '_' is usually placed every four digits.

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You can read more about where this number literal representation comes from here. (Note:

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Anything past the first slide is irrelevant to this course, but will be useful in CSE 140 & 141.)

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The 15-bit FP Format

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The 15-bit floating-point format is similar to, but not the same as, the one we studied in class. It

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has a sign bit, 6 bits of biased exponent, a bias value of 31 (base-10), and 8 bits of mantissa.

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Note that we include special cases to represent infinities and subnormal numbers.

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The following figure illustrates the bit assignments:

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FP Format (15-bit)

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sign

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(1 bit)

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exponent

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(6 bits, bias = 31)

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mantissa

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(8 bits)

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Points to note:

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1. There is an implied “1.” in front of the mantissa, unless it is a subnormal number.

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2. Subnormal numbers have an exponent field of 6’b000000 which represents 2 and

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−30

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implies a “0.” in front of the mantissa.

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3. “Infinite” numbers have an exponent field equal to 6’b111111 with ANY value in the

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mantissa.

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The following table shows how to interpret the exponent fields and mantissa fields.

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Exponent/mantissa represents Notes

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111111/mmmmmmm infinity infinity

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111110/mmmmmmm 2^31 x 1.mmmmmmm normal number

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111100/mmmmmmm 2^29 x 1.mmmmmmm normal number

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111000/mmmmmmm 2^25 x 1.mmmmmmm normal number

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100000/mmmmmmm 2^1 x 1.mmmmmmm normal number

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011111/mmmmmmm 2^0 x 1.mmmmmmm normal number

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001111/mmmmmmm 2^-16 x 1.mmmmmmm normal number

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000011/mmmmmmm 2^-28 x 1.mmmmmmm normal number

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000001/mmmmmmm 2^-30 x 1.mmmmmmm normal number

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000000/mmmmmmm 2^-30 x 0.mmmmmmm subnormal number

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(no leading 1)

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```````````````````````````````````````````````````````````````````````````````````````````````````````````

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Exponent bits are shown in purple below to help you distinguish it from the sign bit and

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mantissa.

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Number Encoding in 15-bits

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+0.0 15’b000_0000_0000_0000 (15 bits of 0 in binary)

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-0.0 15’b100_0000_0000_0000

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Number `15-Bit

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Representation

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Binary

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Representation

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Base-10

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Representation

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+∞

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15’b011_1111_xxxx_xxxx +Inf

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-∞

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15’b111_1111_xxxx_xxxx -Inf

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Most positive # 15’b011_1110_1111_1111 2^31

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9’b1.11111111

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4286578688

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Smallest positive

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#

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(subnormal)

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15’b000_0000_0000_0001 2^-30

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9’b0.00000001

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2^-38 ≅

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3.637978807e-12

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Most negative # 15’b111_1110_1111_1111 -2^31

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9’b1.11111111

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-4286578688

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Smallest

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negative #

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(subnormal)

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15’b100_0000_0000_0001 -2^-30

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9’b0.00000001

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-2^-38 ≅

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-3.637978807e-1

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2

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IEEE-754 Single Precision Format

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IEEE-754 Single Precision Format

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sign

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(1 bit)

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exponent

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(8 bits, bias = 127)

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mantissa

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(23 bits)

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Subnorms

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The bias for the IEEE Format is 127 (base-10) and the format uses an implied “1.” for normal

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numbers, as usual. The smallest possible exponent is -126 represented by 8’b0000_0001 for

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normal numbers, whereas 8’b0000_0000 represents subnormal numbers. For subnormal

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numbers, we prepend the mantissa with “0.” instead of “1.” similar to how subnormal numbers

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are evaluated in our 15-bit FP format.

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Infinities

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In IEEE single precision, any exponent of all 1’s (8’b1111_1111) represents a number too

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large to represent. For example, 0xff80_0000 is a number with a negative sign bit, all 1’s for

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the exponent and all 0’s for the mantissa. This represents negative infinity (-Inf). Similarly,

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0x7f80_0000 represents positive infinity (+Inf). Note that the mantissa bits are all 0. Non-0

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mantissa bits represent another kind of IEEE special number (NaN, “not a number”) which is not

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required in this assignment since our 15-bit floating point format does not use NaN.

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Summary of Select Conversions

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FP 15-bit FP IEEE-754 Single

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+0 15’b000_0000_0000_0000 0x00000000

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-0 15’b100_0000_0000_0000 0x80000000

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2^-38 15’b000_0000_0000_0001 0x2c800000

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-2^-38 15’b100_0000_0000_0001 0xac800000

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4286578688 15’b011_1110_1111_1111 0x4f7f8000

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-4286578688 15’b111_1110_1111_1111 0xcf7f8000

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+Inf 15’b011_1111_xxxx_xxxx 0x7f800000

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-Inf 15’b111_1111_xxxx_xxxx 0xff800000

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Getting Started

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Developing Your Program

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For this class, you MUST compile and run your programs on the pi-cluster.

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Need help or instructions? See the Edstem FAQ. (Do NOT wait until the end to try this. There

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will be limited or no ETS support on the weekends!)

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We’ve provided you with the starter code at the following link:

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https://github.com/cse30-fa23/hw6-starter

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1. Download the files in the repository.

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a. You can either use

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git clone https://github.com/cse30-fa23/hw6-starter.git

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directly from pi-cluster, or download the repo locally and scp the folder to

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pi-cluster if that doesn’t work.

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2. Fill out the fields in the README before turning in.

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Running Your Program

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We’ve provided you with a Makefile so compiling your program should be easy!

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Additionally, the reference solution binary will be placed on Saturday 11/11 morning at:

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/home/linux/ieng6/cs30fa23/public/bin/fpconvert-a6-ref

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Makefile: The Makefile provided will create a fpconvert executable from the source files

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provided. Compile it by typing make into the terminal. Run make clean to remove files

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generated by make.

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How the Program Works

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Your program will take a filename as an argument and read it in. This file is a txt file storing the

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15 bit FP numbers. The main function (implemented for you in main.c) will parse the input file

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and call the fpconvert function which you will implement in assembly on each 15-bit FP

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number to convert it into IEEE floating point format, and print the result to stdout.

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Once you compiled the program with make, you can run it as follows:

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./fpconvert somehexnums.txt

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where somehexnums.txt is the name of the input txt file that holds the hex numbers you

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want to convert.

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Input Guarantees

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● fpconvert will be only given valid 15-bit wide numbers.

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Program Implementation

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Files to Edit

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You need to edit fpconvert.S and convert_inf.S

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Functions to Implement

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You will need to implement 1 function within fpconvert.S:

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● fpconvert(n): This is the function that will do most of the floating-point conversion.

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○ Argument: n the 15-bit FP number to convert

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○ Returns: n’s equivalent IEEE 754 single precision representation.

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○ If n is ±infinity, you MUST call convert_infinity(n) to do the conversion

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instead.

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You need to implement 1 function within convert_inf.S

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● convert_infinity(n):

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○ Argument: n the 15-bit FP number to convert (should only be ±infinity)

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○ Returns: the FP number’s equivalent IEEE 754 single precision representation.

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NOTE:

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● 32-bit ARM stores arguments passed into the function in registers r0-r3; n only

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symbolizes that the function takes in one argument. You cannot directly use n in your

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assembly program to refer to the first argument.

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● As registers are all 32-bits wide, our 15-bit floating point format will always only occupy

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the least significant 15 bits, the upper 17 bits will be padded with 0’s.

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● Return value should be stored in r0.

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Calling a Function in ARM

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To call a function in ARM, you must use the bl “branch and link” instruction. It is not sufficient

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or correct to use a regular branch instruction. Without branch-and-link, the return operations in

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the epilogue of the function will not work and return as expected.

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Developing Your Code

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Development Process

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To make development easier, you should first implement the conversion of normal numbers.

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Test your code on a range of normal numbers (smallest, largest). For the smallest numbers, you

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should familiarize yourself with their scientific notation representations. You can also check the

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IEEE column of the output to see if it matches the expected IEEE version. Additionally, be sure

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to check the special cases of +0.0, -0.0, +Inf, and -Inf.

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After thoroughly testing the functionality of your code, you should consider subnormal numbers.

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Subnormal numbers are represented when the exponent field is 6’b000000.

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After implementing the conversion of subnormal numbers, your code should be able to produce

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all of the values in the Summary of Select Conversions table.

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Development Tips

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Before you write assembly code, think about the algorithm.

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● How are the 15-bit format and the 32-bit IEEE format similar and different?

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● How do I break down the 15-bit format into the 3 individual fields?

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● How does each field convert from the 15-bit format to the 32-bit IEEE format?

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You should find the bitwise instructions useful for this assignment. In particular, you will want to

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make use of bitmasks.

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While an immediate can only be 8 bits wide, you can use left and right shifts to move the mask

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into the right position. For example, if you need the bitmask 0xFF00, you can shift the

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immediate 0xFF left by 8 bits.

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Testing your Code

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To run your code you need a txt file that holds the hex numbers that you want to convert,

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separated by a new line.

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Example text input file, named somehexnums.txt:

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0x0000

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0x4000

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0x3f00

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0x7f00

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0x3eff

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0x0001

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0x7eff

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0x4001

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NOTE: you should make sure each hexadecimal number only has four digits, otherwise you

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may get unexpected results.

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Checking For Exact Output Match

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A common technique is to redirect the outputs to files and compare against the reference

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solution

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1

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:

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./your-program args > output; our-reference args > ref

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diff -s output ref

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This command will output lines that differ with a < or > in front to tell which file the line came

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from.

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Debugging Tips

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The public autograder will only printf test some features. DO NOT RELY ON THE

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AUTOGRADER. (Many CSE 30 students have been burned by this.) Test your code using your

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own test cases!

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GDB treats ARM assembly labels as functions except those that begin with the prefix “.L”. If

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you want to use GDB to debug your ARM code, you will need to prefix your labels with “.L”.

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1 You might want to check out vimdiff on the pi-cluster (https://www.tutorialspoint.com/vim/vim_diff.htm).

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Thus, ARM code for the given C if statement would look like the code snippet on the right, rather

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than the snippet on the left.

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if (r5 == 99) {

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r3 = r3 + 2;

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} else {

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r3 = r3 + 3;

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}

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r4 = r4 - 1;

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GDB will not recognize labels: GDB will recognize labels:

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cmp r5, 99

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bne else

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add r3, r3, 2

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B end_if

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else:

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add r3, r3, 3

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end_if:

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sub r4, r4, 1

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cmp r5, 99

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bne .Lelse

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add r3, r3, 2

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b .Lend_if

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.Lelse:

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add r3, r3, 3

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.Lend_if:

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sub r4, r4, 1

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Allowed ARM Instructions

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You are only allowed to use the instructions provided in the ARM ISA Green Card. Failure to

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comply will result in a score of 0 on the assignment.

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Style Requirements

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Reading raw assembly is hard and debugging will be nigh impossible if you don’t put comments!

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To encourage you to make your life easier, style will be worth 2 points in this assignment on a

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holistic readable/unreadable basis. You will get full style points as long as your code is

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reasonably commented to be readable (so that someone who doesn’t know ARM can still

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roughly understand what it’s doing), so don’t worry if you can’t get all the details right. However,

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you will get no style points if it’s not (e.g. very inconsistent indentation, sparse or unreadable

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comments). In addition, staff won't be able to provide any assistance other than styling

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advice unless code is readable. For reference, here is the Style Guideline for ARM assembly.

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We strongly recommend you to use comments after each instruction to help describe what

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step occurs like what is done in the style guide. Note: style will not be graded for

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resubmission.

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Midpoint (5 Points)

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This part of the assignment is due earlier than the full assignment, on

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Friday 11/10 at 11:59 pm PST. There are no late submissions on the

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Midpoint.

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Complete the Gradescope assignment “HW6: Checkpoint”, an Online Assignment that is done

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entirely through Gradescope. This assignment consists of a few quiz questions and a

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free-response question where you will document your algorithm in plain English or C code.

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Discuss your implementations of the following functions: fpconvert and

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convert_infinity. Your fpconvert should call convert_infinity when appropriate.

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Submission and Grading

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Submitting

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1. Submit your files to Gradescope under the assignment titled “HW6 (Coding): Floating

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Point”. You will submit ONLY the following files:

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fpconvert.S

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convert_inf.S

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README.md

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Submission will open by Saturday morning. You should test your code extensively on

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pi-cluster before submitting to gradescope.

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You can upload multiple files to Gradescope by holding CTRL (⌘ on a Mac) while you

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are clicking the files. You can also hold SHIFT to select all files between a start point

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and an endpoint.

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Alternatively, you can place all files in a folder and upload the folder to the assignment.

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Gradescope will upload all files in the folder. You can also zip all of the files and upload

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the .zip to the assignment. Ensure that the files you submit are not in a nested folder.

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2. After submitting, the autograder will run a few tests:

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a. Check that all required files were submitted.

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b. Check that fpconvert.S and convert_inf.S compiles.

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c. Runs some sanity tests on the resulting fpconvert executable.

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Grading Breakdown [5 + 45 points]

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Make sure to check the autograder output after submitting! We will be running

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additional tests after the deadline passes to determine your final grade. Also, throughout this

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course, make sure to write your own test cases. It is bad practice to rely on the minimal

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public autograder tests as this is an insufficient test of your program.

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To encourage you to write your own tests, we are not providing any public tests that have

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not already been detailed in this writeup.

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The assignment will be graded out of 50 points and will be allocated as follows:

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● Midpoint writeup: 5 points. This part of the assignment is due earlier than the full

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assignment, on Friday 11/10. Complete the Gradescope assignment “Homework 6:

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Checkpoint”.

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● Code compiles with no warnings: 1 point

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● Style: 2 points

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● Public tests with the provided examples.

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● Private tests with hidden test cases.

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NOTE: The tests expect an EXACT output match with the reference binary. There will be

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NO partial credit for any differences in the output. Test your code - do NOT rely on the

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autograder for program validation.

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Make sure your assignment compiles correctly through the provided Makefile on the

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pi-cluster without warnings. Any assignment that does not compile will receive 0 credit.

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[Optional] Bells and Whistles

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2

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(epsilon points)

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Write a new function add_fp(a, b) that takes in 2 numbers a and b that are in the 15-bit

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floating point format. It should add these 2 numbers together and return the value. However,

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what makes this complicated is that a and b may not have the same exponent! You’ll need to

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make the exponents the same first before you can add them.

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This part of the assignment will NOT be graded - and does not need to be submitted. It is

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completely up to you to try writing a program which achieves the above output.

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The Bells and Whistles component may be submitted to a separate Gradescope assignment:

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Homework 6 Optional: Bells and Whistles.

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2

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In our experience, students like extra credit. However, we find that extra credit isn't used by students

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who need it the most. Thus, we have an extra credit where the number of points assigned is epsilon,

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where epsilon is a very small number [0, 1).

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請加QQ：99515681 或郵箱：99515681@qq.com   WX：codehelp

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