Floating Point Multiplication

The float_mult.S contains a RISC-V program that multiplies two bfloat16 floating point numbers. However, the fraction bit of the float is 7 instead of 23.

Skills Involved

• RISC-V Assembly Programming
• C Programming
• Binary Floating Point Multiplication

Development Environment

This is part of a lab in the course, EEC170, from UC Davis. The program is written in RISC_V assembly language with the help of the RISC-V Venus Simulator in Visual Studio Code.

Algorithm

I mostly followed the algorithm from page 221 of this book:

Code Description

There's a C version of the same program, and I will reference the variable names from the main.c.

1. Process X and Y's mantissas
   • Give the mantissa its invisible bit by bit shifting 0x1 seven times and OR it with the mantissa.
   • Use a while loop to shift the mantissa to the right to take away its 0s after the decimal point.
   • The while loop also counts how many shift it has done, which we would use to calculate the decimal point for the mantissa as 7 - count.
2. Perform Multiplication of Mantissas and Perform Normalization
   • 7 - count1 + 7 - count2 = 14 - count1 - count2 is the decimal point of the product mantissa.
   • The key is that the result of mantissa multiplication is either 1.XXXXX or 10.XXXXX because of the invisible 1 being always in the front.
   • Use a variable e to check if the bit to the left of decimal point is 0 or 1.
   • If 1, no need to normalize, clear the bit to hide invisible 1.
   • If 0, normalize the MSB and shift the mantissa to the right once and record exponent offset.
3. Shift the Product Mantissa To Appropriate Position for float16
   • if 7 - (14 - count1 - count2) > 0, this means the product mantissa has less decimal point than the allotted 7 bits, so we shift it left by the difference.
   • Else, we do the opposite.
4. Sign Bit
   • Extract the sign bits of x and y.
5. Exponent
   • Extract the exponent and subtract 127 and add the offset.
6. Assemble
   • Get the sign bit by XOR the two sign.
   • Assemble everything by OR.

Time Complexity

• O(1), the while loop at most loops 7 times to clear the 0 bits.

Space Complexity

• O(1), we always store float16

What I Could Do Better?

• There's no need to use while loops to clear the trailing zeros after the decimal point. I could've directly multiply the two mantissas after adding the invisible bit. Their product will always be 15 to 16 bits, and I could start from there to retrieve the 7 bits of mantissas I need.
• I coded the the more efficient version in main1.cpp in the C_Version folder.

Helpful Links

Use float.exposed to quickly convert binary floating point to decimal floating point and vice versa.