[Math] Finding the mantissa from binary with floating point numbers

binarycomputer sciencefloating point

Here is the example problem slide I am working with:

slide with task

I understand how to get the exponent, its just 2+128=130-127=3

I understand the first bit is the sign bit for positive or negative.

I just get lost with the mantissa. I know each mantissa has to have a 1 before the decimal, but I'm unsure where the .75 came from. 1100000000000000000000 isn't 75 in decimal I mean right?

How do I calculate the mantissa with floating point numbers in binary?

Best Answer

The number after "." is calculated by $$\frac1{2^n}$$ instead of $$2^n$$ So $$1.11 = 2^0 + (1/2^1) + (1/2^2)$$ $$= 1 + 0.5 + 0.25$$ $$=1.75$$

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[Math] How to convert from floating point binary to decimal in half precision(16 bits)

You are right.

You can do that automatically with python and numpy :

import numpy as np
import struct
a=struct.pack("H",int("0101011101010000",2))
np.frombuffer(a, dtype =np.float16)[0]

and you get : 117.0

[Math] Floating point number,Mantissa,Exponent

Usually the mantissa is considered to have a binary point after the first bit, so your mantissa would be $1.1100_2=\frac 74=1.75_{10}$. Sometimes a leading $1$ is assumed, so your mantissa would be $(1).11100_2=\frac{15}8=1.875_{10}$ This gives one more bit of precision. To find the exponent, you subtract the offset from the stored value. You probably meant $2^{k-1}-1$ as the offset. Can you do that?

Best Answer

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