Single-Precision IEEE 754 Floating-Point Standard

Single-Precision IEEE 754 Floating-Point Standard look as follows:

$0^{th}$ bit represent Sign bit. (1 bit)

$1^{th}$ to $8^{th}$ bits represents Exponents. (8 bits)

$9^{th}$ to $31^{st}$ bits represents Mantissa also called Number or Significand. (23 bits)

Now, here the given number = 125.025

1] Sign Bit Calculation:

The given number is Positive.

Therefore,

Sign bit = 0

2] Mantissa or Significand Bits Calculation:

First, Convert 125 into binary format

$$ (125)_{10} = (1111101)_2 $$

Second, look at the fraction part now that is 0.025

Multiply this Fraction part value with 2 and Stop when we get a fractional part that is equal to zero..

Therefore,

1] 0.025 × 2 = $\implies 0 + 0.05 $

2] 0.05 × 2 = $\implies 0 + 0.1 $

3] 0.1 × 2 = $\implies 0 + 0.2 $

4] 0.2 × 2 = $\implies 0 + 0.4 $

5] 0.2 × 2 = $\implies 0 + 0.8 $

6] 0.8 × 2 = 1.6 $\implies 1 + 0.6 $

7] 0.6 × 2 = $\implies 1 + 0.2 $

8] 0.2 × 2 = $\implies 0 + 0.4 $

9] 0.4 × 2 = $\implies 0 + 0.8 $

10] 0.8 × 2 = $\implies 1 + 0.6 $

11] 0.6 × 2 = $\implies 1 + 0.2 $

12] 0.2 × 2 = $\implies 0 + 0.4 $

13] 0.4 × 2 = $\implies 0 + 0.8 $

14] 0.8 × 2 = $\implies 1 + 0.6 $

15) 0.6 × 2 = $\implies 1 + 0.2 $

16] 0.2 × 2 = $\implies 0 + 0.4 $

17] 0.4 × 2 = $\implies 0 + 0.8 $

18] 0.8 × 2 = $\implies 1 + 0.6 $

19] 0.6 × 2 = $\implies 1 + 0.2 $

20] 0.2 × 2 = $\implies 0 + 0.4 $

21] 0.4 × 2 = $\implies 0 + 0.8 $

22] 0.8 × 2 = $\implies 1 + 0.6 $

23] 0.6 × 2 = $\implies 1 + 0.2 $

24] 0.2 × 2 = $\implies 0 + 0.4 $

We didn't get any fractional part that was equal to zero.

But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (losing precision...)

Write numbers from top to bottom direction.

Therefore,

$$ (0.025)_{10} = (0.000001100110011001100110)_2 $$

Complete Binary form of the given number is

$$ (125.025)_{10} = (1111101.000001100110011001100110)_2 $$

Third, Convert this Complete binary format into Exponent format

Therefore,

$$ 1111101.000001100110011001100110 \implies 1.111101000001100110011001100110 \times 2^6 $$

Discard 1 and take 111101000001100110011001100110 as Mantissa or Significand

Now, remove the excess bits and adjust the length of Mantissa to 23 bits.

Therefore,

Mantissa holds = 11110100000110011001100

3] Exponent Bits Calculation:

Because it is a single-precision biased exponent hence 127 + 6 = 133 ......(+6 is the Exponent)

Exponent Number $$ (133)_{10} = (10000101)_2 $$

Exponent holds = 10000101

4] Complete Single-Precision Format:

Hence, the Complete Single-Precision IEEE 754 Floating-Point Representation of 125.025 looks as follows:

$$ Sign\ Bit + Exponent\ Bits + Mantissa\ or\ Significand\ Bits $$

$$ 125.025 = 0\ 10000101\ 11110100000110011001100 $$

This is written in hexadecimal form as 42FACCC.