Can't figure out how to convert this scientific # into IEEE 754

I can't seem to figure out how to convert 2 * 10^33 into IEEE 754 format.

I find the sign bit to be 0 I find the exponent to be 110 + bias (of 127) to be 0xED

But, the mantissa is just killing me.. I can't figure out why I keep getting 0 for this 开发者_StackOverflowpart.

---

You need the first 24 bits of 2*10^33. The first bit is always 1, and the remaining 23 bits form the last 23 bits of the IEEE-754 single-precision floating-point number.

Now, 2*10^33 has 110 binary digits, so it is too large to calculate exactly with most tools (calculators or programming languages). We can make things a little bit easier by noting that 2*10^33 = 2*(2*5)^33 = 2^34*5^33, so the first 24 bits of our number are the same as those of 5^33, which has only 76 bits.

We can further write:

5^33 = (2^7 - 3)^11
     = 2^77 - 11*3*2^70 + 55*9*2^63 - 165*27*2^56 + 330*81*2^49 
           - 462*243*2^42 + 462*729*2^35 - 330*2187*2^27 + ...
     = 2^53 * (2^24 - 33*2^17 + 495*2^10 - 4455*2^3 + 26730/2^4 
           - 112266/2^11 + 336798/2^18 - 721710/2^25 + ...)
     = 2^53 * (16777216 - 4325376 + 506880 - 35640 + 1670.625
           - 54.817... + 1.284... - 0.0215...)
     = 2^53 * 12924697.071
     = 2^53 * 110001010011011100011001b

where we rounded in the last step. So the stored part of the mantissa is 10001010011011100011001. Together with the information you already have, the result is:

0 11101101 10001010011011100011001

or in hex:

76C53719

---

If you want it done automatically, try this website. Type 2e33 into the top text box and hit the Rounded or Not Rounded buttons to get the answer.

---

If you type 2000000000000000000000000000000000 into my decimal/binary converter you will get

110001010011011100011001000100100011011001001100111000110000010101101100001010000000000000000000000000000000000

Rounded to 24 significant bits -- the number of bits in a float -- this is 110001010011011100011001 (the trailing 23 bits of this are the mantissa).