Floating Point Math

>Single precision

32-bit floating point value

Format:
SEEE EEEE EMMM MMMM MMMM MMMM MMMM MMMM

S=sign
E=exponent
M=mantissa

Sign bit is 0 for positive, 1 for negative.

Exponent is biased by 127.
Power	Bit pattern	Decimal value
0	01111111	127
+1	10000000	128
-1	01111110	126
+127	11111110	254
-126	00000001	1

Special conditons:
Sign	Exponent	Mantissa	Meaning
0	All 0's		All 0's		+0
1	All 0's		All 0's		-0
0	All 0's		Nonzero		+Denormal
1	All 0's		nonzero		-Denormal
0	All 1's		All 0's		+Infinity
1	All 1's		All 0's		-Infinity
0	All 1's		nonzero		NaN
1	All 1's		nonzero		NaN

Implied Mantissa bit:
The Mantissa is 23 bits long with an implied bit left of the radix. The implied bit is 1 for almost all circumstances. The implied bit is 0 only if the number is denormal.

Convert from decimal to binary floating point:

173.7 = b?

Sign = 0 (positive)

To get Exponent, repeatedly divide/multiply by two, while counting how many iterations, until there's there's only 1 to the left of the decimal point.
Final result will be a normalised number. The number of iterations will be our exponent (positive for division, negative for multiplication).

173.7/2=86.85			2^1
86.85/2=43.425			2^2
43.425/2=21.7125		2^3
21.7125/2=10.85625		2^4
10.85625/2=5.428125		2^5
5.428125/2=2.7140625		2^6
2.7140625/2=1.35703125		2^7 Finished, power is 7.

Power is 7. Our binary exponent is: 7+127=134=b1000_0110.

To get our binary Mantissa (for normals), we multiply the fractional part of our normalised number by two and allocate a 1 if the result is greater than 1, and 0 otherwise. When we get a 1, we subtract 1 from the number.

Normalised number into Mantissa:

Number		X2 Result	Binary result
0.35703125	0.7140625	0	;Since we didn't get a 1, we only copy the number from the x2 Result column to the Number 					;column of the next row.
0.7140625	1.428125	1	;Since we got a 1 here, when we copy the number over from the x2 Result column to the Number 
0.428125	0.85625		0	;column of the next row, we subtract 1.
0.85625		1.7125		1
0.7125		1.425		1
0.425		0.85		0
0.85		1.7		1
0.7		1.4		1
0.4		0.8		0
0.8		1.6		1
0.6		1.2		1
0.2		0.4		0	;We've just got a number that we got before, meaning a recursive sequence. We can take a 
				0	;shortcut and copy this sequence until we fill up the mantissa.
				1
				1
				0
				0
				1
				1
				0
				0
				1
				1

We now have our full floating point number: 0,100_0011_0,010_1101_1011_0011_0011_0011=0x432DB333 ;commas separate bit fields

Now try a negative number:

-0.75

Sign = 1

We can, from this point, act as if it's a positive number.

Normalise number:

0.75*2=1.5 2^-1

Get exponent: 127-1 = 126 = 01111110

Normalised number, so implied bit is 1.

Get Mantissa bit pattern:

0.5*2 = 1 ;Nothing left, so fill in rest of mantissa with 0.

Our final floating point binary number is: 1,011_1111_0,100_0000_0000_0000_0000_0000=0xBF400000

>Double Precision

64-bit floating point value

Format:

SEEEEEEEEEEEMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM ;1 sign bit, 11 exponent bits, 52 mantissa bits

Exponent has bias of 1,023. Rest of process is exactly the same.

Practice: Convert Pi to single precision floating point

3.1415926535897932384...

Sign is 0, since it's positive.

Normalize and acquire power

'' / 2 = 1.5707963267948966192313216916398 x 2^1 ;done

Get bit pattern of exponent

127+1=128=10000000 ;done

Get Mantissa pattern

Number					x2					Bit
0.5707963267948966192313216916398	1.1415926535897932384626433832795	1
0.1415926535897932384626433832795	0.283185307179586476925286766559	0
0.283185307179586476925286766559	0.566370614359172953850573533118	0
0.566370614359172953850573533118	1.132741228718345907701147066236	1
0.132741228718345907701147066236	0.265482457436691815402294132472	0
0.265482457436691815402294132472	0.530964914873383630804588264944	0
0.530964914873383630804588264944	1.061929829746767261609176529888	1
0.061929829746767261609176529888	0.123859659493534523218353059776	0
0.123859659493534523218353059776	0.247719318987069046436706119552	0
0.247719318987069046436706119552	0.495438637974138092873412239104	0
0.495438637974138092873412239104	0.990877275948276185746824478208	0
0.990877275948276185746824478208	1.981754551896552371493648956416	1
0.981754551896552371493648956416	1.963509103793104742987297912832	1
0.963509103793104742987297912832	1.927018207586209485974595825664	1
0.927018207586209485974595825664	1.854036415172418971949191651328	1
0.854036415172418971949191651328	1.708072830344837943898383302656	1
0.708072830344837943898383302656	1.416145660689675887796766605312	1
0.416145660689675887796766605312	0.832291321379351775593533210624	0
0.832291321379351775593533210624	1.664582642758703551187066421248	1
0.664582642758703551187066421248	1.329165285517407102374132842496	1
0.329165285517407102374132842496	0.658330571034814204748265684992	0
0.658330571034814204748265684992	1.316661142069628409496531369984	1
0.316661142069628409496531369984	0.633322284139256818993062739968	1 ;Final bit, next bit would be a 1, so we round up

We now have our mantissa pattern: 10010010000111111011011

Concatenate them all together: 0,100_0000_0,100_1001_0000_1111_1101_1011=0x40490FDB

Compare with Pi value from Propeller manual Pg. 93: 0x40490FDB