6-bit floating point numbers
============================

uses a 6-bit number to represent fractional values from -14 to +14.
there are 64 distinct float6 values.

this is not a real specification but the ideas and layout directly
translate to other floating point types (float32 and float64, float16
and bfloat16, and so on).

BIT LAYOUT:
1 bit  sign      (0 is positive, 1 is negative)
3 bits exponent  (the value's magnitude)
2 bits mantissa  (the value's closest interval)

BIT PATTERNS:
s eee mm  value
= === ==  ============================
1 111 00  -inf, "negative infinity"
0 111 00  +inf, "positive infinity"
_ 111 __  NaN, "not a number" (m != 0)
_ ___ __  normal number (1 <= e <= 6)
1 000 00  -0, "negative zero"
0 000 00  0, "zero"
_ 000 __  subnormal number (m != 0)

RANGE:
-14 to +14 (-1.75 * 8 to +1.75 * 8)

FORMULA FOR NORMAL NUMBERS (1 <= e <= 6):
(1 - 2*sign) * (1 + mantissa/4) * 2^(e-3)

FORMULA FOR SUBNORMAL NUMBERS (e=0):
(1 - 2*sign) * (0 + mantissa/4) * 2^-2

SENTINAL VALUES (special values that aren't numbers):
-inf (s=1 e=7 m=0)
+inf (s=0 e=7 m=0)
 NaN (e=7 m!=0)

CHART SHOWING ALL 64 VALUES (symbolic)
e | s=1,m=3 | s=1,m=2 | s=1,m=1 | s=1,m=0 | s=0,m=0 | s=0,m=1 | s=0,m=2 | s=0,m=3 |
===================================================================================
7 |  NaN    |  NaN    |  NaN    | -inf    | +inf    |  NaN    |  NaN    |  NaN    |
6 | -1.75*8 | -1.50*8 | -1.25*8 | -1.00*8 | +1.00*8 | +1.25*8 | +1.50*8 | +1.75*8 |
5 | -1.75*4 | -1.50*4 | -1.25*4 | -1.00*4 | +1.00*4 | +1.25*4 | +1.50*4 | +1.75*4 |
4 | -1.75*2 | -1.50*2 | -1.25*2 | -1.00*2 | +1.00*2 | +1.25*2 | +1.50*2 | +1.75*2 |
3 | -1.75*1 | -1.50*1 | -1.25*1 | -1.00*1 | +1.00*1 | +1.25*1 | +1.50*1 | +1.75*1 |
2 | -1.75/2 | -1.50/2 | -1.25/2 | -1.00/2 | +1.00/2 | +1.25/2 | +1.50/2 | +1.75/2 |
1 | -1.75/4 | -1.50/4 | -1.25/4 | -1.00/4 | +1.00/4 | +1.25/4 | +1.50/4 | +1.75/4 |
0 | -0.75/4 | -0.50/4 | -0.25/4 | -0.00/4 | +0.00/4 | +0.25/4 | +0.50/4 | +0.75/4 |

CHART SHOWING ALL 64 VALUES (literal)
e | s=1,m=3 | s=1,m=2 | s=1,m=1 | s=1,m=0 | s=0,m=0 | s=0,m=1 | s=0,m=2 | s=0,m=3 |
===================================================================================
7 |  NaN    |  NaN    |  NaN    |  -inf   | +inf    |  NaN    |  NaN    |  NaN    |
6 | -14     | -12     | -10     |  -8     |  8      |  10     |  12     |  14     |
5 |  -7     |  -6     |  -5     |  -4     |  4      |   5     |   6     |   7     |
4 |  -3.5   |  -3     |  -2.5   |  -2     |  2      |   2.5   |   3     |   3.5   |
3 |  -1.75  |  -1.5   |  -1.25  |  -1     |  1      |   1.25  |   1.5   |   1.75  |
2 |  -0.875 |  -0.75  |  -0.625 |  -0.5   |  0.5    |   0.625 |   0.75  |   0.875 |
1 |  -0.4375|  -0.375 |  -0.3125|  -0.25  |  0.25   |   0.3125|   0.375 |   0.4375|
0 |  -0.1875|  -0.125 |  -0.0625|  -0     |  0      |   0.0625|   0.125 |   0.1875|

NOTES:
- exact integer values are only available from -8 to +8
- there are the same number of positive and negative values
- there are two zero values (zero and negative zero)
- negative zero and zero are typically considered equal (except during specific numerical algorithms)
- there are six different NaN values
- each value actually represents an interval of possible numbers
- when doing math with floating point, the "nearest" value is chosen as the result