Floating Point Arithmetic, Part One

A month ago I was discussing some of the issues in integer arithmetic, and I said that issues in floating point arithmetic were a good subject for another day. Over the weekend I got some questions from a reader about floating point arithmetic, so this seems like as good a time as any to address them.

To distinguish between decimal and binary numbers, I'm going to do all binary numbers in blue fixed-width.

Here's how floating point numbers work.  A float is 64 bits.  Of that, one bit represents the sign: 0 is positive, 1 is negative.  

Eleven bits represent the exponent.  To determine the exponent value, treat the exponent field as an eleven-bit unsigned integer, then subtract 1023.  However, note that the exponent fields 00000000000 and 11111111111 have special meaning, which we'll come to later.

To compute the value of a float, here's what you do.  You take the mantissa, and you stick a "1." onto its left hand side.  Then you compute that value as a 53 bit fraction with 52 fractional places.  Then you multiply that by two to the power of the given exponent value, and sign it appropriately.

(1, 10000000001, 0110000000000000000000000000000000000000000000000000)

The sign is 1, so its a negative number.  The exponent is 1025 - 1023 = 2.  Put a 1. on the top of the mantissa and you get 1.0110000000000000000000000000000000000000000000000000 = 1.375 and sure enough, -1.375 x 2² = -5.5

However, you might be wondering how zero is represented, since every bit pattern has 1. plunked onto the beginning.  That's where the special values for the exponent come in.  If the exponent is 00000000000, then the float is considered a "denormal".  It gets 0. plunked onto the beginning, not 1., and the exponent is assumed to be -1022.  This has the nice property that if all bits in the float are zero, it's representing zero. Note that this lets you represent smaller numbers than you would be able to otherwise, as we'll see, though you pay the price of lower precision.  Essentially, denormals exist so that the chip can do "graceful underflow" -- represent tiny values without having to go straight to zero.

If the exponent 11111111111 and the fraction is all zeros, that's Infinity.  If the exponent is 11111111111 and the fraction is not all zeros, that's considered to be Not A Number -- this is a bit pattern reserved for errors.

which is 1.1111111111111111111111111111111111111111111111111111 x 2¹⁰²³, and

which is  1.000 x 2^-1022

which is 0.0000000000000000000000000000000000000000000000000001 x 2^-1022  = 2^-1074, and

which is 0.1111111111111111111111111111111111111111111111111111 x 2^-1022