### Numerical Errors

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Floating point arithmetic is only carried out to a finite accuracy by computer hardware. This means that there is

1. a limited number of significant digits (about 6 decimal digits for 32-bit floats, more, perhaps 16 digits, for 64-bits etc.)
2. a maximum floating point number
3. a minimum +ve floating point number >0

### 1.0 = 1.0 - δ

Because of the finite accuracy, there is a floating point number, δ > 0.0, such that 1.0 - δ equals 1.0 as far as the computer hardware can tell.

Click on `go' in the form below to see what δ is for the implementation of JavaScript on your computer:

δ is quite small, but it is certainly greater than zero.

### Underflow

Mathematically, the following two equations are equal:

``` big - (big'  - δ)           -- 1
(big -  big') + δ            -- 2
```
However, if big~big' then then equation #1 may loose accuracy. Consider the extreme case in which big=big'=1.0. Then equation #1 will evalute to zero, but equation #2 will evaluate to δ, not zero!
-- L.A., 1999

 Coding Ockham's Razor, L. Allison, Springer A Practical Introduction to Denotational Semantics, L. Allison, CUP

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 © L. Allison   http://www.allisons.org/ll/   (or as otherwise indicated), Faculty of Information Technology (Clayton), Monash University, Australia 3800 (6/'05 was School of Computer Science and Software Engineering, Fac. Info. Tech., Monash University, was Department of Computer Science, Fac. Comp. & Info. Tech., '89 was Department of Computer Science, Fac. Sci., '68-'71 was Department of Information Science, Fac. Sci.) Created with "vi (Linux + Solaris)",  charset=iso-8859-1,  fetched Thursday, 02-Feb-2023 05:59:25 AEDT.