Machine epsilon computation issue

问题

I stumbled on the difference between the result of Machine epsilon calculation. When compared to 0 PHP yields 4.9406564584125E-324. While for 1 it pops up with 1.1102230246252E-16.

Quite a difference. Guess it's something with the type of data initially set by default in PHP.

The code is:

<?php
//Machine epsilon calculation

$e = 1;
$eTmp = null;

for ($i = 0; 0 != 0 + $e; $i++){ //Changing 0 by 1 produces absolutely different result
     $e = $e/2;
    if ($e != 0) {$eTmp = $e;}
}

echo $eTmp;
//var_dump($eTmp);

?>

Any clarification on the difference between the two? And how can a variable or value by assigned manually to float in PHP? Many thanks for your ideas!

回答1:

Your PHP implementation appear to be using the common IEEE 754 64-bit binary format. In this format, finite values are represented, effectively, as a sign, an integer M less than 2⁵³, and an integer exponent e between −1074 and 971, inclusive. The value represented is +M•2^e or −M•2^e, according to the sign. (This format will often be described with M being a fraction between 1 and 2 with a certain number of bits after the radix point. The descriptions are mathematically equivalent, just useful for different purposes.)

Given this, we can easily see that the next representable number after 0 is +1•2⁻¹⁰⁷⁴, which is approximately 4.94065645841246544•10⁻³²⁴.

In this format as stated, 1 can be represented as +1•2⁰. However, to see what the smallest change that can be made to 1 is, we must normalize it by making M as large as possible. 1 can also be represented with M = 2⁵² and e = −52, resulting in +2⁵²•2⁻⁵². In this form, we can see the next representable value greater than 1 is achieved by adding 1 to M, mkaing the number +(2⁵²+1)•2⁻⁵².

The difference between 1 and +(2⁵²+1)•2⁻⁵² is of course 1•2⁻⁵², which is exactly 2.220446049250313080847263336181640625•10⁻¹⁶.

Note that your code records $eTmp (if $e is not zero) after reducing $e with $e = $e/2, which means it reports the first value of $e that does not cause a change. Thus it reports 1.11e-16 rather than 2.22e-16, which is the last value that does cause a change to 1.

来源：https://stackoverflow.com/questions/48830402/machine-epsilon-computation-issue