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The Inner Float applet, included below, lets you experiment with the IEEE 754 floating point format. This applet accompanies Chapter 14, "Floating Point Arithmetic," of Inside the Java 2 Virtual Machine.
The Inner Float applet displays the value of a float
in several formats. The top three lines show each of the three parts of a float
the sign, exponent, and mantissa individually in binary. The fourth line, labeled "binary," shows the entire 32bit image of the float
in binary (with all three pieces together). The fifth line, labeled "hex," shows the 32bit image of the float
in hex.
The last two lines show the value of the float
in radix two and ten. Note that the radix two scientific notation format shows the mantissa
and exponent in base ten. Before being displayed, the actual mantissa is multiplied by 2^{24},
which yields an integral number, and the unbiased exponent is decremented by 24. Both the
integral mantissa and exponent are then easily converted to base ten and displayed.
A few keypress sequences that illustrate interesting characteristics of IEEE 754 floating point numbers are given below:
Too Big For Its Increments
Try this keypress sequence:
*=0 ++ *=2, *=2, *=2, ... repeat until radix 2 == 8388608e1 ++, ++, ...
Press the *=2
button repeatedly until the exponent of the radix two display is a one.
When you then press the ++
button, note that the value of the float
doesn't change. The float
's value is too large to be affected by adding only one.
Too Small For Its Increments
Try this keypress sequence:
Min ++  note that you can't get back to Min
When you then press the ++
button, you loose precision because Min is so much smaller than one. When you then press the 
button, you get zero.
Equivalence of Radix 2 and Radix 10
Try this keypress sequence:
*=0 ++ *=2, *=2, *=2, ... repeat until radix 2 == 8388608
Press the *=2
button repeatedly until the exponent of the radix two display is a zero (it will just be shown as "8388608" without an "e0"). At that point,
the value displayed in radix 10 is 8.3361e+06, which is the same as the radix 2 number.
Underflow to Zero
Try this keypress sequence:
Min /=2 *=2, *=2, ... note that you can't get back to Min again
When you press the /=2
button, the value of the float
underflows to zero. When a floating point operation underflows to zero, all bits of precision are lost.
Underflow to Negative Zero
Try this keypress sequence:
Min *=1 /=2 *=2, *=2, ... note that you can't get back to (Min * 1) again
When you press the /=2
button, the value of the float
underflows to negative zero. When a floating point operation underflows to negative zero, as when it underflows to positive zero, all bits of precision are lost.
Overflow to Infinity
Try this keypress sequence:
Max *=2 /=2, /=2, ... note that you can't get back to Max again
When you press the *=2
button, the value of the float
overflows to infinity. When a floating point operation overflows to infinity, all bits of precision are lost.
Overflow to Negative Infinity
Try this keypress sequence:
Max *=1 *=2 /=2, /=2, ... note that you can't get back to (Max * 1) again
When you press the *=2
button, the value of the float
overflows to negative infinity. When a floating point operation overflows to negative infinity, as when it overflows to positive infinity, all bits of precision are lost.
Infinity Times Zero is NaN
Try this keypress sequence:
+Inf *=0
Infinity Times 1 Changes Sign of Infinity
Try this keypress sequence:
+Inf *=(1)
Infinity Times 2 is Still Infinity
Try this keypress sequence:
+Inf *=(2), *=(2), *=(2), ... repeat until you're convinced
For a float
, infinity is as big as it gets.
Click here to view a page of links to the source code of the Inner Float applet.
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