fp: The Floating Point Explorer

Note: This is a work-in-progress. It is not currently complete.

The screen is divided into several sections. The upper part of the screen deals with the representation of a double. The 64 check boxes represent the 64 bits of a double, shown in binary. The high-order bit is the sign. The next 11 bits are the exponent value, and the remaining 52 bits are the mantissa.

The Exponent Box

The Set All and Clear All buttons will set or clear all the bits of the exponent. The 2^0 button sets the exponent to represent 2^0.

The edit control displays the exponent in hexadecimal. You can type the desired hexadecimal exponent, which ranges from 0000 to 07FF.

The Value edit control and associated spin control show and allow you to modify the exponent value, which is allowed to be in the range of 0..2047. However, the more natural represention would be the Excess-1023 representation. Thus 0 given in the Excess-1023 control represents the absolute numeric Value in the exponent of 1023.

TheSignificand Box

The Set All and Clear All buttons will set or clear all the bits of the significand. The representation also allows you to see the hexadecimal value of the significand, and edit it by changing the hex value displayed.

The Show Nybbles check box displays markers every four bits, making it somewhat easier to read the bits.

There are four buttons to select specific representations, such as +inf, -inf, +Nan and -NaN. For the NaN representations, an exponent of all 1s and a nonzero mantissa is a NaN, and I arbitrarily chose to set the low-order bit of the mantissa to represent these values. You can experiment with other bits.

The Number

Together, all of these controls allow you to manipulate what is called "The Number".

The Working Registers

There are three "working registers" designated n1, n2 and n3.

Transferring to and from The Number

The large vertical arrow buttons pointing downward transfer a value from The Number to one of the three working registers. The large upward pointing arrows transfer a value from one of the working registers to The Number, so it can be examined in detail.

Transferring Working Registers

The horizontal arrow buttons transfer a value from the working register below them to one of the other working registers.

The Operation buttons

The operation buttons perform the indicated operation n3 = n1 op n2.

A typical sort of test might involve typing value into n1 (for example, 80), typing a value into n2 (for example, 100) and performing an operation such as division, seeing the result in n3. Then the large up-arrow above n3 is clicked to transfer the result back to The Number so its representation can be examined.

Formatting

The Format box allows you to select a printf-style format.

In terms of formatting, this also allows you to see the effects of the various formatting options you might use in a program. The Format options allow selection of the flags, the width, the precision, and the type. The formatting is shown to the left of the Format box. The format can be associated with the display of n1, n2, n3 or The Number by clicking one of the small arrow buttons.

The FPU ops buttons

An FPU has a number of built-in operations. These instructions will be executed based on which button is clicked. Each button specifies where it gets its operand(s) and where it puts its result. The instructions used appear in the box on the right.

The buttons are arranged in alphabetical order, left-to-right, top-to-bottom.

The convention used is that when the instruction references the two top stack elements, referred to in the manuals as ST(0) and ST(1), n1 represents ST(0)`` and 'n2' represents ST(1)`.

Operation	Source	Destination	Function
F2XM1	n1	n3	n3 = 2^n1 - 1
FABS	n1	n3	n3 = abs(n1)
FCHS	n1	n3	n3 = -n1
FCOMP	n1	n2
FCOS	n1	n3	n3 = cos(n1)
FLD1		n3	n3 = 1.0
FLDPI		n3	n3 = π
FLDL2E		n3	n3 = log2(e)
FLDL2T		n3	n3 = log2(10)
FLDLG2		n3	n3 = log10(2)
FLDLN2		n3	n3 = ln(2)
FLDZ		n3	n3 = 0.0
FPATAN	n1,n2	n3	n3 = arctan(n2 / n1)
FPREM	n1,n2	n3
FPREM1	n1,n2	n3	n3 = ?(rnd(n1 / n2) * n2)
FPTAN	n1	n3	n3 = tan(n1)
FRNDINT	n1	n3	n2 = ⌈(int)n1 + 0.5⌉
FSCALE	n1,n2	n3	n3 = n1 * 2^(rnd0(n2))
FSIN	n1	n3	n3 = sin(n1)
FSINCOS	n1	n2,n3	n2 = sin(n1); n3 = cos(n1)
FSQRT	n1	n3	n3 = √n1
FXAM	n1		See table
FXTRACT	n1	n2,n3
FYL2X	n1,n2	n3	n3 = n1 * log2(n2)
FYL2XP1	n1,n2	n3	n3 = n1 * log2(n2 + 1)

rnd0() is round-towards-zero, that is, the fractional part is truncated and only the integer part is considered.

F2XM1 results

Computes 2^(n1) - 1. 'n1' must be in the range -1.0..+1.0

n1	n3
-1.0 to -0	-0.5 to -0
-0	-0
+0	+0
+0 to +1.0	+0 to 1.0

Note that values other than 2 can be exponentiated according to the formula x^y = 2^(y * log2(x))

FABS results

n1	n3
-∞	+∞
-F	+F
-0	+0
+0	+0
+F	+F
+∞	+∞
NaN	NaN

FCHS results

n1	n3
-∞	+∞
-F	+F
-0	+0
+0	-0
+F	-F
+∞	-∞
NaN	NaN

FCOS results

n1	n3
-∞	*
-F	-1 to +1
-0	+1
+0	+1
+F	-1 to +1
+∞	*
NaN	NaN

• indicates an invalid-operand exception, which is indicated by the IE bit in the FPU Status Word

FPREM/FPREM1 results

n1↓ n2→	-∞	-F	-0	+0	+F	+∞	NaN
-∞	*	*	*	*	*	*	NaN
-F	n1	±F or -0	**	**	±F or -0	n1	NaN
-0	-0	-0	*	*	-0	-0	Nan
+0	+0	+0	*	*	-0	-0	NaN
+F	n1	±F or +0	**	**	±F or +0	n1	NaN
+∞	*	*	*	*	*	*	NaN
Nan	NaN	NaN	NaN	NaN	NaN	Nan	Nan

"*" indicates an invalid-operand exception, which is indicated by the IE bit in the FPU Status Word

FSIN results

n1	n3
-∞	*
-F	-1 to +1
-0	-0
+0	+0
+F	-1 to +1
+∞	*
NaN	NaN

FSINCOS results

n1	n2 sin	n3 cos
-∞	*	*
-F	-1 to +1	-1 to +1
-0	-0	+1
+0	+0	+1
+F	-1 to +1	-1 to +1
+∞	*	*
NaN	NaN	Nan

"*" indicates an invalid-operand exception, which is indicated by the IE bit in the FPU Status Word

FSQRT results

n1	n3
-∞	*
-F	*
-0	-0
+0	+0
+F	√F
+∞	+∞
NaN	NaN

FXTRACT results

This instruction extracts the exponent and the significand from the value in 'n1'. The exponent, which is the power of 2, is stored in n2, and the significand is stored in n3. The significand is essentially identical to the significand in n1, except its exponent is 0, which means, in excess-1023 notation, it is 1023.

The x87 FPU status word

For compatibilitly with existing libraries, there is an equivalent of the x87 FPU status register. This appears on the right of the Explorer. These bits are read-only boxes and cannot be set by the user. However, they should be cleared before performing an operation you need to use them after, since they are not cleared automatically.

The small box under the "Top" bits decodes the stack pointer depth.

Bit	Mnemonic	Meaning
15	B
14	C3
13-11	Top	Stack pointer
10	C2
9	C1
8	C0
7	ES
6	SF
5	PE	Inexact Precision
4	UE	Underflow
3	OE	Overflow
2	ZE	Divide by zero
1	DE	A Denormalized result occurred
0	IE	Invalid result

FXAM condition codes

Condition	C3	C2	C1	C0
Unsupported	0	0		0
NaN	0	0		1
Normal finite number	0	1		0
Infinity	0	1		1
Zero	1	0		0
Empty	1	0		1
Denormal number	1	1		0
Negative			1

Floating point condition codes

C0

Instruction	Condition
FCOS	undefined
FRNDINT	undefined
FSCALE	undefined
FSIN	undefined
FSINCOS	undefined
FSQRT	undefined
FSUB	undefined
FXAM	See FXAM table

C1

| Instruction | Condition | | FRNDINT | 0: Stack underflow | | | 1: Result was rounded up | | FSCALE | 0: Stack underflow or result was not rounded up | | | 1: Result was rounded up | | FSIN | 0: Stack underflow or result was not rounded up | | | 1: Result was rounded up | | FSINCOS | 0: Stack underflow or result was not rounded up | | | 1: Result was rounded up | | FSQRT | 0: Stack underflow or result was not rounded up | | | 1: Result was rounded up | | FSUB | 0: Stack underflow or result was not rounded up | | | 1: Result was rounded up | | FXAM | 0: Positive number | | | 1: Negative number |

C2

Instruction	Condition
FPTAN	1: Operand outside acceptable range
FRNDINT	undefined
FSIN	0: Source operand in valid range
	1: Source operand outside range (-2^63 < n1 < 2^63)
FSINCOS	0: Source operand in valid range
	1: Source operand outside range (-2^63 < n1 < 2^63)
FSCALE	undefined
FSQRT	undefined
FSUB	undefined
FXAM	See FXAM table

C3

Instruction	Condition
FCOS	undefined
FRNDINT	undefined
FSCALE	undefined
FSIN	undefined
FSINCOS	undefined
FSQRT	undefined
FSUB	undefined
FXAM	See FXAM table

32-bit application

This is written as a 32-bit application. It uses embedded _asm directives, which are not supported in the 64-bit compiler. If you want to create a 64-bit version, please do it with #ifdef directives since the 64-bit intrinsics will not compile in the 32-bit version.

Microsoft Foundation Class Output

MICROSOFT FOUNDATION CLASS LIBRARY : fp Project Overview

The application wizard has created this fp application for you. This application not only demonstrates the basics of using the Microsoft Foundation Classes but is also a starting point for writing your application.

This is a summary of what you will find in each of the files that make up your fp application.

Files

• fp.vcproj

  • This is the main project file for VC++ projects generated using an application wizard. It contains information about the version of Visual C++ that generated the file, and information about the platforms, configurations, and project features selected with the application wizard.

• fp.h

  • This is the main header file for the application. It includes other project specific headers (including Resource.h) and declares the CfpApp application class.

• fp.cpp

  • This is the main application source file that contains the application class CfpApp.

• fp.rc

  • This is a listing of all of the Microsoft Windows resources that the program uses. It includes the icons, bitmaps, and cursors that are stored in the RES subdirectory. This file can be directly edited in Microsoft Visual C++. Your project resources are in 1033.

• res\fp.ico

  • This is an icon file, which is used as the application's icon. This icon is included by the main resource file fp.rc.

• res\fp.rc2

  • This file contains resources that are not edited by Microsoft Visual C++. You should place all resources not editable by the resource editor in this file.

Classes

The application wizard creates one dialog class:

• fpDlg.h, fpDlg.cpp - the dialog

  • These files contain your CfpDlg class. This class defines the behavior of your application's main dialog. The dialog's template is in fp.rc, which can be edited in Microsoft Visual C++.

• StdAfx.h, StdAfx.cpp

  • These files are used to build a precompiled header (.PCH) file named fp.pch and a precompiled types file named StdAfx.obj.

• Resource.h

  • This is the standard header file, which defines new resource IDs. Microsoft Visual C++ reads and updates this file.

• fp.manifest

  • Application manifest files are used by Windows XP to describe an applications dependency on specific versions of Side-by-Side assemblies. The loader uses this information to load the appropriate assembly from the assembly cache or private from the application. The Application manifest maybe included for redistribution as an external .manifest file that is installed in the same folder as the application executable or it may be included in the executable in the form of a resource.

Other notes:

If your application uses MFC in a shared DLL, and your application is in a language other than the operating system's current language, you will need to copy the corresponding localized resources MFC70XXX.DLL from the Microsoft Visual C++ CD-ROM under the Win\System directory to your computer's system or system32 directory, and rename it to be MFCLOC.DLL. ("XXX" stands for the language abbreviation. For example, MFC70DEU.DLL contains resources translated to German.) If you don't do this, some of the UI elements of your application will remain in the language of the operating system.