MantissaAndExponent

New to Analytica 5.2

The MantissaAndExponent function is used to examine the internals of a floating-point number. This is sometimes useful when debugging difficult numeric round-off problems.

MantissaAndExponent( x, integerMantissa)

Returns the mantissa and base-2 exponent of a floating-point number. The function has two return values, so you need to capture them using the syntax

Local (mantissa, exponent) := MantissaAndExponent(x)

The return values are such that «x» is exactly equal to mantissa * 2^exponent. The sign of the mantissa matches the sign of «x».

When the optional «integerMantissa» is omitted or False, then exponent is an integer and for any finite non-zero «x», [math]\displaystyle{ 0.5 \le mantissa \lt 1 }[/math]. This variation is equivalent to the frexp(x) function found in many other programming languages.

When «integerMantissa» is True, the mantissa is an integer, [math]\displaystyle{ -2^{53} \lt mantissa \lt 2^{53} }[/math], and exponent in an integer or is NaN when «x» is NaN. The exponent returned will usually be smaller by 53 (the number of bits in the mantissa) than the exponent returned when integerMantissa:True is not specified.

Floating point from Mantissa and Exponent

To obtain a floating point number from a mantissa and exponent, the simple expression

mantissa * 2^exponent

returns the corresponding floating point number except in the case where exponent=1024. The problem in the 1024 case is that 2^1024 exceeds the maximum representable finite floating-point number, so 2^exponent overflows to INF, resulting in INF. Because 0.5 <= mantissa < 1, the true number is actually finite. The 1024 case does not occur when using integerMantissa:True, so for that usage the simple expression is sufficient.

To handle the full case, you can introduce the following User-Defined Function:

Function Ldexp(mantissa, exponent)

Definition:

If Not IsNaN(exponent) And exponent=1024

Then (mantissa*2) * 2^1023

Else mantissa * 2^exponent

Edge cases

The following table describes the various special cases that can arise, and the representation returned in both usage cases.

	Real mantissa		Integer mantissa
«x»	mantissa	exponent	mantissa	exponent
0 (zero)	0	0	0	0
Inf	Inf	-1	2^52	972
-Inf	-Inf	-1	-2^52	972
NaN	NaN	-1	«nan code»	NaN
Ordinary (normalized) number	[math]\displaystyle{ 0.5 \le m \lt 1.0 }[/math]	[math]\displaystyle{ -1021 \le e \le 1024 }[/math]	[math]\displaystyle{ 2^{52} \le |m| \lt 2^{53} }[/math]	[math]\displaystyle{ -971 \le e \le -1074 }[/math]
Sub-normalized number	[math]\displaystyle{ 0.5 \le m \lt 1.0 }[/math]	[math]\displaystyle{ -1073 \le e \lt -1021 }[/math]	[math]\displaystyle{ -2^{-52} \lt m \lt 2^{52} }[/math]	[math]\displaystyle{ -1074 }[/math]