# IBM Hexadecimal Floating Point

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Our technical support group recently received a request for a tool that would convert IBM System/360 hexadecimal floating point numbers to the IEEE-754 format. I am probably the only one left at MathWorks that actually used IBM mainframe computers. I thought we had seen the last of hexadecimal arithmetic years ago. But, it turns out that the hexadecimal floating point format is alive and well.

### Contents

#### IBM System/360

The System/360 is a family of mainframe computers that IBM introduced in 1965 and that dominated the computer industry until PCs came along twenty years later. They range in size from desk-sized to systems that fill a large room.

Here is a photo of a mid-sized model.

*System/360, Model 60.* *Photo from Ken Shirrif's blog, IBM 360/System Summary*.

The System/360 architecture is byte-oriented, so it can handle business data processing as well as scientific and engineering computation. This leads to base-16, rather than base-2 or base-10, floating point arithmetic.

* Binary f*2^e 1/2<=f<1 * Decimal f*10^e 1/10<=f<1 * Hexadecimal f*16^e 1/16<=f<1

#### Formats

Floating point formats played an important role in technical computing in the early days. This table from FMM lists formats that were in use in the 1970s, before IEEE-754 was introduced in 1985.

The System/360 hexadecimal format is used in many industries for the preservation of data files.

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Enthought. Python wrappers for C.

#### Hex_ieee

Hex_ieee. I have two twenty-line MATLAB functions, ieee2ibm and ibm2ieee, that convert IEEE-754 floating point to and from IBM hexadecimal format.

Three statements in the middle of ieee2ibm are the key to the entire operation. The first statement is

[~,e] = log2(x)

With two output arguments, log2 returns the mantissa and exponent of an IEEE-754 floating point number. The mantissa is not needed here.

The second key statement

e = ceil(e/4)

makes e divisible by 4. This turns e into the appropriate hexadecimal exponent so that the third statement

f = x.*16.^(-e)

can produce the hexadecimal mantissa.

#### ieee2ibm

function z = ieee2ibm(x) Convert IEEE-754 to IBM System 360 hexadecimal. z = ieee2ibm(x) Input x, real column vector. Output z, length(x)-by-16 char. Example: ieee2ibm(-118.625) = 'C276A00000000000'.

s = sign(x); % -1, 0, or 1 x = abs(x); x(x < 16^(-65)) = 0; % Underflow x(x >= 16^63) = (1-eps/2)*16^63; % Overflow

[~,e] = log2(x); % base 2 exponent e = ceil(e/4) % base 16 exponent f = x.*16.^(-e); % base 16 mantissa

E = uint64((e+64)*2^56); % Assemb1e output F = uint64(f*2^56); S = uint64((1-s)*2^62); % 1 or 0 z = dec2hex(S + E + F); % z = 'ZZFFFFFFFFFFFFFF' end

#### ibm2ieee

function x = ibm2ieee(z) Convert IBM System 360 hexadecimal to IEEE-754. x = ibm2ieee(z) Input z, n-by-16 char. Output x, n-by-1 double. Example: ibm2ieee('C276A00000000000') = -118.625.

E = hex2dec(z(:,1:2)); % Disassemble input F1 = hex2dec(z(:,3:8)); % < 16^6 F2 = hex2dec(z(:,9:end)); % < 16^8 s = sign(128-E); % -1 or 1

e = E-(s>0)*64-(s<0)*192; % base 16 exponent f = F1/16^6 + F2/16^14; % base 16 mantissa x = s.*f.*16.^e; end

#### Examples

Underflow. Anything < 16^(-65) is too small and is flushed to zero. There are no denormals.

Overflow. Anything >= 16^63 is too large. There is no inf or NaN.

* 1.0 4110000000000000 * 0.1 401999999999999A * -pi C13243F6A8885A30 * 5.3976e-79 0010000000000000 * 7.2370e+75 7FFFFFFFFFFFFFF8

#### Comparison

S/360 hexadecimal has 7 exponent bits, while IEEE-754 has 11. Consequently, hexadecimal has a much smaller range, 5.4e-79 to 7.2e+75 versus 2.2e-308 to 1.8e+308.

The base-16 normalization implies that hexadecimal effectively has between 53 and 56 mantissa bits. Counting the hidden bit, IEEE-754 also has 53. So, the accuracy of the two is pretty much the same.

#### Software

My functions ieee2ibm and ieee2ibm described above, modified to handle both single and double, plus hex_test, which does what its name implies, are available at Hex_ieee.

Homework: What happens?

ok = 0; for k = 1:10 x = single(k/10); ok(k) = hex_test(x); end ok

Published with MATLAB® R2024a

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