# Quadruple Precision, 128-bit Floating Point Arithmetic

The floating point arithmetic format that occupies 128 bits of storage is known as binary128 or quadruple precision. This blog post describes an implementation of quadruple precision programmed entirely in the MATLAB language. ### Contents

#### Background

The IEEE 754 standard, published in 1985, defines formats for floating point numbers that occupy 32 or 64 bits of storage. These formats are known as binary32 and binary64, or more frequently as single and double precision. For many years MATLAB used only double precision and it remains our default format. Single precision has been added gradually over the last several years and is now also fully supported.

A revision of IEEE 754, published in 2008, defines two more floating point formats. One, binary16 or half precision, occupies only 16 bits and was the subject of my previous blog post. It is primarily intended to reduce storage and memory bandwidth requirements. Since it provides only "half" precision, its use for actual computation is problematic.

The other new format introduced in IEEE 754-2008 is binary128 or quadruple precision. It is intended for situations where the accuracy or range of double precision is inadequate.

I see two descriptions of quadruple precision software implementations on the Web.

I have not used either package, but judging by their Web pages, they both appear to be complete and well supported.

The MATLAB Symbolic Math Toolbox provides vpa, arbitrary precision decimal floating point arithmetic, and sym, exact rational arithmetic. Both provide accuracy and range well beyond quadruple precision, but do not specifically support the 128-bit IEEE format.

My goal here is to describe a prototype of a MATLAB object, fp128, that implements quadruple precision with code written entirely in the MATLAB language. It is not very efficient, but is does allow experimentation with the 128-bit format.

#### Beyond double

There are other floating point formats beyond double precision. Long double usually refers to the 80-bit extended precision floating point registers available with the Intel x86 architecture and described as double extended in IEEE 754. This provides the same exponent range as quadruple precision, but much less accuracy.

Double double refers to the use of a pair of double precision values. The exponent field and sign bit of the second double are ignored, so this is effectively a 116-bit format. Both the exponent range and the precision are more than double but less than quadruple.

#### Floating point anatomy

The format of a floating point number is characterized by two parameters, p, the number of bits in the fraction and q, the number of bits in the exponent. I will compare four precisions, half, single, double, and quadruple. The four pairs of characterizing parameters are

   p = [10, 23, 52 112];

   q = [5, 8, 11, 15];


With these values of p and q, and with one more bit for the sign, the total number of bits in the word, w, is a power of two.

   format shortg
w = p + q + 1

w =
16    32    64   128


Normalized numbers

Most floating point numbers are normalized, and are expressed as

$$x = \pm (1+f)2^e$$

The fraction $f$ is in the half open interval

$$0 \leq f < 1$$

The binary representation of $f$ requires at most p bits. In other words $2^p f$ is an integer in the range

$$0 \leq 2^p f < 2^p$$

The exponent $e$ is an integer in the range

$$-b+1 \leq e \leq b$$

The quantity $b$ is both the largest exponent and the bias.

$$b = 2^{q-1} - 1$$

   b = 2.^(q-1)-1

b =
15         127        1023       16383


The fractional part of a normalized number is $1+f$, but only $f$ needs to be stored. That leading $1$ is known as the hidden bit.

Subnormal

There are two values of the exponent $e$ for which the biased exponent, $e+b$, reaches the smallest and largest values possible to represent in q bits. The smallest is

$$e + b = 0$$

The corresponding floating point numbers do not have a hidden leading bit. These are the subnormal or denormal numbers.

$$x = \pm f 2^{-b}$$

Infinity and Not-A-Number

The largest possible biased exponent is

$$e + b = 2^q-1$$.

Quantities with this exponent field represent infinities and NaN, or Not-A-Number.

The percentage of floating point numbers that are exceptional because they are subnormal, infinity or NaN increases as the precision decreases. Exceptional exponents are only $2$ values out of $2^q$. For quadruple precision this is $2/2^{15}$, which is less than a one one-thousandth of one percent.

Encode the sign bit with s = 0 for nonnegative and s = 1 for negative. And encode the exponent with an offsetting bias, b. Then a floating point number can be packed in w bits with

  x = [s e+b 2^p*f]

#### Precision and range

epsilon

If a real number cannot be expressed with a binary expansion requiring at most p bits, it must be approximated by a floating point number that does have such a binary representation. This is roundoff error. The important quantity characterizing precision is machine epsilon, or eps. In MATLAB, eps(x) is the distance from x to the next larger (in absolute value) floating point number (of that class). With no argument, eps is simply the difference between 1 and the next larger floating point number.

    format shortg
eps = 2.^(-p)

eps =
0.00097656   1.1921e-07   2.2204e-16   1.9259e-34


This tells us that quadruple precision is good for about 34 decimal digits of accuracy, double for about 16 decimal digits, single for about 7, and half for about 3.

realmax

If a real number, or the result of an arithmetic operation, is too large to be represented, it overflows and is replaced by infinity. The largest floating point number that does not overflow is realmax. When I try to compute quadruple realmax with double precision, it overflows. I will fix this up in the table to follow.

    realmax = 2.^b.*(2-eps)

realmax =
65504   3.4028e+38  1.7977e+308          Inf


realmin

Underflow and representation of very small numbers is more complicated. The smallest normalized floating point number is realmin. When I try to compute quadruple realmin it underflows to zero. Again, I will fix this up in the table.

    realmin = 2.^(-b+1)

realmin =
6.1035e-05   1.1755e-38  2.2251e-308            0


tiny

But there are numbers smaller than realmin. IEEE 754 introduced the notion of gradual underflow and denormal numbers. In the 2008 revised standard their name was changed to subnormal.

Think of roundoff in numbers near underflow. Before 754, floating point numbers had the disconcerting property that x and y could be unequal, but their difference could underflow, so x-y becomes 0. With 754 the gap between 0 and realmin is filled with numbers whose spacing is the same as the spacing between realmin and 2*realmin. I like to call this spacing, and the smallest subnormal number, tiny.

    tiny = realmin.*eps

tiny =
5.9605e-08   1.4013e-45  4.9407e-324            0


#### Floating point integers

flintmax

It is possible to do integer arithmetic with floating point numbers. I like to call such numbers flints. When we write the numbers $3$ and $3.0$, they are different descriptions of the same integer, but we think of one as fixed point and the other as floating point. The largest flint is flintmax.

    flintmax = 2./eps

flintmax =
2048   1.6777e+07   9.0072e+15   1.0385e+34


Technically all the floating point numbers larger than flintmax are integers, but the spacing between them is larger than one, so it is not safe to use them for integer arithmetic. Only integer-valued floating point numbers between 0 and flintmax are allowed to be called flints.

#### Table

Let's collect all these anatomical characteristics together in a new MATLAB table. I have now edited the output and inserted the correct quadruple precision values.

   T = [w; p; q; b; eps; realmax; realmin; tiny; flintmax];

T = table(T(:,1), T(:,2), T(:,3), T(:,4), ...
'rownames',{'w','p','q','b','eps','realmax','realmin', ...
'tiny','flintmax'});

type Table.txt

                 half         single        double       quadruple
__________    __________    ___________    __________

w                 16            32             64           128
p                 10            23             52           112
q                  5             8             11            15
b                 15           127           1023         16383
eps       0.00097656    1.1921e-07     2.2204e-16    1.9259e-34
realmax        65504    3.4028e+38    1.7977e+308   1.190e+4932
realmin   6.1035e-05    1.1755e-38    2.2251e-308   3.362e-4932
tiny      5.9605e-08    1.4013e-45    4.9407e-324   6.475e-4966
flintmax        2048    1.6777e+07     9.0072e+15    1.0385e+34


#### fp128

I am currently working on code for an object, @fp128, that could provide a full implementation of quadruple-precision arithmetic. The methods available so far are

   methods(fp128)

Methods for class fp128:

abs         eq          le          mtimes      realmax     subsref
cond        fp128       lt          ne          realmin     svd
diag        frac        lu          norm        shifter     sym
disp        ge          max         normalize   sig         times
display     gt          minus       normalize2  sign        tril
double      hex         mldivide    plus        sqrt        triu
ebias       hypot       mpower      power       subsasgn    uminus
eps         ldivide     mrdivide    rdivide     subsindex



The code that I have for quadrule precision is much more complex than the code that I have for half precision. There I am able to "cheat" by converting half precision numbers to doubles and relying on traditional MATLAB arithmetic. I can't do that for quads.

The storage scheme for quads is described in the help entry for the constructor.

   help @fp128/fp128

  fp128 Quad precision constructor.
z = fp128(x) has three fields.
x = s*(1+f)*2^e, where
z.s, one uint8, s = (-1)^sg = 1-2*sg, sg = (1-s)/2.
z.e, 15 bits, biased exponent, one uint16.
b = 2^14-1 = 16383,
eb = e + b,
1 <= eb <= 2*b for normalized quads,
eb = 0 for subnormal quads,
eb = 2*b+1 = 32767 for infinity and NaN.
z.f, 112 bits, nonnegative fraction,
4-vector of uint64s, each with 1/4-th of the bits,
0 <= f(k) < 2^28, 4*28 = 112.
z.f represents sum(f .* pow2), pow2 = 2.^(-28*(1:4))

Reference page in Doc Center
doc fp128



Breaking the 112-bit fraction into four 28-bit pieces makes it possible to do arithmetic operations on the pieces without worrying about integer overflow. The core of the times code, which implements x.*y, is the convolution of the two fractional parts.

   dbtype 45:53 @fp128/times

45            % The multiplication.
46            % z.f = conv(x.f,y.f);
47            % Restore hidden 1's.
48            xf = [1 x.f];
49            yf = [1 y.f];
50            zf = zeros(1,9,'uint64');
51            for k = 1:5
52                zf(k:k+4) = zf(k:k+4) + yf(k)*xf;
53            end


The result of the convolution, zf, is a uint64 vector of length nine with 52-bit elements. It must be renormalized to the fit the fp128 storage scheme.

Addition and subtraction involve addition and subtraction of the fractional parts after they have been shifted so that the corresponding exponents are equal. Again, this produces temporary vectors that must be renormalized.

Scalar division, y/x, is done by first computing the reciprocal of the denominator, 1/x, and then doing one final multiplication, 1/x * y. The reciprocal is computed by a few steps of Newton iteration, starting with a scaled reciprocal, 1/double(x).

#### Examples

The output for each example shows the three fields in hexadecimal -- one sign field, one biased exponent field, and one fraction field that is a vector with four entries displayed with seven hex digits. This is followed by a 36 significant digit decimal representation.

One

   clear
format long
one = fp128(1)


one =
0 3fff  0000000  0000000  0000000  0000000
1.0


eps

   eps = eps(one)


eps =
0 3f8f  0000000  0000000  0000000  0000000
0.000000000000000000000000000000000192592994438723585305597794258492732


1 + eps

   one_plus_eps = one + eps


one_plus_eps =
0 3fff  0000000  0000000  0000000  0000001
1.00000000000000000000000000000000019


2 - eps

   two_minus_eps = 2 - eps


two_minus_eps =
0 3fff  fffffff  fffffff  fffffff  fffffff
1.99999999999999999999999999999999981


realmin

   rmin = realmin(one)


rmin =
0 0001  0000000  0000000  0000000  0000000
3.3621031431120935062626778173217526e-4932


realmax

   rmax = realmax(one)


rmax =
0 7ffe  fffffff  fffffff  fffffff  fffffff
1.18973149535723176508575932662800702e4932


Compute 1/10 with double, then convert to quadruple.

   dble_tenth = fp128(1/10)


dble_tenth =
0 3ffb  9999999  99999a0  0000000  0000000
0.100000000000000005551115123125782702


   quad_tenth = 1/fp128(10)


0 3ffb  9999999  9999999  9999999  9999999
0.0999999999999999999999999999999999928


Double precision pi converted to quadruple.

   dble_pi = fp128(pi)


dble_pi =
0 4000  921fb54  442d180  0000000  0000000
3.14159265358979311599796346854418516


   quad_pi = fp128(sym('pi'))


0 4000  921fb54  442d184  69898cc  51701b8
3.1415926535897932384626433832795028


#### Matrix operations

The 4-by-4 magic square from Durer's Melancholia II provides my first matrix example.

   clear
M = fp128(magic(4));


Let's see how the 128-bit elements look in hex.

   format hex
M


M =
0 4003  0000000  0000000  0000000  0000000
0 4001  4000000  0000000  0000000  0000000
0 4002  2000000  0000000  0000000  0000000
0 4001  0000000  0000000  0000000  0000000
0 4000  0000000  0000000  0000000  0000000
0 4002  6000000  0000000  0000000  0000000
0 4001  c000000  0000000  0000000  0000000
0 4002  c000000  0000000  0000000  0000000
0 4000  8000000  0000000  0000000  0000000
0 4002  4000000  0000000  0000000  0000000
0 4001  8000000  0000000  0000000  0000000
0 4002  e000000  0000000  0000000  0000000
0 4002  a000000  0000000  0000000  0000000
0 4002  0000000  0000000  0000000  0000000
0 4002  8000000  0000000  0000000  0000000
0 3fff  0000000  0000000  0000000  0000000


Check that the row sums are all equal. This matrix-vector multiply can be done exactly with the flints in the magic square.

   e = fp128(ones(4,1))
Me = M*e


e =
0 3fff  0000000  0000000  0000000  0000000
0 3fff  0000000  0000000  0000000  0000000
0 3fff  0000000  0000000  0000000  0000000
0 3fff  0000000  0000000  0000000  0000000

Me =
0 4004  1000000  0000000  0000000  0000000
0 4004  1000000  0000000  0000000  0000000
0 4004  1000000  0000000  0000000  0000000
0 4004  1000000  0000000  0000000  0000000


I've overloaded mldivide, so I can solve linear systems and compute inverses. The actual computation is done by lutx, a "textbook" function that I wrote years ago, long before this quadruple-precision project, followed by the requisite solution of triangular systems. But now the MATLAB object system insures that every individual arithmetic operation is done with IEEE 754 quadruple precision.

Let's generate a 3-by-3 matrix with random two-digit integer entries.

   A = fp128(randi(100,3,3))


A =
0 4002  0000000  0000000  0000000  0000000
0 4001  8000000  0000000  0000000  0000000
0 4004  b000000  0000000  0000000  0000000
0 4005  3800000  0000000  0000000  0000000
0 4005  7800000  0000000  0000000  0000000
0 4002  a000000  0000000  0000000  0000000
0 4004  c800000  0000000  0000000  0000000
0 4004  7800000  0000000  0000000  0000000
0 4000  0000000  0000000  0000000  0000000


I am going to use fp128 backslash to invert A. So I need the identity matrix in quadruple precision.

   I = fp128(eye(size(A)));


Now the overloaded backslash calls lutx, and then solves two triangular systems to produce the inverse.

   X = A\I


X =
0 3ff7  2fd38ea  bcfb815  69cdccc  a36d8a5
1 3ff9  c595b53  8c842ee  f26189c  a0770d4
0 3ffa  c0bc8b7  4adcc40  4ea66ca  61f1380
1 3ff7  a42f790  e4ad874  c358882  7ff988e
0 3ffa  12ea8c2  3ef8c17  01c7616  5e03a5a
1 3ffa  70d4565  958740b  78452d8  f32d866
0 3ff9  2fd38ea  bcfb815  69cdccc  a36d8a7
0 3ff3  86bc8e5  42ed82a  103d526  a56452f
1 3ff6  97f9949  ba961b3  72d69d9  4ace666


Compute the residual.

   AX = A*X
R = I - AX;
format short
RD = double(R)


AX =
0 3fff  0000000  0000000  0000000  0000000
0 3f90  0000000  0000000  0000000  0000000
1 3f8d  0000000  0000000  0000000  0000000
0 0000  0000000  0000000  0000000  0000000
0 3fff  0000000  0000000  0000000  0000000
0 3f8d  8000000  0000000  0000000  0000000
1 3f8c  0000000  0000000  0000000  0000000
1 3f8d  0000000  0000000  0000000  0000000
0 3ffe  fffffff  fffffff  fffffff  ffffffb
RD =
1.0e-33 *
0         0    0.0241
-0.3852         0    0.0481
0.0481   -0.0722    0.4815


Both AX and R are what I expect from arithmetic that is accurate to about 34 decimal digits.

Although I get a different random A every time I publish this blog post, I expect that it has a modest condition number.

   kappa = cond(A)


kappa =
0 4002  7e97c18  91278cd  8375371  7915346
11.9560249020758193065358323606886569


Since A is not badly conditioned, I can invert the computed inverse and expect to get close to the original integer matrix. The elements of the resulting Z are integers, possibly bruised with quadruple precision fuzz.

   format hex
Z = X\I


Z =
0 4002  0000000  0000000  0000000  0000000
0 4001  8000000  0000000  0000000  0000000
0 4004  b000000  0000000  0000000  0000004
0 4005  37fffff  fffffff  fffffff  ffffffc
0 4005  77fffff  fffffff  fffffff  ffffffe
0 4002  a000000  0000000  0000000  0000001
0 4004  c7fffff  fffffff  fffffff  ffffffc
0 4004  77fffff  fffffff  fffffff  ffffffc
0 3fff  fffffff  fffffff  fffffff  ffffffc


I have just nonchalantly computed cond(A). Here is the code for the overloaded cond.

   type @fp128/cond.m

function kappa = cond(A)
sigma = svd(A);
kappa = sigma(1)/sigma(end);
end



So it is correctly using the singular value decomposition. I also have svd overloaded. The SVD computation is handled by a 433 line M-file, svdtx, that, like lutx, was written before fp128 existed. It was necessary to modify five lines in svdtx. The line

u = zeros(n,ncu);


u = fp128(zeros(n,ncu));


Similarly for v, s, e and work. I should point out that the preallocation of the arrays is inherited from the LINPACK Fortran subroutine DSVDC. Without it, svdtx would not have required any modification to work correctly in quadruple precision.

Let's compute the full SVD.

   [U,S,V] = svd(A)


U =
1 3ffe  57d9492  76f3ea4  dc14bb3  15d42c1
1 3ffe  75a77c4  8c7b469  2cac695  59be7fe
1 3ffc  0621737  9b04c78  1c2109d  8736b46
1 3ffb  38214c0  d75c84c  4bcf5ff  f3cffd7
1 3ffb  a9281e3  e12dd3a  d632d61  c8f6e60
0 3ffe  fbbccdc  a571fa1  f5a588b  fb0d806
1 3ffe  79587db  4889548  f09ae4b  cd0150c
0 3ffe  59fae16  17bcabb  6408ba4  7b2a573
0 3ff8  cde38fc  e952ad5  8b526c2  780c2e5

S =
0 4006  1f3ad79  d0b9b08  18b1444  030e4ef
0 0000  0000000  0000000  0000000  0000000
0 0000  0000000  0000000  0000000  0000000
0 0000  0000000  0000000  0000000  0000000
0 4004  a720ef6  28c6ec0  87f4c54  82dda2a
0 0000  0000000  0000000  0000000  0000000
0 0000  0000000  0000000  0000000  0000000
0 0000  0000000  0000000  0000000  0000000
0 4002  8061b9a  0e96d8c  c2ef745  9ea4c9a

V =
1 3ffb  db3df03  9b5e1b3  5bf4478  0e42b0d
1 3ffe  b540007  4d4bc9e  dc9461a  0de0481
1 3ffe  03aaff4  d9cea2c  e8ee2bc  2eba908
0 3ffe  fa73e09  9ef8810  a03d2eb  46ade00
1 3ffa  b316e2f  fe9d3ae  dfa9988  fbca927
1 3ffc  184af51  f25fece  97bc0da  5ff13a2
1 3ffb  706955f  a877cbb  b63f6dd  4e2150e
0 3ffe  08fc1eb  7b86ef7  4af3c6c  732aae9
1 3ffe  b3aaead  ef356e2  7cd2937  94b22a7


Reconstruct A from its quadruple precision SVD. It's not too shabby.

   USVT = U*S*V'


USVT =
0 4001  fffffff  fffffff  fffffff  fffffce
0 4001  7ffffff  fffffff  fffffff  fffffc7
0 4004  b000000  0000000  0000000  000000a
0 4005  37fffff  fffffff  fffffff  ffffff1
0 4005  77fffff  fffffff  fffffff  ffffff6
0 4002  9ffffff  fffffff  fffffff  fffffd2
0 4004  c7fffff  fffffff  fffffff  ffffff1
0 4004  77fffff  fffffff  fffffff  ffffff4
0 3fff  fffffff  fffffff  fffffff  ffffe83


#### Rosser matrix

An interesting example is provided by a classic test matrix, the 8-by-8 Rosser matrix. Let's compare quadruple precision computation with the exact rational computation provided by the Symbolic Math Toolbox.

First, generate quad and sym versions of rosser.

   R = fp128(rosser);
S = sym(rosser)

S =
[  611,  196, -192,  407,   -8,  -52,  -49,   29]
[  196,  899,  113, -192,  -71,  -43,   -8,  -44]
[ -192,  113,  899,  196,   61,   49,    8,   52]
[  407, -192,  196,  611,    8,   44,   59,  -23]
[   -8,  -71,   61,    8,  411, -599,  208,  208]
[  -52,  -43,   49,   44, -599,  411,  208,  208]
[  -49,   -8,    8,   59,  208,  208,   99, -911]
[   29,  -44,   52,  -23,  208,  208, -911,   99]


R is symmetric, but not positive definite, so its LU factorization requires pivoting.

   [L,U,p] = lutx(R);
format short
p

p =
1
2
3
7
6
8
4
5


R is singular, so with exact computation U(n,n) would be zero. With quadruple precision, the diagonal of U is

   format long e
diag(U)


ans =
611.0
836.126022913256955810147299509001582
802.209942588471107300640276546225738
99.0115741407236314604636000423592687
-710.481057851148425133280246646085002
579.272484693223512196223933017062413
-1.2455924519190846395771824210276321
0.000000000000000000000000000000215716190833522835766351129431653015


The relative size of the last diagonal element is zero to almost 34 digits.

   double(U(8,8)/U(1,1))

ans =
3.530543221497919e-34


Compare this with symbolic computation, which, in this case, can compute an LU decomposition with exact rational arithmetic and no pivoting.

   [L,U] = lu(S);
diag(U)

ans =
611
510873/611
409827400/510873
50479800/2049137
3120997/10302
-1702299620/3120997
255000/40901
0


As expected, with symbolic computation U(8,8) is exactly zero.

   r = svd(R)


r =
1020.04901842999682384631379130551006
1020.04901842999682384631379130550858
1019.99999999999999999999999999999941
1019.90195135927848300282241090227735
999.999999999999999999999999999999014
999.999999999999999999999999999998817
0.0980486407215169971775890977220345302
0.0000000000000000000000000000000832757192990287779822645036097560521


The Rosser matrix is atypical because its characteristic polynomial factors over the rationals. So, even though it is of degree 8, the singular values are the roots of quadratic factors.

   s = svd(S)

s =
10*10405^(1/2)
10*10405^(1/2)
1020
10*(1020*26^(1/2) + 5201)^(1/2)
1000
1000
10*(5201 - 1020*26^(1/2))^(1/2)
0


The relative error of the quadruple precision calculation.

   double(norm(r - s)/norm(s))

ans =
9.293610246879066e-34


#### Postscript

Finally, verify that we've been working all this time with fp128 and sym objects.

   whos

  Name       Size            Bytes  Class     Attributes

A          3x3              3531  fp128
AX         3x3              3531  fp128
I          3x3              3531  fp128
L          8x8                 8  sym
M          4x4              6128  fp128
Me         4x1              1676  fp128
R          8x8             23936  fp128
RD         3x3                72  double
S          8x8                 8  sym
U          8x8                 8  sym
USVT       3x3              3531  fp128
V          3x3              3531  fp128
X          3x3              3531  fp128
Z          3x3              3531  fp128
ans        1x1                 8  double
e          4x1              1676  fp128
kappa      1x1               563  fp128
p          8x1                64  double
r          8x1              3160  fp128
s          8x1                 8  sym



Published with MATLAB® R2017a

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