partial-walk
At its defaults it draws the error of a reduction is a walk, and the number of pieces changes which walk. Running error of two reductions of the same 2048 numbers, against how many terms have been consumed: one accumulator, and 8 pieces combined at the end. Each is the distance between what the arithmetic has accumulated so far and the exact sum of the same terms, carried in a compensated pair so the instrument does not round. The two separate within the first few terms and never meet again, ending at 7.56·10⁻¹⁴ and 2.41·10⁻¹⁴. The mechanism is in the horizontal axis: a rounding is committed at every partial sum and its size is that partial sum's spacing, and this vector's running total reaches 32.1 on the way to an answer of 1.63·10⁻⁵. Dividing the work shortens each walk, so the mean error over eight vectors falls from 4.78·10⁻¹⁴ at one piece to 8.14·10⁻¹⁵ at sixty-four: more workers is more accurate, which is the opposite of what the word *parallel* suggests here.
partial-walk is one function in lib/figures/repro.js —
one sum, several answers — what a reduction's division costs, and the two policies that remove it. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Running error of two reductions of the same 2048 numbers, against how many terms have been consumed: one accumulator, and 8 pieces combined at the end. Each is the distance between what the arithmetic has accumulated so far and the exact sum of the same terms, carried in a compensated pair so the instrument does not round. The two separate within the first few terms and never meet again, ending at 7.56·10⁻¹⁴ and 2.41·10⁻¹⁴. The mechanism is in the horizontal axis: a rounding is committed at every partial sum and its size is that partial sum's spacing, and this vector's running total reaches 32.1 on the way to an answer of 1.63·10⁻⁵. Dividing the work shortens each walk, so the mean error over eight vectors falls from 4.78·10⁻¹⁴ at one piece to 8.14·10⁻¹⁵ at sixty-four: more workers is more accurate, which is the opposite of what the word *parallel* suggests here.
p: 8
The arguments are the ones A bound every answer satisfies passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Running error of two reductions of the same 2048 numbers, against how many terms have been consumed: one accumulator, and 8 pieces combined at the end. Each is the distance between what the arithmetic has accumulated so far and the exact sum of the same terms, carried in a compensated pair so the instrument does not round. The two separate within the first few terms and never meet again, ending at 7.56·10⁻¹⁴ and 2.41·10⁻¹⁴. The mechanism is in the horizontal axis: a rounding is committed at every partial sum and its size is that partial sum's spacing, and this vector's running total reaches 32.1 on the way to an answer of 1.63·10⁻⁵. Dividing the work shortens each walk, so the mean error over eight vectors falls from 4.78·10⁻¹⁴ at one piece to 8.14·10⁻¹⁵ at sixty-four: more workers is more accurate, which is the opposite of what the word *parallel* suggests here.
p: 2
The arguments are the ones Where the disagreement comes from passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Running error of two reductions of the same 2048 numbers, against how many terms have been consumed: one accumulator, and 2 pieces combined at the end. Each is the distance between what the arithmetic has accumulated so far and the exact sum of the same terms, carried in a compensated pair so the instrument does not round. The two separate within the first few terms and never meet again, ending at 7.56·10⁻¹⁴ and 1.13·10⁻¹³. The mechanism is in the horizontal axis: a rounding is committed at every partial sum and its size is that partial sum's spacing, and this vector's running total reaches 32.1 on the way to an answer of 1.63·10⁻⁵. Dividing the work shortens each walk, so the mean error over eight vectors falls from 4.78·10⁻¹⁴ at one piece to 8.14·10⁻¹⁵ at sixty-four: more workers is more accurate, which is the opposite of what the word *parallel* suggests here.
p: 128
The arguments are the ones Where the disagreement comes from passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Running error of two reductions of the same 2048 numbers, against how many terms have been consumed: one accumulator, and 128 pieces combined at the end. Each is the distance between what the arithmetic has accumulated so far and the exact sum of the same terms, carried in a compensated pair so the instrument does not round. The two separate within the first few terms and never meet again, ending at 7.56·10⁻¹⁴ and 1.86·10⁻¹⁴. The mechanism is in the horizontal axis: a rounding is committed at every partial sum and its size is that partial sum's spacing, and this vector's running total reaches 32.1 on the way to an answer of 1.63·10⁻⁵. Dividing the work shortens each walk, so the mean error over eight vectors falls from 4.78·10⁻¹⁴ at one piece to 8.14·10⁻¹⁵ at sixty-four: more workers is more accurate, which is the opposite of what the word *parallel* suggests here.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
5 distinct claims across 4 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a partition count a machine might have
a vector the walk can be drawn over
and dividing the reduction further makes it more accurate, not less
the running total dwarfs the answer, which is what cancellation is
two partitionings end at two different errors
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 197
of 363 generators —
179 print a residual and
18 are exempt with a published reason;
166 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A bound every answer satisfies
The classical bound on a summation error is correct, it covers all twenty-six answers one vector produced, and it is 7,932 times larger than the difference between them. A statement true of every ordering cannot say which ordering you got.
The answer that depends on the machineAccuracy and agreement are different properties
The most accurate policy on this site's summation figure returns 119 different answers, and the one that returns a single answer is four orders less accurate. Neither property implies the other, and the vocabulary has one word for both.
The answer that depends on the machineAn inner product with no fixed sign
‖QᵀQ − I‖ is how this site turns "orthogonal" into a number, and across ten partitionings it moves by 2.4%. The entries it is built from are not so lucky: 125 of the 1,128 off-diagonal pairs take both signs, and one of them takes five different values including zero.
Two errors, and whose fault they areThe fifth author
Four authors of a wrong answer have been named on this site and each is a statement about one computation. The fifth is not: it is what separates two computations that are both correct, it is a backward error of measurable size, and no residual, bound or condition number contains it.
The answer that depends on the machineThe length that changes the kernel
A dot product's accuracy steps by a factor of 1.57 between 63 and 64 terms, on vectors drawn identically at both lengths. Nothing about the problem changes there. A library switches from one accumulator to four, at a constant in somebody else's source file.
The answer that depends on the machineThe same program, twice
One vector of 4,096 numbers, one summation algorithm, one precision, twenty-six runs — and twenty-one different answers. Nothing in the program chose between them, every one of them satisfies the textbook bound, and the exactly rounded answer is not among them.
The answer that depends on the machineThe sum that cannot be wrong
Snap every addend to a common multiple before adding, and every partial sum is exact — so the order stops mattering, by construction rather than by luck. Four hundred permutations return one value where an ordinary reduction returns three hundred and three.
The answer that depends on the machineThe variation that comes with a seed
A randomised low-rank approximation's error moves by 31% between draws and by 10⁻¹⁵ between partitionings of one draw. In the one field on this site whose answer already comes as a band, the machine is inside the width of the line — and it is still there.
The answer that depends on the machineThe vector that hides it
Every quick demonstration of a parallel sum uses positive numbers, and positive numbers are the one family where the effect is absent. Measured on six inner products this site already computes, the summation condition number runs from exactly 1 to 10¹⁷ — and the safe end is where nobody makes a decision.
The answer that depends on the machineTwo machines, one certificate
Nothing a solver returns says which of its answers you got. Four things could be reported instead — the summation condition number, the partition count, an exactly accumulated residual and a directed-rounding interval — and each costs about one pass over data the routine already has in hand.
The answer that depends on the machineWhat determinism costs
Six ways to add up a vector, priced in operations per element and in accuracy. Nothing sits in the bottom left of the figure — an answer that is the same on every machine costs between three and twelve operations where an answer that is not costs one.
The answer that depends on the machineWhere the disagreement comes from
The error of a reduction is a walk whose step length is the spacing of the running total, not of the answer. That one sentence predicts the size of the disagreement to a factor of two, explains why dividing the work makes it smaller, and explains why the value cannot be predicted at all.