accumulation-excess

At its defaults it draws what a chain of truncations costs, against the best rank-4 answer, in two regimes. Both curves are the error of a running rank-4 object divided by the error of the best rank-4 approximation of the exact sum — the answer nothing in a real code has and none could beat. The standard reading of formatted arithmetic is that a truncation after every operation must accumulate, so a chain of 32 of them is dangerous. Measured, the worst point on either curve is 1.040×. A naive bound multiplies the per-step error by the number of steps and would put the line at 32. The roundings are doing something — the energy they have discarded, which a truncation can only remove, grows from 1.8·10⁻⁷ to 0.03 across the sweep — they are just not doing it to the answer. What does limit the answer is whether the exact sum is still nearly rank 4, and the two regimes disagree about that by a factor of 1022 at the second term.

accumulation-excess is one function in lib/figures/recompress.js — arithmetic in the format — the one operation it is not closed under, and what a hundred roundings cost. 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.

Both curves are the error of a running rank-4 object divided by the error of the best rank-4 approximation of the exact sum — the answer nothing in a real code has and none could beat. The standard reading of formatted arithmetic is that a truncation after every operation must accumulate, so a chain of 32 of them is dangerous. Measured, the worst point on either curve is 1.040×. A naive bound multiplies the per-step error by the number of steps and would put the line at 32. The roundings are doing something — the energy they have discarded, which a truncation can only remove, grows from 1.8·10⁻⁷ to 0.03 across the sweep — they are just not doing it to the answer. What does limit the answer is whether the exact sum is still nearly rank 4, and the two regimes disagree about that by a factor of 1022 at the second term.

Both curves are the error of a running rank-4 object divided by the error of the best rank-4 approximation of the exact sum — the answer nothing in a real code has and none could beat. The standard reading of formatted arithmetic is that a truncation after every operation must accumulate, so a chain of 32 of them is dangerous. Measured, the worst point on either curve is 1.040×. A naive bound multiplies the per-step error by the number of steps and would put the line at 32. The roundings are doing something — the energy they have discarded, which a truncation can only remove, grows from 1.8·10⁻⁷ to 0.03 across the sweep — they are just not doing it to the answer. What does limit the answer is whether the exact sum is still nearly rank 4, and the two regimes disagree about that by a factor of 1022 at the second term.

Both curves are the error of a running rank-8 object divided by the error of the best rank-8 approximation of the exact sum — the answer nothing in a real code has and none could beat. The standard reading of formatted arithmetic is that a truncation after every operation must accumulate, so a chain of 32 of them is dangerous. Measured, the worst point on either curve is 1.058×. A naive bound multiplies the per-step error by the number of steps and would put the line at 32. The roundings are doing something — the energy they have discarded, which a truncation can only remove, grows from 1.7·10⁻⁷ to 0.036 across the sweep — they are just not doing it to the answer. What does limit the answer is whether the exact sum is still nearly rank 8, and the two regimes disagree about that by a factor of 1005 at the second term.

Both curves are the error of a running rank-2 object divided by the error of the best rank-2 approximation of the exact sum — the answer nothing in a real code has and none could beat. The standard reading of formatted arithmetic is that a truncation after every operation must accumulate, so a chain of 32 of them is dangerous. Measured, the worst point on either curve is 1.034×. A naive bound multiplies the per-step error by the number of steps and would put the line at 32. The roundings are doing something — the energy they have discarded, which a truncation can only remove, grows from 1.8·10⁻⁷ to 0.031 across the sweep — they are just not doing it to the answer. What does limit the answer is whether the exact sum is still nearly rank 2, and the two regimes disagree about that by a factor of 1109 at the second term.

Both curves are the error of a running rank-6 object divided by the error of the best rank-6 approximation of the exact sum — the answer nothing in a real code has and none could beat. The standard reading of formatted arithmetic is that a truncation after every operation must accumulate, so a chain of 32 of them is dangerous. Measured, the worst point on either curve is 1.049×. A naive bound multiplies the per-step error by the number of steps and would put the line at 32. The roundings are doing something — the energy they have discarded, which a truncation can only remove, grows from 1.8·10⁻⁷ to 0.033 across the sweep — they are just not doing it to the answer. What does limit the answer is whether the exact sum is still nearly rank 6, and the two regimes disagree about that by a factor of 1005 at the second term.

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.

13 distinct claims across 5 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.

It draws a decomposition and prints its residual. It calls accumulationSlopes, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 141 of 264 generators — 126 print a residual and 15 are exempt with a published reason; 123 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.