Five summation policies, 400 permutations each, and how many answers each returned

At its defaults it draws five summation policies, 400 permutations each, and how many answers each returned. The same 1,024 numbers, summation condition number 10⁸, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 303 values and eight pieces return 72; Kahan's compensated loop is the most accurate of the three at 4.96·10⁻¹⁰ and still returns 119. Pre-rounded summation returns one value, at an error of 4.91·10⁻⁶ — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

permutation-census 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.

The same 1,024 numbers, summation condition number 10⁸, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 303 values and eight pieces return 72; Kahan's compensated loop is the most accurate of the three at 4.96·10⁻¹⁰ and still returns 119. Pre-rounded summation returns one value, at an error of 4.91·10⁻⁶ — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

The same 1,024 numbers, summation condition number 10⁸, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 303 values and eight pieces return 72; Kahan's compensated loop is the most accurate of the three at 4.96·10⁻¹⁰ and still returns 119. Pre-rounded summation returns one value, at an error of 4.91·10⁻⁶ — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

The same 1,024 numbers, summation condition number 101, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 62 values and eight pieces return 37; Kahan's compensated loop is the most accurate of the three at 6.35·10⁻¹⁶ and still returns 6. Pre-rounded summation returns one value, at an error of 5.57·10⁻¹² — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

The same 1,024 numbers, summation condition number 10⁴, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 316 values and eight pieces return 73; Kahan's compensated loop is the most accurate of the three at 5.49·10⁻¹⁴ and still returns 122. Pre-rounded summation returns one value, at an error of 4.56·10⁻¹⁰ — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

The same 1,024 numbers, summation condition number 10⁶, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 290 values and eight pieces return 69; Kahan's compensated loop is the most accurate of the three at 5.76·10⁻¹² and still returns 117. Pre-rounded summation returns one value, at an error of 4.58·10⁻⁸ — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

The same 1,024 numbers, summation condition number 10¹⁰, presented in 400 different orders. The bar is the number of distinct binary64 values the policy returned; the figure beside it is the worst relative error it made. A single accumulator returns 310 values and eight pieces return 72; Kahan's compensated loop is the most accurate of the three at 5.75·10⁻⁸ and still returns 122. Pre-rounded summation returns one value, at an error of 4.6·10⁻⁴ — four orders worse than the compensated loop and the same on every machine. Exact accumulation returns one value and the right one. The comparison is bitwise because the claim is bitwise: a tolerance here would pass everything.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks 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.

8 distinct claims across 6 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.

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 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 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.