The same 4096 numbers, added up 26 ways, 21 different answers

At its defaults it draws the same 4096 numbers, added up 26 ways, 21 different answers. Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 1.01·10⁸: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 21 of the 26 runs returned distinct values, spanning 1.55·10⁻¹³ — 2.3·10⁷ ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.58·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

partition-answers 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.

Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 1.01·10⁸: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 21 of the 26 runs returned distinct values, spanning 1.55·10⁻¹³ — 2.3·10⁷ ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.58·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 1.01·10⁸: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 21 of the 26 runs returned distinct values, spanning 1.55·10⁻¹³ — 2.3·10⁷ ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.58·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 100: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 15 of the 26 runs returned distinct values, spanning 1.56·10⁻¹³ — 22 ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.47·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 1.01·10⁴: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 21 of the 26 runs returned distinct values, spanning 1.55·10⁻¹³ — 2800 ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.58·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 1.01·10⁶: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 20 of the 26 runs returned distinct values, spanning 1.55·10⁻¹³ — 3.6·10⁵ ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.58·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

Each dot is one reduction of one vector of 4096 binary64 numbers whose summation condition number is 1.01·10¹⁰: the vector is cut into p contiguous pieces, each piece summed left to right, and the pieces combined in a tree. The vertical position is the distance from the exactly rounded sum, relative. Filled dots are the static split a fixed thread count gives; open dots are unequal splits, which is what a work-stealing scheduler produces at the same p. 20 of the 26 runs returned distinct values, spanning 1.55·10⁻¹³ — 2.9·10⁹ ulps of the answer. The dashed line is the classical bound γ₍ₙ₋₁₎Σ|xᵢ|, which every one of them satisfies with 2.58·10⁴ to spare, and which is the same number for all of them: it contains n and Σ|xᵢ| and nothing about the order.

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.

5 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.