A determinant of one, computed as zero by the expression that is written down

At its defaults it draws a determinant of one, computed as zero by the expression that is written down. The matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

contraction-determinant is one function in lib/figures/fma.js — one multiply the compiler removed — a determinant of one computed as zero, and a verdict that is a build's. 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 matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

The matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

The matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

The matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

The matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

The matrix [[x+2, x+1], [x+3, x+2]] at x = 2ᵏ has determinant exactly 1 at every k. The two products are x² + 4x + 4 and x² + 4x + 3, so they differ by one part in x², and above k = 26 they round to the same binary64 number. Relative error of two evaluations: ad − bc as written, which is exact until k = 27 and then returns zero at every larger size; and Kahan's form, w = bc, e = bc − w recovered by the fusion, (ad − w) − e, which is exact throughout. The difference between them is one rounding of one product — a relative 10⁻¹⁶ — and it is the whole of the answer, because the subtraction that follows removes everything else. Both forms are IEEE-754 conforming and a compiler may emit either from the same source.

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.

4 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.