(x − 1)⁶ near x = 1, computed twice, negative in both and not in the same places

At its defaults it draws (x − 1)⁶ near x = 1, computed twice, negative in both and not in the same places. The polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 0.001 of x = 1, in units of 2.68·10⁻¹⁵ — the size of the rounding, since the true value here is under 10⁻¹⁸ and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 179 of the 401 points, the second at 196, and the two disagree about the sign at 98. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.44·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

sign-window 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 polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 0.001 of x = 1, in units of 2.68·10⁻¹⁵ — the size of the rounding, since the true value here is under 10⁻¹⁸ and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 179 of the 401 points, the second at 196, and the two disagree about the sign at 98. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.44·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

The polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 0.001 of x = 1, in units of 2.68·10⁻¹⁵ — the size of the rounding, since the true value here is under 10⁻¹⁸ and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 179 of the 401 points, the second at 196, and the two disagree about the sign at 98. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.44·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

The polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 0.1 of x = 1, in units of 10·10⁻⁷ — the size of the rounding, since the true value here is under 10⁻⁶ and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 7 of the 401 points, the second at 6, and the two disagree about the sign at 2. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.54·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

The polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 0.01 of x = 1, in units of 10⁻¹² — the size of the rounding, since the true value here is under 10⁻¹² and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 53 of the 401 points, the second at 55, and the two disagree about the sign at 29. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.38·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

The polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 10⁻⁴ of x = 1, in units of 2.66·10⁻¹⁵ — the size of the rounding, since the true value here is under 10⁻²⁴ and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 184 of the 401 points, the second at 196, and the two disagree about the sign at 106. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.43·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

The polynomial x⁶ − 6x⁵ + 15x⁴ − 20x³ + 15x² − 6x + 1 evaluated by Horner's rule at 401 points within 10⁻⁵ of x = 1, in units of 2.66·10⁻¹⁵ — the size of the rounding, since the true value here is under 10⁻³⁰ and is a flat line on this axis. The upper trace rounds the multiply and the add separately; the lower one fuses them. A sixth power cannot be negative and the first is negative at 186 of the 401 points, the second at 205, and the two disagree about the sign at 97. Neither is more nearly right: both are the accumulated rounding of six cancelling terms, and they differ by at most 1.41·10⁻¹⁵ — one rounding of one product, six times. Any test on this value — a bisection, a definiteness check, a sign — gets a different answer from the two builds at a quarter of the points here.

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.

2 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.