WEBVTT

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This overview explains how ordered Boolean outputs
occupy bit positions in an integer or packed word.

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Confusing assignment order with printed binary order can make
a correct value appear reversed or put it in the wrong position.

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For this packing example, assign bit zero to input zero zero, bit one to zero one, bit two to one zero, and bit three to one one.

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We pack the output bits of or.

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The input labels identify positions; they are not the values being packed.

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Multiply each output by its bit weight.

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The zero contributes nothing, and the three ones contribute two, four, and eight, totaling fourteen.

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Written as a binary numeral, the highest bit appears on the left, so the word reads one one one zero.

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The displayed truth-vector order is still zero one one one.

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For input one zero, select position two.

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The four-bit numeral one one one zero shifts right twice to zero zero one one.

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A mask names the positions to keep: zero zero zero one keeps only the low bit.

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Pairwise and gives zero and zero, zero and zero, one and zero, and one and one: zero zero zero one.

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The retrieved answer is one.

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The retained example uses hexadecimal notation.

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The prefix zero x marks base sixteen, where the digit E means fourteen.

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Each hexadecimal digit represents four binary positions.

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One E E E therefore spells zero zero zero one, then three groups of one one one zero.

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It is a different spelling of the same number, not a new logical operator.

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That number is seven thousand nine hundred eighteen in decimal.

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Its meaning as a truth word still depends on the declared valid width and assignment-to-bit order.

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Assign each truth output a bit position.

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For this example, assignments A B C D run from 0000 to 1111;

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output for assignment index j occupies bit j.

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Written binary numerals normally print their most significant bit first,

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so do not confuse that print order with assignment order.

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First compute A AND B and C OR D, then
take the exclusive-or of those results.

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In ascending assignment order, the outputs are 0111 0111 0111 1000.

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Packing with bit j assigned to output j gives hexadecimal 0x1EEE.

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A word array can carry those same bits in lanes.

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For a toy six-bit result, bitwise NOT must be followed
by mask 111111 to discard unused high bits.

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Packing changes storage and execution layout, not the function.

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Current direct BitSet evaluation memoizes
expression identities and supports fixed bindings;

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calling it raw unshared AST would misdescribe this implementation.

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Big integers and machine words need separate
measurements before choosing an engine.

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Return to the four-bit word for F equals A or B.

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The numeric word is fourteen.

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Its valid width is four, with inputs zero zero, zero one, one zero and one one assigned to bit positions zero through three.

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Here A and B are both zero.

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The numeric word fourteen has its lowest bit clear.

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Because input zero zero was assigned to bit zero, masking that bit gives zero.

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Reading bit one instead would give one and would answer the neighboring input zero one.

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An off-by-one address can produce a plausible but incorrect answer.

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This case establishes a specific contrast.

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Bit position comes from the declared order.

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Packed values still require exact addressing.

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Keep the input, intermediate values, and conclusion together when comparing it with the previous case.

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The calculated answer belongs to this complete assignment under the stated rule.

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The calculated answer belongs to this complete assignment under the stated rule.

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The word stores four valid truth outputs.

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A machine integer may contain many more bit positions, so its full machine-word complement

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does not by itself identify the valid outputs.

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Keep the width and the valid-bit mask alongside the data.

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Complementing and then masking to four bits gives one.

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In assignment order, that is one zero zero zero, the outputs of not or.

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The mask removes unused high bits instead of counting them

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as extra satisfying assignments.

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With four Boolean inputs, there are sixteen assignments and therefore

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sixteen outputs to pack.

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There are not merely four output bits because there are four input variables.

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Packing can reduce storage overhead, but whether it improves a particular request depends

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on construction, operations, and output delivery.

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on construction, operations, and output delivery.

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Declare the assignment-to-bit mapping and requested width.

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Mask negation to that width and measure
engines under the same output contract.