Published
Edited
Apr 27, 2021
1 star
m = 9
n = 2
divide = (a, b) => {
let isNegative = false;
let dividend = a;
let divisor = b;
// check for signed bit of dividend
if (AND(dividend, LSHIFT(1, 31)) == LSHIFT(1, 31)) {
isNegative = !isNegative;
// Negation
dividend = addOne(NOT(dividend));
}
// check for signed bit of divisor
if (AND(divisor, LSHIFT(1, 31)) === LSHIFT(1, 31)) {
isNegative = !isNegative;
divisor = addOne(NOT(divisor));
}
let quotient = 0;
let r = null;
for (let i = 30; i >= 0; i = subtractOne(i)) {
// left shift divisor until it's smaller than dividend
r = LSHIFT(divisor, i);
if (r < Number.MAX_VALUE && r >= 0) {
if (r <= dividend) {
quotient = OR(quotient, LSHIFT(1, i));
dividend = subtract(dividend, r);
}
}
}
if (isNegative) {
quotient = addOne(NOT(quotient));
}
return quotient;
}
divideExample = divide(m, n)
divideProof = parseInt(m / n) === divide(m, n)
add = (a, b) => {
let x = a;
let y = b;
do {
const partialSum = XOR(x, y);
const carry = LSHIFT(AND(x, y), 1);
x = partialSum;
y = carry;
} while (!!y);
return x;
}
addExample = add(m, n)
addProof = (m + n) === add(m, n)
md`## \`addOne(a)\`
Adds \`1\` to \`a\``
addOne = a => add(a, 1)
add1example = addOne(m)
add1proof = (m + 1) === addOne(m)
subtract = (a,b) => add(a, addOne(NOT(b)))
subtractExample = subtract(m, n)
subtractProof = (m - n) === subtract(m,n)
subtractOne = a => subtract(a, 1)
subtractOneExample = subtractOne(m)
subtractOneProof = (m - 1) === subtractOne(m)
NOT = a => ~a
NOTexample = NOT(m)
NOTproof = (~m) === NOT(m)
AND = (a, b) => a & b
ANDexample = AND(m, n)
ANDproof = (m & n) === AND(m, n)
OR = (a, b) => a | b
ORexample = OR(m,n)
ORproof = (m | n) === OR(m,n)
XOR = (a,b) => a ^ b
XORexample = XOR(m, n)
XORproof = (m ^ n) === XOR(m, n)
LSHIFT = (a,b) => a << b
LSHIFTexample = LSHIFT(m, n)
LSHIFTproof = m << n === LSHIFT(m, n)
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