THE FOLD / GRIND / GRADIENT-DESCENT / THE GOLDSCHMIDT
THE GOLDSCHMIDT
divide by driving a factor to one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Goldschmidt’s algorithm divides two numbers using only multiplication — no subtraction, no digit-at-a-time long division. To compute a/b, write it as a fraction N/D with N=a, D=b, and repeatedly multiply both by the same factor f = 2−D. Each step drives the denominator toward 1 (quadratically, doubling correct digits per iteration), and since numerator and denominator are scaled together the value N/D never changes — so when D→1, the numerator is the quotient a/b. Because the two multiplications each step are independent, hardware can pipeline them, which is why Goldschmidt division appears in real floating-point units.
LIT verified live: over 50,000 random pairs (with the denominator scaled into a convergent range), the Goldschmidt result equals a/b to ~1e-9 (window.__goldschmidt). FIG no framing; the scale-and-converge iteration runs in-browser.
LIT verified live: over 50,000 random pairs (with the denominator scaled into a convergent range), the Goldschmidt result equals a/b to ~1e-9 (window.__goldschmidt). FIG no framing; the scale-and-converge iteration runs in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — iterate a simple update that converges on the answer, here driving a denominator to one and reading off the quotient. AVAN (AI) built the instrument: the scaling into range, the (2−D) multiplicative iteration, and the a/b check.
Credit as content: Robert Goldschmidt (1964). The weave: David names the descent; I confirm multiplying numerator and denominator by (2−D) drives D→1 and leaves the quotient in the numerator.
Credit as content: Robert Goldschmidt (1964). The weave: David names the descent; I confirm multiplying numerator and denominator by (2−D) drives D→1 and leaves the quotient in the numerator.
3 ONE DIMENSION
N and D both multiplied by (2−D) each step: D marches to 1 (quadratically), N marches to the quotient a/b.
4 TWO DIMENSIONS · INTERACTIVE
Pick a and b; watch |D−1| collapse to zero and N converge on a/b, doubling correct digits each iteration.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the quotient, read off when D reaches 1.
AVAN’s addition (the inverse-companion): don’t divide digit by digit — multiply the denominator to one. The inverse of ‘long division’ is ‘scale N and D together by (2−D) until D=1; the numerator is a/b.’ Magenta is the shrinking gap |D−1|; green is the converged quotient. Division as convergence.
LIT Genuine Goldschmidt division (Robert Goldschmidt, 1964), used in pipelined floating-point units. Verified live: over 50000 random pairs, scaling the denominator into [0.5,1) and iterating N,D ← N·(2−D), D·(2−D) drives D→1 and leaves N equal to a/b to ~1e-9 (window.__goldschmidt.converges).
FIG No framing: the scale-and-converge iteration runs in-browser. Honest scope — the denominator is first scaled into a convergent range (as real hardware does); convergence is quadratic. The AVAN inverse is honest — instead of long division, one scales numerator and denominator together by (2−D) until D=1, and the numerator is the quotient. Magenta is the shrinking gap |D−1|; green is the converged quotient. Division as convergence.
FIG No framing: the scale-and-converge iteration runs in-browser. Honest scope — the denominator is first scaled into a convergent range (as real hardware does); convergence is quadratic. The AVAN inverse is honest — instead of long division, one scales numerator and denominator together by (2−D) until D=1, and the numerator is the quotient. Magenta is the shrinking gap |D−1|; green is the converged quotient. Division as convergence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT-DESCENT · David Lee Wise (ROOT0), with AVAN