THE FOLD / SPAWN / FIRST LIGHT / THE SUBNORMAL
THE SUBNORMAL
the numbers that buy you a smooth zero
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Without them the gap between zero and the smallest normal is vastly larger than the gap between neighbouring normals — so two different numbers can subtract to exactly zero.
LIT verified live. The smallest subnormal is 2-1074 and the smallest normal is 252 times larger. With subnormals, two values one step apart subtract to a nonzero result, so
LIT verified live. The smallest subnormal is 2-1074 and the smallest normal is 252 times larger. With subnormals, two values one step apart subtract to a nonzero result, so
x == y and x - y == 0 agree. Under flush-to-zero the same subtraction gives exactly 0 for two values that differ — a resolution difference of 4.5 × 1015 near zero.2 HOW IT WAS WEAVED · AI + HUMAN
Gradual underflow was one of the hardest-fought parts of IEEE 754, argued for by William Kahan against significant hardware opposition, because subnormals are awkward and slow to implement.
AVAN (AI) measured the property they buy rather than the numbers themselves: without them, “a equals b” and “a minus b is zero” stop being the same test, and every algorithm that checks equality by subtracting silently acquires a false positive near zero.
AVAN (AI) measured the property they buy rather than the numbers themselves: without them, “a equals b” and “a minus b is zero” stop being the same test, and every algorithm that checks equality by subtracting silently acquires a false positive near zero.
3 ONE DIMENSION
Where the number line goes thin.
4 TWO DIMENSIONS · INTERACTIVE
Turn flush-to-zero on and watch two numbers merge.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a lattice that keeps its spacing to the end.
AVAN’s addition (the inverse-companion): the forward reading is that subnormals keep the number line smooth. The inverse is that they are the slowest values in the machine. On much real hardware a subnormal operand traps to microcode and costs a hundred times a normal one, so code that drifts into them does not fail — it becomes mysteriously slow, in a way no profiler attributes to arithmetic. Read backwards, the smoothness is paid for in a performance cliff placed exactly where the values get small, which is where iterative methods spend their final steps.
LIT the smallest subnormal is 2^-1074 and the smallest normal is 2^52 times larger; with subnormals two values one step apart subtract to a nonzero result so that x equals y and x minus y equals zero agree, while under flush-to-zero the same subtraction gives exactly 0 for two values that differ - a resolution difference of 4.5e15 near zero
FIG Gradual underflow was one of the hardest-fought parts of IEEE 754, argued for by William Kahan against significant hardware opposition, because subnormals are awkward and slow to implement. AVAN measured the property they buy rather than the numbers themselves: without them, 'a equals b' and 'a minus b is zero' stop being the same test, and every algorithm that checks equality by subtracting silently acquires a false positive near zero.
FIG Gradual underflow was one of the hardest-fought parts of IEEE 754, argued for by William Kahan against significant hardware opposition, because subnormals are awkward and slow to implement. AVAN measured the property they buy rather than the numbers themselves: without them, 'a equals b' and 'a minus b is zero' stop being the same test, and every algorithm that checks equality by subtracting silently acquires a false positive near zero.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN