Floating Point

Run this anywhere:

>>> 0.1 + 0.2
0.30000000000000004
>>> 0.1 + 0.2 == 0.3
False

And in real training runs, this happens:

step 39,998   loss 2.41
step 39,999   loss 2.40
step 40,000   loss nan

Both outputs have the same cause: the way computers store numbers. By the end of this page you will be able to predict both, explain them to someone else, and read a line like "MXFP4, E2M1, block 32, E8M0 scale" part by part. No prior knowledge is assumed. Every measured number on this page comes from a small script you can run, linked where the number appears, measured on an Apple M3 Max with torch 2.11.0 [1].

A fixed set of points

A computer stores a number in a fixed number of bits. Each bit has 2 states, so 32 bits can form 2 x 2 x ... x 2 = 2^32 = 4,294,967,296 different patterns. A number format assigns one number to each pattern. So for a 32-bit machine only those 4,294,967,296 numbers exist; anything else you compute is moved to the nearest one of them, and the distance moved is the rounding error. A format is exactly this: a choice of which numbers get a pattern. This page is about how that choice is made.

First, see the size of the problem. Between 1 and 2 there are infinitely many numbers. Zoom in anywhere and you find more. The line is never empty, at any depth. A format is a finite list of points. So almost no computed number is on the list, and almost every operation ends with a move to the nearest point:

The simplest choice is equal spacing: 0, 1, 2, 3, and so on, the same distance 1 between every pair of neighbors. That is what integers are, and for counting it works. Now take two numbers that one real neural network produced in one training step: an activation of 10.5 (a value flowing through the network) and a gradient of 0.0000004 (a value that steers learning). Store both with equally spaced points:

Those 2 numbers are not invented. They come from 1 training step of a small network, and this page keeps returning to that step, so here it is, drawn:

3 kinds of numbers move in that step. Activations are the values flowing forward through the layers. Weights are the numbers the network is learning; multiplying by them is what a layer does. Gradients are computed backward from the loss, 1 for every weight: each one says how much that weight should move. Train the network for a few seconds and measure all 3:

"""Proof: where a real network's numbers live. Train a small MLP for
two seconds, then measure the spans of its weights, activations, and
gradients in one step. The same model needs numbers nine orders of
magnitude apart; a uniform grid cannot serve both ends."""
import torch, torch.nn as nn

torch.manual_seed(0)
net = nn.Sequential(nn.Linear(64, 256), nn.ReLU(),
                    nn.Linear(256, 256), nn.ReLU(), nn.Linear(256, 1))
opt = torch.optim.Adam(net.parameters(), lr=1e-3)
X = torch.randn(4096, 64); y = (X[:, :3].prod(1, keepdim=True) * 10)
for step in range(300):
    opt.zero_grad()
    loss = ((net(X) - y) ** 2).mean()
    loss.backward(); opt.step()

acts = {}
h = X
tensors = {"weights": torch.cat([p.detach().flatten() for p in net.parameters()]),
           "grads": torch.cat([p.grad.flatten() for p in net.parameters()])}
with torch.no_grad():
    a = net[1](net[0](X))
    tensors["activations"] = net[2](a).flatten()

print(f"final loss {loss.item():.3f}")
for name, t in tensors.items():
    nz = t[t != 0].abs()
    q = torch.quantile(nz, torch.tensor([0.001, 0.5, 0.999]))
    print(f"{name:12s} |min..max| {nz.min():.2e} .. {nz.max():.2e}   "
          f"p0.1/median/p99.9  {q[0]:.2e} / {q[1]:.2e} / {q[2]:.2e}")
span = tensors["grads"].abs()[tensors["grads"] != 0]
import math
print(f"gradient span: {math.log2(span.max() / span.min()):.1f} powers of two "
      f"in one tensor family, this small a net, this early")

print("\ndecade histogram (percent of values per log10 decade; "
      "the drawn silhouettes):")
edges = list(range(-8, 2))
for name, t in tensors.items():
    nz = t[t != 0].abs().log10()
    counts = [(100.0 * ((nz >= a) & (nz < a + 1)).sum() / nz.numel())
              for a in edges]
    row = "  ".join(f"{c:4.1f}" for c in counts)
    print(f"{name:12s} [{edges[0]}..{edges[-1] + 1}]  {row}")

Read the two computed lines in the drawing. Why divide? The ratio says how many of the small number fit into the big one. With equal spacing, that is the number of steps between them. The gradients alone run from 0.00000037 to 0.71, and 0.71 / 0.00000037 is about 1,900,000, which is 2^20.9. So if the spacing is fine enough to keep the smallest gradient, the largest one sits 1,900,000 steps away. Across kinds it is worse: 10.5 / 0.00000037 = 28,000,000 steps. A larger model makes both numbers bigger. Equal spacing cannot hold both ends. Set the spacing to 0.0000004 so the smallest gradient survives, and even 2^31 = 2,147,483,648 points in the positive direction reach only 0.0000004 x 2,147,483,648 = 859: the format ends before 1,000. Set the spacing to 1 so that large values fit, and every gradient becomes 0, as the first drawing showed. The spacing itself has to change with the size of the number: points packed close near zero, spread out far from it.

Floating the point

One small tool first: what a dot means in base two, because from here on every number is bits. In decimal, each place is worth a tenth of the place before it: the 7 in 204.75 means seven tenths. Binary uses the same rule with 2 in place of 10. Left of the dot the places are worth 8, 4, 2, 1; right of it they are worth a half, a quarter, an eighth:

Read 0010.1100 with those place values: the 1s sit on 2, on a half and on a quarter, and 2 + 0.5 + 0.25 = 2.75. That is all a string of bits with a dot in it can mean, and you can now read any of them.

One question hides here, and it decides half of this page: which fractions can these places write exactly? The places are halves, quarters, eighths. Any finite sum of them is a fraction whose denominator is a power of 2: 1/2, 3/4, 5/8. Those end cleanly. Now take 1/3, 1/5 or 1/10. Their denominators carry a 3 or a 5. No finite sum of halves, quarters and eighths equals them. Watch both cases run:

Now a first attempt that fails. Take 1 byte, which is 8 bits, and fix the dot in the middle: 4 bits for the whole part, 4 for the fraction:

This is called fixed point, and the drawing shows its problem: the dot's position decides the largest value and the smallest step at the same time, and no single position makes both good. The repair is in the name of this page. Let the dot move, and store where it went.

Where should the machine write down "where the dot went"? You already know the answer, because you already do this on paper. To write a very small or a very large number, you split it into its digits and its size: 0.00000037 becomes 3.7 x 10^-7. The 3.7 says what the digits are; the 10^-7 says how big the number is. 2 separate jobs, written separately:

A float makes exactly this split, in base 2. Any positive number is some power of 2 times a number between 1 and 2: 10.5 is 1.3125 x 2^3 (check it: 2^3 = 8, and 1.3125 x 8 = 10.5), and 0.00000037 is about 1.552 x 2^-22. The power of 2 goes into one field, named the exponent. The 1.something goes into the other field, named the mantissa. The 2 words are only names for the 2 halves of the split: the exponent is the number's size, the mantissa is its digits.

What does each field buy on the number line? Draw them [2]:

Where do the intervals come from? From the split itself. Fix the power at 8. The digits part runs from 1.00 to 1.99. So the values run from 8 to just under 16: every number with power 8 lives between 8 and 16. Fix the power at 16 and you fill 16 to 32. That stretch starts twice as high, so it is twice as wide. That is all the doubling is. Choosing the exponent is choosing the interval. 8 exponent bits count 2^8 = 256 intervals, and because every interval doubles, 256 of them span an enormous distance: that is how 8 bits reach from 1e-38 to 3e38. The mantissa then cuts the chosen interval into equal steps: 3 bits make 2 x 2 x 2 = 8 steps, binary32's 23 bits make 8,388,608. And the growing step we asked for at the start falls out by itself: [4, 8) is twice as wide as [2, 4), both are cut into the same number of steps, so every step in [4, 8) is twice as long. Small numbers get fine steps, large numbers get coarse ones, with nobody managing it.

Now store one number end to end. Move 1: pick the interval. 6.1 = 1.525 x 4, and 1.525 is between 1 and 2, so the interval is [4, 8). Move 2: say where inside. 6.1 sits 2.1 past 4, and 2.1 / 4 = 0.525. That fraction, 0.525, is what the mantissa holds:

In a real 32-bit float the split gets 1 bit for the sign, 8 for the exponent, 23 for the mantissa. Here is 6.1, every field decoded:

Two details in that card deserve their own sentences, because every format on this page inherits both. First, the decode row says "1 + 0.525", but the mantissa field only stored the .525. The leading 1 costs nothing: every number in a power-of-two interval is 1.something times a power of two, so the format does not store the 1 and gets 24 bits of position for the price of 23. (The subnormals, coming 2 sections from now, are 0.something and are the one exception.) Second, the bias. The exponent field is unsigned: 8 bits, 0 to 255, no minus sign. But interval numbers need minus signs: 0.00000037 lives at interval -22. The fix is one constant, 127, the middle of the range. When the machine stores, it adds 127: the -22 above went in as -22 + 127 = 105. When it reads, it subtracts 127: this card's 129 means 129 - 127 = 2. Same 127, two directions. Every interval number from -126 to 127 now fits in 1 to 254. (Stored 0 and 255 are reserved; they build the edge cases 2 sections ahead.)

The unsigned choice has a quiet payoff. A bigger float always carries a bigger bit pattern, so a chip can compare 2 floats with the integer circuits it already owns [3].

Notice what the last decode row admits: the machine does not store 6.1. It cannot. 6.1 is 61/10, and the 10 carries a 5. The fractions figure showed what a 5 does in base 2: the bits repeat forever. The mantissa keeps 23 of them and cuts the rest. What remains decodes to 6.0999999..., the nearest float32. The gap is the rounding error. The next three sections are about the size and the consequences of that gap. This layout, the bias and the field widths, was standardized in 1985 as IEEE 754 [3], and it is the format your float32 tensors use today, unchanged.

The map

You cannot draw all 4,294,967,296 points of float32. But the same design at 8 bits has only 2^8 = 256 patterns, and its entire positive half fits in one picture. The 8-bit format is called E4M3, and the name is just its recipe: E4 means 4 exponent bits, M3 means 3 mantissa bits. Every format name on this page reads the same way. This is the most important drawing on this page, because every format you will ever meet is this drawing with different counts:

"""Proof: every FP8 E4M3 value, decoded from its bits by hand, and
checked bit-for-bit against torch.float8_e4m3fn. 256 codes: this is
the entire format, and the decoder is fifteen lines."""
import torch

def decode_e4m3(byte):
    s = (byte >> 7) & 1
    e = (byte >> 3) & 0xF
    m = byte & 0x7
    if e == 0xF and m == 0x7:
        return float("nan")            # the single NaN code per sign
    if e == 0:                          # subnormal: no hidden 1
        return (-1)**s * 2**(1 - 7) * (m / 8)
    return (-1)**s * 2**(e - 7) * (1 + m / 8)

agree, values = 0, []
for byte in range(256):
    mine = decode_e4m3(byte)
    ref = torch.tensor([byte], dtype=torch.uint8).view(torch.float8_e4m3fn).float().item()
    ok = (mine != mine and ref != ref) or mine == ref
    agree += ok
    if not ok:
        print(f"DISAGREE byte {byte:08b}: mine {mine}, torch {ref}")
    values.append(mine)
print(f"{agree}/256 bit patterns agree with torch.float8_e4m3fn")

finite = sorted(v for v in values if v == v)
pos = [v for v in finite if v > 0]
print(f"largest value          : {max(finite)}")
print(f"smallest positive      : {min(pos)}")
print(f"positive values        : {len(pos)}")
print(f"step just above 1.0    : {min(v for v in pos if v > 1.0) - 1.0}")
print(f"step just below 448    : {448 - max(v for v in pos if v < 448)}")

Read the map slowly; every later section builds on it. Every interval holds exactly 8 points, one per mantissa pattern. Neighbors inside a interval are 1/8 of the interval's starting value apart, so rounding moves a number by at most half of that: 1/16 of its own size, about 6%, and this is the same in every interval, from 0.02 to 400. The step inside [1, 2) is 0.125; the step inside [256, 448] is 32. So the absolute error grows with the number, but the relative error does not. Every float format works this way.

The local step also has a proper name, and you will meet it in every numerics discussion: the ulp, the unit in the last place [4]. "The ulp at x" means the step between neighboring points where x lives. E4M3's ulp at 1.0 is 0.125; its ulp at 300 is 32. One number, the ulp at 1.0, summarizes a format's whole precision, because every other interval's step is that number times a power of two; the field guide later on this page prints it on every format's card.

The proof under the map is 15 lines of Python that decode all 256 bit patterns from first principles, and torch agrees with every single one. The largest value is 448. The smallest positive value is 0.001953125. Divide them: 448 / 0.001953125 = 229,376, about 2^17.8, so the top of this format is about 18 doublings above the bottom. Now compare that with the second drawing: one tensor's gradients spanned 20.9 doublings. The whole format is narrower than one tensor's spread, and that gap runs the entire training half of this page.

The edges of the map

Four places on the map need special handling, and every format must decide all four.

Zero. The position field always means "1.something", so no pattern naturally means zero. The all-zeros pattern is simply assigned the value zero, as a special case. The sign bit still exists, so there is a +0 and a -0, and they compare equal.

The ramp. Just above zero there is trouble. Every normal point is 1.something x a power of 2. The exponent has a lowest power, so there is a smallest normal point: in E4M3 it is 1.0 x 2^-6 = 0.015625. Below it, a naive format has nothing until 0. A cliff. The fix: in the lowest interval only, drop the hidden 1. Read the mantissa as 0.something instead of 1.something. Then 0.000 x 2^-6 is 0 itself, and the other 7 patterns walk up from 0 in the same equal steps. These points are the subnormals. They turn the cliff into a ramp:

Infinity. When a result overflows the ceiling, float32 returns a dedicated pattern called inf, which then behaves lawfully: anything finite divided by inf is 0, and inf - inf is the next special value. E4M3 made a harder choice: it has no infinity at all. Those patterns were used for one more doubling of range, and overflow stops at 448 instead. Keep that choice in mind; it is the first sign of how little room 8 bits leave.

NaN. Not a Number is the pattern returned when no answer is defensible: 0/0, inf - inf, the square root of -1. It has one deliberately strange law: nan == nan is false, so x != x is the honest test for it. When your loss prints nan, this value is what you are looking at, and it arrived through an overflow or an undefined step somewhere upstream. We will catch it in the act in a few sections.

Rounding

Between any 2 neighboring points of the map lies everything the format cannot say. When a result lands there, one rule decides its fate: go to the nearer point. When it lands exactly halfway, go to the point whose last mantissa bit is 0, so that ties break upward and downward equally often and a long chain of them does not drift. This pair of rules is round-to-nearest, ties-to-even, the default of every machine you will touch [3]:

The worst this rule can do is move a value by half the local step: half an ulp. For float32 the ulp at 1.0 is 2 to the power -23, which is about 1.2e-7 (read e-7 as: move the decimal point 7 places left, so 0.00000012), and that number is called the machine epsilon (torch.finfo(torch.float32).eps prints it). So every operation lands within about 6e-8 of the true result, measured relative to the result's size. Both numbers read straight off the map: find the interval that starts at 1, take its step, halve it. The standard error analysis of floating point is this one bound, applied once per operation [4].

How can hardware apply the rule without computing every bit of the exact answer first? It keeps 3 extra bits, and that is all it ever needs:

This explains the first code block at the top of the page, and you can check every step of it by hand. To write 0.1 in binary you double it, again and again, and each time the whole part of the result is the next bit. 0.1 doubles to 0.2: the first bit is 0. Then 0.4, 0.8: two more 0s. Then 0.8 doubles to 1.6: the first 1, and the 0.6 carries on. 0.6 doubles to 1.2: another 1, and 0.2 carries on. But 0.2 is where the second step started. The process is in a loop, and the bits repeat forever:

1/3 does the same thing in decimal, and for the same reason: the denominator has a prime factor the base does not. Ten is 2 times 5, so 0.1 is 1/(2 times 5), and base two has no 5. The format cuts the repetition at its mantissa width and stores the nearest representable number, which for float64 is exactly 0.1000000000000000055511151231257827... (proof prints every digit). The same happens to 0.2 and to 0.3, and the sum of the two stored numbers is not the stored number nearest 0.3. Nothing malfunctioned. 3 numbers you typed do not exist, and the printed digits show exactly which nearby numbers were stored instead.

"""Proof: the two riddles. 0.1 + 0.2 is not 0.3 because 0.1 and 0.2
do not exist in binary; the sum of the two nearest citizens is not
the nearest citizen to 0.3. And at 1e8, float32's local step is 8,
so adding 1 moves nothing."""
from fractions import Fraction
import struct, torch

# the exact value a float64 stores for 0.1
exact = Fraction(struct.unpack("<q", struct.pack("<d", 0.1))[0] & ((1<<52)-1) | (1<<52), 1)
# simpler and fully honest: print the stored values to 30 digits
print(f"stored 0.1  = {0.1:.30f}")
print(f"stored 0.2  = {0.2:.30f}")
print(f"stored sum  = {0.1 + 0.2:.30f}")
print(f"stored 0.3  = {0.3:.30f}")
print(f"0.1 + 0.2 == 0.3 : {0.1 + 0.2 == 0.3}")
print()
t = torch.tensor(1e8, dtype=torch.float32)
print(f"float32: 1e8 + 1 - 1e8 = {((t + 1) - t).item()}")
print(f"the local step at 1e8  = {torch.nextafter(t, torch.tensor(2e8)).item() - t.item()}")

One more piece of standard equipment: the fused multiply-add, 1 instruction computing a times b plus c with a single rounding at the end instead of 2. The result is as if the product had been computed exactly and only the final sum were rounded, and on modern chips it costs about as much as 1 multiply [27]. The difference is not small print. Choose a = b = 1 + 2^-27 and c = -(1 + 2^-26): the separate path rounds the product first and returns exactly 0, while the fused path returns the true answer, 2^-54 (proof runs both on this machine). Every tensor core in the training half of this page is a lattice of these fused units, and the sums they accumulate stay in float32 even when the products arrive in 8 bits. Keep that sentence; it is why low-precision training is possible at all.

"""Proof: the fused multiply-add really is one rounding instead of
2, and the difference is visible. Choose a, b and c so that the
exact product a*b carries a tail the separate path must round away:
  a = b = 1 + 2^-27, so a*b = 1 + 2^-26 + 2^-54 exactly
  c = -(1 + 2^-26)
The separate path rounds a*b to 1 + 2^-26 (the 2^-54 tail is under
half an ulp), then adds c and returns exactly 0.0: it reports that
a*b + c is zero when it is not. The fused path computes the exact
product, adds c, rounds once, and returns the true answer, 2^-54.
Both answers are legal for the source line "a*b + c", which is why
a compiler's fusion choice can change a program's bits.
"""
import math
import torch

a = b = 1.0 + 2.0 ** -27
c = -(1.0 + 2.0 ** -26)

separate = a * b + c
if hasattr(math, "fma"):
    fused = math.fma(a, b, c)          # Python 3.13+
    how = "math.fma"
else:
    import ctypes, ctypes.util
    libm = ctypes.CDLL(ctypes.util.find_library("m"))
    libm.fma.restype = ctypes.c_double
    libm.fma.argtypes = [ctypes.c_double] * 3
    fused = libm.fma(a, b, c)          # the C library's fma
    how = "libm fma"

print(f"a = b = 1 + 2^-27, c = -(1 + 2^-26)")
print(f"separate (a*b) + c : {separate}")
print(f"fused fma(a, b, c) : {fused}   (via {how})")
print(f"true answer        : {2.0 ** -54}")
assert separate == 0.0
assert fused == 2.0 ** -54

t = (torch.tensor(a, dtype=torch.float64) *
     torch.tensor(b, dtype=torch.float64) +
     torch.tensor(c, dtype=torch.float64)).item()
print(f"torch eager a*b+c  : {t}   (separate ops: also 0.0)")
print("\nthe separate path says the answer is exactly 0 when it is")
print("not; the fused path keeps it. both are legal readings of the")
print("same source line, which is why fusion decisions (a compiler")
print("flag, an inlined function, torch.compile) can change a")
print("program's last bits without anyone touching the math.")

The unit itself is worth one look, because the training half of this page keeps returning to it:

Where arithmetic goes wrong

Three rules of ordinary arithmetic stop holding once every result is rounded to a point, and all three cause real bugs.

Absorption. At 100,000,000, which is 1e8, float32's step is 8, so adding 1 offers a move smaller than half the gap, and the sum rounds straight back (proof):

"""Proof: the two classic accidents. Absorption: below half the
local step, an addend vanishes. Cancellation: subtracting two close
numbers deletes their shared leading digits and promotes the noise."""
import torch
t = torch.tensor(1e8, dtype=torch.float32)
print(f"float32 step at 1e8      : {(torch.nextafter(t, t*2) - t).item()}")
print(f"1e8 + 1                  : {(t + 1).item():.1f}   (absorbed)")
print(f"1e8 + 5                  : {(t + 5).item():.1f}   (rounds up: past half the step)")
a = torch.tensor(1.0000001, dtype=torch.float32)
b = torch.tensor(1.0000000, dtype=torch.float32)
d = (a - b).item()
print(f"\n(1.0000001 - 1.0) in float32 = {d:.10e}")
print(f"true answer                  = 1.0000000e-07")
print(f"relative error               = {abs(d - 1e-7)/1e-7:.1%}")
print("seven leading digits matched and left; only the rounding")
print("noise of the inputs remained.")

Cancellation. Subtract two nearly equal numbers and their shared leading digits leave together, promoting whatever rounding noise the inputs carried into the leading digits of the answer. The proof above subtracts 1.0000001 from 1.0 in float32 and gets 1.19e-7 where the truth is 1.0e-7: a 19% error from one subtraction of two almost-exact inputs. The strange part: the subtraction itself commits no error at all. When 2 numbers are within a factor of 2 of each other, their difference is always exactly representable (Sterbenz's lemma [4]; proof checks it on this very case). Cancellation never creates error; it exposes the error the inputs already carried.

Order. Because every addition rounds, (a + b) + c and a + (b + c) are different numbers. Sum 50,000 values forward and backward and the answers disagree (proof); a parallel machine that splits the same sum across cores picks yet another order, and another answer. torch's own .sum() adds pairwise in a tree, which is both faster and about 130 times closer to the true sum than a one-by-one loop (measured: 0.24 error against 0.0018), and compensated summation (Kahan's trick of carrying the rounding error in a second variable) closes most of the rest [4].

"""Proof: addition order changes the answer. One million float32
values, three orders, three sums; then the repairs: pairwise
(what torch does), Kahan compensation, and fsum as ground truth."""
import math, torch

torch.manual_seed(0)
x = (torch.randn(1_000_000) * 100).float()
truth = math.fsum(x.double().tolist())

fwd = torch.tensor(0.0)
for c in x.split(100_000):          # sequential in chunks, forward
    for v in c: pass
# a plain python loop over 1e6 floats is slow; do exact fp32 fold in torch
def fold(t):
    s = torch.tensor(0.0, dtype=torch.float32)
    for v in t.split(4096):
        for u in v.tolist():
            s = s + torch.tensor(u, dtype=torch.float32)
    return s.item()

seq_fwd = fold(x[:50_000])          # smaller slice keeps runtime sane
seq_rev = fold(x[:50_000].flip(0))
tsum    = x[:50_000].sum().item()   # torch's pairwise-style reduction
truth50 = math.fsum(x[:50_000].double().tolist())
print(f"ground truth (fsum)       : {truth50:.6f}")
print(f"sequential, forward       : {seq_fwd:.6f}   err {seq_fwd-truth50:+.6f}")
print(f"sequential, reversed      : {seq_rev:.6f}   err {seq_rev-truth50:+.6f}")
print(f"torch .sum() (pairwise)   : {tsum:.6f}   err {tsum-truth50:+.6f}")
print(f"forward == reversed       : {seq_fwd == seq_rev}")

One more fact turns this whole section from a list of failures into a toolbox: the rounding error is catchable, exactly. The error of an addition is itself a number the format can hold, and 3 operations recover it whole: s = a + b, z = s - a, e = b - z, and now s + e equals the true sum of a and b with no error at all. Run it on this page's opening example and the 3 operations hand back the exact rounding error of 0.1 + 0.2 (proof checks it against exact rational arithmetic). A fused multiply-add does the same for a product in 1 instruction. Kahan's compensated sum is this trick run in a loop, and the double-word arithmetic that stretches precision in software is this trick kept instead of thrown away [4].

"""Proof: the rounding error is catchable, exactly. 3 facts, each
checked against exact rational arithmetic (fractions.Fraction):
1. Fast2Sum: for |a| >= |b|, 3 operations recover the rounding
   error of a + b perfectly: s = a+b; z = s-a; e = b-z. Then
   s + e equals the true sum of the stored a and b, exactly.
   Run on the page's opening example, 0.1 + 0.2.
2. Sterbenz: when x and y are within a factor of 2 of each
   other, x - y is computed exactly: cancellation commits no
   error of its own; it only exposes error the inputs carried.
3. 2MultFMA: 1 fused multiply-add recovers a product's whole
   rounding error: e = fma(a, b, -a*b).
"""
from fractions import Fraction as F
import ctypes, ctypes.util

# ---- 1. Fast2Sum on 0.1 + 0.2
a, b = 0.2, 0.1
s = a + b
z = s - a
e = b - z
print(f"s = a + b        = {s!r}")
print(f"e = (b - (s - a)) = {e!r}")
assert F(a) + F(b) == F(s) + F(e)
print("checked: stored 0.2 + stored 0.1 == s + e, EXACTLY.")
print("the 3 float operations caught the rounding error whole.\n")

# ---- 2. Sterbenz: close subtraction is exact
x, y = 1.0000001, 1.0
d = x - y
assert F(x) - F(y) == F(d)
print(f"x - y = {d!r}")
print("checked: equals the exact difference of the stored inputs.")
print("the 19% error of the cancellation section was already in")
print("the inputs; the subtraction added nothing.\n")

# ---- 3. 2MultFMA: a product's error in one instruction
libm = ctypes.CDLL(ctypes.util.find_library("m"))
libm.fma.restype = ctypes.c_double
libm.fma.argtypes = [ctypes.c_double] * 3
u = v = 1.0 + 2.0 ** -27
p = u * v
e2 = libm.fma(u, v, -p)
assert F(u) * F(v) == F(p) + F(e2)
print(f"p = u * v       = {p!r}")
print(f"e = fma(u,v,-p) = {e2!r}")
print("checked: stored u times stored v == p + e, exactly.")
print("\nKahan's compensated sum is fact 1 run in a loop; the")
print("double-word arithmetic that stretches precision in software")
print("is facts 1 and 3 kept instead of thrown away.")

Keep the order fact in mind. It returns at the end of this page as the reason two identical training runs on two GPUs never match to the last bit.

The field guide

The formats in this guide carry nearly all of the world's floating-point arithmetic, and you now own every idea needed to read them. Each gets the same treatment: its card, the bit counts and the numbers that follow from them; its map, where its points land; where it came from; where it runs today; and how it fails. Every number on every card is derived from the 2 bit counts alone, with the formulas you used on binary32, and checked against torch's own tables (proof runs the check). The guide covers the 4 wide formats, the ones that stand alone. The 8-, 6- and 4-bit formats enter later, each at the exact place in the training story that needs it, and a closing sheet gathers the whole family on one drawing.

"""Proof: the field guide's numbers, derived by hand from each
format's two counts (E exponent bits, M mantissa bits), then checked
against torch. Nothing on the cards is typed from memory:
  bias        = 2^(E-1) - 1
  ceiling     = (2 - 2^-M) * 2^bias
  floor       = 2^(1 - bias)          (smallest normal)
  ramp floor  = 2^(1 - bias - M)      (smallest subnormal)
  step at 1   = 2^-M                  (the ulp of 1.0; finfo calls it eps)
  cells       = 2^E - 2               (power-of-two intervals)
  points/cell = 2^M
Also prints the torch defaults the float32 section quotes."""
import math
import torch

FORMATS = [
    ("float64", 11, 52, torch.float64, torch.int64),
    ("float32", 8, 23, torch.float32, torch.int32),
    ("float16", 5, 10, torch.float16, torch.int16),
    ("bfloat16", 8, 7, torch.bfloat16, torch.int16),
]

for name, E, M, dt, it in FORMATS:
    bias = 2 ** (E - 1) - 1
    ceiling = (2 - 2.0 ** -M) * 2.0 ** bias
    floor = 2.0 ** (1 - bias)
    ramp_floor = 2.0 ** (1 - bias - M)
    step_at_1 = 2.0 ** -M
    cells = 2 ** E - 2
    digits = (M + 1) * math.log10(2)

    fi = torch.finfo(dt)
    # smallest subnormal: bit pattern 0...01 viewed as this dtype
    sub = torch.tensor([1], dtype=it).view(dt).double().item()

    assert ceiling == fi.max, (name, ceiling, fi.max)
    assert floor == fi.tiny, (name, floor, fi.tiny)
    assert step_at_1 == fi.eps, (name, step_at_1, fi.eps)
    assert ramp_floor == sub, (name, ramp_floor, sub)

    print(f"{name:9s} E={E:2d} M={M:2d}  bias {bias:4d}   "
          f"ceiling {ceiling:.4g}")
    print(f"          floor {floor:.4g}   ramp floor {ramp_floor:.4g}   "
          f"step at 1 = {step_at_1:.4g}")
    print(f"          {cells} cells of {2**M:,} points; "
          f"about {digits:.1f} decimal digits   "
          f"(torch.finfo agrees on all four numbers)")

# absorption never leaves; it only moves. float64's step at 1e16 is 2.
x = torch.tensor(1e16, dtype=torch.float64)
print(f"\nfloat64: (1e16 + 1) - 1e16 = {((x + 1) - x).item():g}"
      f"   (the step at 1e16 is 2; the 1 is below half of it)")

# the defaults the float32 section quotes, read from this install
print(f"\ntorch {torch.__version__} defaults:")
print(f"  default dtype                  {torch.get_default_dtype()}")
print(f"  float32_matmul_precision       "
      f"{torch.get_float32_matmul_precision()!r}")
print(f"  backends.cuda.matmul.allow_tf32  "
      f"{torch.backends.cuda.matmul.allow_tf32}")
print(f"  backends.cudnn.allow_tf32        "
      f"{torch.backends.cudnn.allow_tf32}")
print(f"  autocast dtype on cpu          "
      f"{torch.get_autocast_dtype('cpu')}")

float64

Double binary32's byte count and you get the format the 1985 standard called double precision:

The bias is 1023, so the intervals run from 2.2e-308 up to 1.8e308: 2,046 of them, each holding 4,503,599,627,370,496 points. The ulp at 1.0 is 2.22e-16, which is about 16 decimal digits of position. A map that wide can only be drawn with breaks:

Where it came from: IEEE 754 began as the arithmetic William Kahan designed with Intel for the 8087, the floating-point coprocessor sold beside the 8086; the standards committee turned that chip's arithmetic into everyone's law [3][18]. float64 was its wide grade, sized so ordinary science could chain millions of operations and still trust the leading digits.

Where it runs today: everywhere you did not choose a dtype. A Python float is a float64. NumPy builds float64 arrays unless told otherwise. The digits of 0.1 printed in the rounding section were float64 digits.

Its proof: p16 derives the whole card from E and M and checks every row against torch.finfo. One extra line in it repeats the absorption experiment at this format's scale: in float64, (1e16 + 1) - 1e16 = 0. The ulp at 1e16 is 2, and the added 1 fell under half of it. 16 digits push the cliff out of sight; no digit count removes it.

How it fails in a training loop: float64 rarely gives training a wrong answer. Its problem is cost. 8 bytes per number is 2 times float32 and 4 times bfloat16, spent on digits training cannot use: the sections ahead show training running where the step at 1.0 is 0.0078, so digits 4 through 16 add nothing. On GPUs there is a second cost: most chips carry few float64 units, and NVIDIA's own throughput tables list float64 operations far below float32 on most of its hardware [19]. torch's default is float32, and float64 appears only when you ask (proof prints the default).

float32, and the format hiding inside it

binary32 you have already decoded by hand; the card collects its numbers:

254 intervals from 1.2e-38 to 3.4e38, 8,388,608 points in each, about 7 digits:

Where it came from: the same 1985 standard's single precision [3]. Where it runs today: it is the default dtype of torch; torch.tensor(1.0) is a float32 unless you say otherwise (proof prints the default from this install). Even when matmuls run narrow, the master weights of mixed-precision training stay float32 and the tensor cores accumulate in float32, so the training half of this page keeps this format at its center.

Now the format hiding inside it: TensorFloat-32. Since the A100, NVIDIA's tensor cores can take an ordinary float32 matmul, round each factor to 10 mantissa bits, multiply, and accumulate in full float32 [20]. The tensors going in and out are ordinary float32; only the multiply is narrow, reading 19 of the 32 bits (1 + 8 + 10). TF32 is that mode, not a storage type you can give a tensor. NVIDIA measured large speedups at matching accuracy on deep-learning workloads [20]; for code that needs all seven digits it is a silent cut, and that is why torch ships with matmul TF32 off since version 1.12 [21]. On this install, torch.backends.cuda.matmul.allow_tf32 is False, the cudnn convolution flag is True, and torch.set_float32_matmul_precision("high") is the one-line opt-in (proof prints all three). None of this can execute on the Apple machine this page is measured on; the flags are read here, the behavior is cited [20][21].

How it fails: you have watched it fail all page. The step of 8 at 1e8 is float32's. The 7 digits that turned a subtraction of near-equal inputs into a 19% error are float32's. In training, its failure is price: 4 bytes per number on the memory bus that a later section shows is the bottleneck.

float16

Cut 32 bits in half and something must go. There are two ways to choose, and the choice split the 16-bit world in two:

float16 is the left fork: keep digits, pay with reach.

5 exponent bits give bias 15 and only 30 intervals, from 6.1e-5 to 65504. 10 mantissa bits give 1,024 points per interval and about 3 digits. 30 intervals fit in one drawing, so this is the one wide format whose map you can see whole:

Where it came from: film and games, not a numerics committee. NVIDIA and Microsoft made half a type in the Cg shading language in 2002, and Industrial Light & Magic built its OpenEXR film format on the same 1+5+10 layout, in production from 2000 and released as open source in 2003 [22]. IEEE 754 adopted the layout as binary16 in its 2008 revision [3]. The film industry shipped the format years before the standard named it.

Where it runs today: graphics APIs and image pipelines still, inference engines, and mixed-precision training on hardware from before bfloat16 spread. In torch it is the half in .half().

Its proof and its failure are the same three numbers. The ceiling: 60000 times 1.2 is inf in float16 (proof). The digits: 1.001 stores as 1.000977. The floor: normal numbers end at 6.1e-5, and the small half of a gradient histogram lives below that, which is why the loss-scaling section exists. When float16 dies in a run, it dies at one of these three numbers.

"""Proof: the 16-bit fork, from finfo, nothing typed from memory.
float16 spent its bits on precision and dies at 65504; bfloat16 kept
float32's exponent and survives, with two decimal digits left."""
import torch

for dt in (torch.float32, torch.float16, torch.bfloat16):
    fi = torch.finfo(dt)
    print(f"{str(dt):16s} max {fi.max:>12.4g}   tiny {fi.tiny:.3g}   "
          f"eps {fi.eps:.3g}")
x = torch.tensor(60000.0)
print(f"\n60000 * 1.2 in float16 : {(x.half() * 1.2).item()}")
print(f"60000 * 1.2 in bfloat16: {(x.bfloat16() * torch.tensor(1.2, dtype=torch.bfloat16)).item()}")
print(f"1.001 in float16       : {torch.tensor(1.001, dtype=torch.float16).item():.6f}")
print(f"1.001 in bfloat16      : {torch.tensor(1.001, dtype=torch.bfloat16).item():.6f}")

bfloat16

The right fork: keep reach, pay with digits. Google's brain float keeps all 8 of float32's exponent bits and 7 of mantissa:

The reach is float32's (the ceiling prints 3.39e38 rather than 3.4e38 only because the coarser last step lands the top point lower), the intervals are the same 254 as float32's, and each holds 128 points: about 2 decimal digits.

Where it came from: Google built it into the TPU's matrix units, which multiply in bfloat16 and accumulate in float32, and chose the layout so that range would never be the problem and conversion from float32 would be nearly free [23]. The name is Brain Floating Point, after the Google Brain team.

Nearly free is provable, and on this machine it is proven at the bit level: bfloat16 is float32's top half. Take any of the 65,536 possible bfloat16 bit patterns, cast it to float32, and the result is the same 16 bits with 16 zeros appended; proof checks all 65,536, bit for bit. The downward cast is the same move with rounding: keep the top 16 bits, round by the low 16. The proof checks torch's cast against that hand rule on 1,000,010 values (a million random ones plus the edge cases), and they agree on every one. Here is 1.7014 making the trip:

float32:  0 01111111 1011001 1100011101111010
bfloat16: 0 01111111 1011010   (the top half, rounded up by the rest)
stored:   1.703125

"""Proof: bfloat16 is the top 16 bits of float32, at the bit level.
Two directions, both exhaustive where exhaustion is possible:
  up:   every one of the 65,536 bfloat16 bit patterns, cast to
        float32, must produce exactly (pattern << 16): the same 16
        bits on top, sixteen zeros appended below.
  down: casting float32 to bfloat16 must equal keeping the top 16
        bits and rounding by the low 16 (nearest, ties to even),
        checked on a million random values plus the edge cases.
"""
import torch

# ---- up: bf16 -> fp32 is "append 16 zero bits", all 65,536 patterns
pats = torch.arange(65536, dtype=torch.int32).to(torch.int16)
as_bf16 = pats.view(torch.bfloat16)
up_bits = as_bf16.float().view(torch.int32)
want = pats.to(torch.int32) << 16
nan_mask = torch.isnan(as_bf16)
ok = (up_bits == want) | nan_mask
print(f"up-cast bit check: {ok.sum().item()} / 65536 patterns match "
      f"(pattern << 16)")
print(f"  of the 65536, {nan_mask.sum().item()} are NaN codes, "
      f"checked separately:")
nan_ok = torch.isnan(as_bf16.float()[nan_mask]).all().item()
print(f"  every NaN pattern up-casts to a float32 NaN: {nan_ok}")
exact = (up_bits == want).sum().item()
print(f"  bit-exact including NaN payloads: {exact} / 65536")

# ---- down: fp32 -> bf16 is "keep the top 16 bits, round by the rest"
def truncate_rne(f32):
    """The hand rule: add the rounding bias, then keep the top half.
    bias = 0x7FFF + (bit 16), the classic nearest-even truncation."""
    u = f32.view(torch.int32)
    bias = 0x7FFF + ((u >> 16) & 1)
    out = ((u + bias) >> 16).to(torch.int16).view(torch.bfloat16)
    return torch.where(torch.isnan(f32), torch.nan, out.float())

torch.manual_seed(0)
x = torch.randn(1_000_000) * torch.logspace(-38, 38, 1_000_000)
edges = torch.tensor([0.0, -0.0, 1.0, 1.0078125, 65504.0, 3.4e38,
                      1e-40, float("inf"), float("-inf"), float("nan")])
x = torch.cat([x, edges])
ours = truncate_rne(x)
torchs = x.bfloat16().float()
agree = (ours == torchs) | (torch.isnan(ours) & torch.isnan(torchs))
print(f"\ndown-cast rule check: {agree.sum().item()} / {len(x)} "
      f"values agree with torch's cast")

# ---- the picture, on one number
v = torch.tensor(1.7014, dtype=torch.float32)
u32 = v.view(torch.int32).item() & 0xFFFFFFFF
b16 = v.bfloat16().view(torch.int16).item() & 0xFFFF
print(f"\n1.7014 in float32 bits : {u32:032b}")
print(f"1.7014 in bfloat16 bits: {b16:016b} (the top half, rounded)")
print(f"bfloat16 value stored  : {v.bfloat16().item():.6f}")

Where it runs today: TPUs since their second generation [23], NVIDIA GPUs since the A100 [20], and this machine: torch's CPU autocast picks bfloat16 by default (proof prints it). In large-model training it is the usual compute half of the mixed-precision loop, with FP8 taking over the matmuls on the newest chips (the FP8 section returns to this).

How it fails: 2 digits. 1.001 stores as exactly 1.0 (proof). The step at 1.0 is 0.0078, so a healthy update of 0.001 sits under half a step and rounds away: the disaster that opens the training sections ahead, and the reason master weights exist. bfloat16 did not make training precise. It made training's failures the quiet, repairable kind, and the machinery that repairs them is most of the rest of this page.

The guide pauses here, because at 8 bits and below no format stands alone. Each narrow format arrives later on this page, at the moment the training story needs it, and a closing sheet gathers the whole family in one drawing. First, a payoff: with the wide formats on the table, the small constants in every model's source code finally have their explanation.

The little epsilons, explained

Open any model's source code and small constants appear: an eps=1e-5 in the layer norm, an eps=1e-8 in Adam, a logits - logits.max() in every attention implementation. None of them is superstition; each one is a patch over a place where a formula's intermediate value leaves the map.

The largest number float32 can hold is about 3.4e38, and exp reaches it at 88.73 (measured below). Softmax, the function that turns a model's raw output scores, its logits, into probabilities, applies exp to every logit. Logits above 88.73 exist in every large model, so a naive softmax returns inf, then inf divided by inf, and your loss prints nan. The repair costs nothing: softmax only sees differences, so subtract the maximum first and the largest input becomes 0 (proof):

"""Proof: the stability patterns. exp overflows at 88.73 in float32
and 11.09 in float16, so a naive softmax dies on logits your model
produces every day; subtracting the max moves nothing and fixes it
(softmax is shift-invariant)."""
import torch

print(f"float32 exp cliff: exp(88.7)={torch.tensor(88.7).exp().item():.3e}, "
      f"exp(88.8)={torch.tensor(88.8).exp().item()}")
h = torch.tensor([11.0, 11.1], dtype=torch.float16)
print(f"float16 exp cliff: exp(11.0)={h.exp()[0].item()}, exp(11.1)={h.exp()[1].item()}")

logits = torch.tensor([120.0, 119.0, 115.0])
naive = logits.exp() / logits.exp().sum()
stable = (logits - logits.max()).exp() / (logits - logits.max()).exp().sum()
print(f"naive  softmax([120,119,115]) = {naive.tolist()}")
print(f"stable softmax([120,119,115]) = {[round(v,4) for v in stable.tolist()]}")
print(f"torch.softmax agrees with stable: "
      f"{torch.allclose(torch.softmax(logits, 0), stable)}")

# logsumexp, same illness, same cure
print(f"naive  log(sum(exp)) = {logits.exp().sum().log().item()}")
print(f"stable logsumexp     = {torch.logsumexp(logits, 0).item():.4f}")

The same pattern explains the rest of the family. logsumexp slides by the max for the same reason. log1p(x) and expm1(x) exist because near zero, 1 + x absorbs x (the absorption figure again, at 1.0 instead of 1e8). The layer-norm epsilon keeps a near-zero variance from turning rsqrt into inf; Adam's epsilon does the same for its denominator; and an attention mask uses a large negative number so that after the slide it underflows to a clean zero instead of poisoning the row with nan. One picture, many patches: keep the intermediate values where the points are.

Why smaller floats

The guide paused at 16 bits. Here is why the family keeps going down anyway.

Training a network is mostly moving numbers, not multiplying them: weights travel from memory to the arithmetic and gradients travel back, billions per step, and the wire is slower than the multiplier. Halve the bits per number and the same wire carries twice the numbers, the same memory holds twice the model, and the tensor cores, built to process the narrower type, double their arithmetic too. Here is the wire, drawn to scale:

The only question is whether the surviving points still cover the numbers the network produces with small enough error. Networks tolerate rounding noise unusually well: they are trained on noisy batches and judged over many outputs, so the answer stays yes far below 32 bits if the format is chosen carefully. The fork figure already showed the first halving's two options, and training mostly picked bfloat16: reach first. The cost is a step of 0.0078 at 1.0, and the next section shows what that step does to learning.

The update that vanished

At 16 bits the map's steps are wide enough to stop training itself. A bfloat16 weight sitting at 1.0 has neighbors 0.0078 away. A healthy update of 0.001 is an arrow one eighth of that step:

The proof runs it: 1,000 consecutive updates of 0.001 applied to a bfloat16 weight leave it at exactly 1.0 (proof). This is why every mixed-precision recipe since 2017 keeps a float32 master copy of the weights: the forward and backward passes run narrow, but the optimizer adds the update into a copy whose local step is 65536 times finer, where 0.001 lands easily [5]. The other repair is stochastic rounding, which we will need again at 4 bits: round up with probability proportional to how far you got. In the same proof, the stochastically rounded weight reaches 2.0 alongside the master copy.

"""Proof: the update that vanished. In bf16, a weight near 1.0 has a
local step of about 0.0078; an update smaller than half that step
rounds away, and training silently stops for that weight. The fp32
master copy is the fix; stochastic rounding is the other fix."""
import torch

w = torch.tensor(1.0, dtype=torch.bfloat16)
u = 1e-3                                   # lr * grad, a normal size
print(f"bf16 next after 1.0 : {torch.nextafter(torch.tensor(1.0, dtype=torch.bfloat16), torch.tensor(2.0, dtype=torch.bfloat16)).item():.6f}")
print(f"w + {u} in bf16     : {(w + torch.tensor(u, dtype=torch.bfloat16)).item():.6f}  (unchanged: {(w + torch.tensor(u, dtype=torch.bfloat16)) == w})")

# 1000 such updates, three ways
steps = 1000
wb = torch.tensor(1.0, dtype=torch.bfloat16)
for _ in range(steps):
    wb = wb + torch.tensor(u, dtype=torch.bfloat16)
wm = torch.tensor(1.0, dtype=torch.float32)     # master copy
for _ in range(steps):
    wm = wm + u
g = torch.Generator().manual_seed(0)            # stochastic rounding, emulated
ws = torch.tensor(1.0, dtype=torch.float32)
for _ in range(steps):
    hi = torch.nextafter(ws.to(torch.bfloat16), torch.tensor(2.0, dtype=torch.bfloat16)).float()
    lo = ws.to(torch.bfloat16).float()
    x = ws + u
    p = ((x - lo) / (hi - lo)).clamp(0, 1)
    ws = hi if torch.rand((), generator=g) < p else lo
print(f"after {steps} updates of {u}:")
print(f"  bf16 accumulate    : {wb.item():.4f}   (true answer 2.0)")
print(f"  fp32 master        : {wm.item():.4f}")
print(f"  bf16 + stochastic  : {ws.item():.4f}")

So the loop that trains every model you use runs like this:

The products accumulate in float32 inside the tensor cores: that is the fused multiply-add from the rounding section doing its job. Precision goes exactly where the numbers need it and nowhere else. Stations 6 and 8, the S dial and the inf check, are the subject of the next section.

Loss scaling

One group of numbers still fails inside this loop. Gradients are the smallest numbers in training, and float16's floor is high: its subnormals end near 6e-8 and its normal range starts at 6e-5. The left tail of the gradient distribution simply falls below every point of the format:

The rescue is one multiplication. Scale the loss by S before the backward pass and, because differentiation is linear, every gradient arrives multiplied by S; unscale after casting and nothing has changed except which gradients survived. In practice S adapts: overflow detected, halve it and skip the step; a run of clean steps, double it. That is the whole of GradScaler (proof measures the cast; on our toy net only 0.1% of gradients die, and the fraction grows with depth and training time, which is why the recipe exists [5]). bfloat16 mostly retired this machinery: it keeps float32's reach, so its normal floor sits at 1.2e-38 against float16's 6.1e-5, about 34 powers of 10 lower, and the scaler became optional.

"""Proof: loss scaling, measured. Take real float32 gradients from a
small net, cast them to float16 as mixed precision must, and count
the ones that flush to zero; then scale by 1024 first, cast, unscale.
The cast is where gradients die; the scale is the rescue."""
import torch, torch.nn as nn
torch.manual_seed(0)
net = nn.Sequential(nn.Linear(64, 256), nn.ReLU(),
                    nn.Linear(256, 256), nn.ReLU(), nn.Linear(256, 1))
X = torch.randn(2048, 64); y = X[:, :3].prod(1, keepdim=True)
loss = ((net(X) - y) ** 2).mean()
loss.backward()
g = torch.cat([p.grad.flatten() for p in net.parameters()])
nz = g[g != 0]
plain = (nz.half() == 0).float().mean()
scaled = ((nz * 1024).half().float() / 1024 == 0).float().mean()
print(f"nonzero float32 gradients            : {len(nz)}")
print(f"flushed to zero by the float16 cast  : {plain:.1%}")
print(f"same cast after scaling by 1024      : {scaled:.1%}")
print(f"smallest surviving gradient          : {nz.abs().min().item():.2e}")
print(f"float16's smallest subnormal         : {torch.finfo(torch.float16).smallest_normal * 2**-10:.2e}")

FP8

Halve again and there are not enough bits left for one format to serve both directions of training. So at 8 bits the fork appears one more time, and this time both forks ship, as a pair [6]: E4M3 keeps digits, E5M2 keeps reach.

E4M3 you have known since the map section: its 15 intervals are this page's teaching map, and its 256 patterns are decoded in proof. The card records its one non-IEEE choice: no infinity. Those codes were used for one more doubling of range, overflow stops at 448, and a single pattern means NaN.

E5M2 is float16 with the bottom byte cut, and that is a bit-level fact, provable the same way bfloat16 was: all 256 E5M2 patterns cast to float16 as the same 8 bits with 8 zeros appended (proof checks every one).

"""Proof: FP8 E5M2 is float16's top byte, at the bit level. The
same fact as bfloat16 being float32's top half, one floor down:
E5M2 keeps float16's 5 exponent bits and the top 2 of its 10
mantissa bits. Cast any of the 256 E5M2 bit patterns to float16
and the result is the same 8 bits with 8 zeros appended.
"""
import torch

pats = torch.arange(256, dtype=torch.uint8)
as_e5m2 = pats.view(torch.float8_e5m2)
up_bits = as_e5m2.half().view(torch.int16) & 0xFFFF
want = pats.to(torch.int16) << 8
nan_mask = torch.isnan(as_e5m2)
ok = (up_bits == want) | nan_mask
print(f"up-cast bit check: {ok.sum().item()} / 256 patterns match "
      f"(pattern << 8)")
print(f"  {nan_mask.sum().item()} NaN codes, all up-cast to "
      f"float16 NaN: {torch.isnan(as_e5m2.half()[nan_mask]).all().item()}")
exact = (up_bits == want).sum().item()
print(f"  bit-exact including NaN payloads: {exact} / 256")

v = torch.tensor(1.7014, dtype=torch.float16)
b8 = v.to(torch.float8_e5m2).view(torch.uint8).item()
u16 = v.view(torch.int16).item() & 0xFFFF
print(f"\n1.7014 in float16 bits: {u16:016b}")
print(f"1.7014 in E5M2 bits   : {b8:08b} (the top byte, rounded)")
print(f"E5M2 value stored     : "
      f"{v.to(torch.float8_e5m2).float().item():.4f}")

NVIDIA, Arm and Intel proposed the pair for deep learning in 2022 [6], and hardware arrived with the Hopper chips: E4M3 carries the forward pass, E5M2 carries the gradients, whose tails need reach more than digits. Each tensor also carries a float32 scale, chosen from the tensor's recent maximum so that its histogram slides into the band: exactly the loss-scaling picture, applied per tensor. Recipes differ only in whether the maximum is tracked from history (delayed scaling) or measured on the spot:

And not everything is quantized: attention's and the MLP's big matmuls run in FP8, while softmax, the normalizations, the first embedding and the final projection stay wide [11].

Wired into a transformer through NVIDIA's Transformer Engine, the published result is training runs 30 to 40 percent faster at matching loss curves [6], and DeepSeek-V3 trained a 671-billion-parameter model with FP8 matmuls by scaling 128-element tiles instead of whole tensors [7], a step toward what comes next. One caution before the next halving: E4M3 carries about 1.2 decimal digits and E5M2 about 0.9, so at 8 bits no single stored number is trustworthy; only averages over many are.

The outlier problem

Before the next halving, meet the format waiting at the bottom, because its 2 numbers explain everything that follows. 4 bits:

16 codes, 15 values (0 appears twice, once with each sign), no inf, no NaN, and a total width of 6 / 0.5 = 12x, about 3.6 doublings. The 2023 OCP standard defined E2M1 as the 4-bit element of the block family these sections are building [9], and the newest NVIDIA chips run it at twice the FP8 rate [24]. Alone it is unusable, and even a per-tensor scale cannot save it, because per-tensor scaling has one enemy. Transformer activations grow a few channels whose values run hundreds of times larger than the rest, systematically, past about 6 billion parameters [8]. One scale must now serve the spike and the bell at once:

Read the measured numbers. FP8 with a single scale loses 1% of the ordinary values next to a 500x outlier, because a float grid keeps its relative error at any scale until values fall below its smallest positive number, and FP8's smallest is 229,376x under its largest (448 / 0.00195, from the map section). FP4's smallest is only 12x under its largest (6 / 0.5, computed above). One spike takes the whole 12x, and 100% of the ordinary values quantize to zero (proof). At 4 bits, blocks are not an improvement; nothing works without them.

"""Proof: why blocks had to be invented. A float grid keeps its
relative precision at any scale, so FP8 with one per-tensor scale
survives an outlier surprisingly well. FP4 does not: its largest
positive value is 6 and its smallest is 0.5, only 12x apart, so one
spike pushes every ordinary value below the smallest, and the
tensor quantizes to zeros. A scale per 32 elements
rescues it."""
import torch

E2M1 = torch.tensor([0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 6.0])

def round_e2m1(v):
    """Round |v| to the nearest E2M1 magnitude, keep the sign."""
    idx = (v.abs().unsqueeze(-1) - E2M1).abs().argmin(-1)
    return E2M1[idx] * v.sign()

def quant(x, block):
    q = torch.empty_like(x)
    for i in range(0, len(x), block):
        b = x[i:i+block]
        s = b.abs().max() / 6.0
        s = s if s > 0 else torch.tensor(1.0)
        q[i:i+block] = round_e2m1(b / s) * s
    return q

torch.manual_seed(0)
x = torch.randn(128) * 0.1
x[40] = 50.0
mask = torch.ones(128, dtype=torch.bool); mask[40] = False

for name, block in (("per-tensor", 128), ("per-block-32", 32)):
    q = quant(x, block)
    zeros = (q[mask] == 0).float().mean()
    rel = ((q - x)[mask].abs() / x[mask].abs()).median()
    print(f"{name:14s} ordinary values quantized to zero: {zeros:.0%}   "
          f"median rel err of the rest: {rel:.1%}")
q = quant(x, 128)
print(f"the outlier itself survives either way: {q[40]:.1f}")
print()
print("the same tensor in FP8 E4M3, one per-tensor scale, for honesty:")
s = x.abs().max() / 448.0
q8 = (x / s).to(torch.float8_e4m3fn).float() * s
print(f"per-tensor FP8    zeros: {(q8[mask]==0).float().mean():.0%}   "
      f"median rel err: {((q8-x)[mask].abs()/x[mask].abs()).median():.1%}")
print("a float grid holds its relative error at any scale, until its")
print("smallest positive value; FP4's smallest (0.5) is only 12x below")
print("its largest (6), and that is the whole reason microscaling exists.")

Microscaling

So the scale moved into the data type. The Open Compute Project's MX formats, standardized in 2023 by AMD, Arm, Intel, Meta, Microsoft, NVIDIA and Qualcomm [9], cut every tensor into blocks of 32 and give each block one 8-bit scale:

Every idea on this page meets here. The E8M0 scale is a float with no mantissa at all, the interval choice from the floating-dot figure with the position deleted, so applying it is exponent addition, free in silicon. The 32-element block is the histogram cut fine enough that an outlier only damages its own block. The accounting is 136 bits per 32 numbers: 4.25 bits each, 136 / 128 = 1.06, so 6% more than raw 4-bit storage. And the block size is a measured compromise: smaller blocks mean less error and more storage spent on scales, and our own sweep across sizes 8, 16, 32, 64, 128 shows both moving (proof). The same chassis carries 8-, 6- and 4-bit elements as MXFP8, MXFP6, MXFP4.

What the tensor core does with 2 of these blocks is 1 more drawing:

The block chassis also changed who does what inside FP8: with a scale for every 32 values, a block rarely spans more than E4M3's 18 doublings, so E4M3, with its 8 points per doubling against E5M2's 4, measured better for the gradients too, and the largest published MXFP8 pretraining run (8 billion parameters, 15 trillion tokens) quantizes every tensor as E4M3 [11].

The 6-bit elements deserve their own cards, because they are the family's open seats: E2M3 keeps digits, E3M2 keeps reach, one more run of the fork.

Their maps are small enough to draw complete: 3 intervals of 8 points, 7 intervals of 4. Blackwell-class chips execute them at full speed as MXFP6 [9][24]; almost no published recipe uses them yet. A format survives by having a job nobody else does. FP8 is safer, FP4 is faster on paper, and FP6 is still looking for its job.

NVIDIA's NVFP4 variant tightens it further: 16-element blocks, a fractional E4M3 scale instead of a power-of-two one, plus one float32 scale per tensor, buying a finer fit for a quarter bit more [10]. The bytes carry the quarter bit, and the deeper difference sits in the scale byte itself:

Here is one NVFP4 block quantized by hand, every scale shown:

The largest of the 16 values, 15.011, sets the scales and comes back exactly; 0.25 and 0.5 fall below half of the smallest step at that scale and come back as 0; 3.2002 comes back as 3.7528, about 17% off (proof runs every number). A single 4-bit value is coarse. The bet of 4-bit arithmetic is that billions of such errors, kept unbiased, cancel on average.

"""Proof: one NVFP4 block, quantized by hand, every scale shown.
Re-derives the worked 1 x 16 example from NVIDIA's recipe (as
walked through by Radical Numerics): a global FP32 scale brings
the tensor's largest value into range, each 16-value block gets an
FP8 E4M3 scale, and the elements land on the 16 values of E2M1.
"""
import torch

E2M1_MAX = 6.0
E4M3_MAX = 448.0
BLOCK = 16
E2M1_GRID = torch.tensor([0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 6.0])

def rne_e2m1(x):
    """Round each value to the nearest E2M1 value, ties to even."""
    grid = torch.cat([E2M1_GRID, -E2M1_GRID.flip(0)])
    d = (x.unsqueeze(-1) - grid).abs()
    return grid[d.argmin(dim=-1)]

x = torch.tensor([0.0, 0.25, 0.5, 0.75356, 1.251245, 3.2002,
                  4.5032, 15.011, 0.012, -0.312, -5.50055, 10.06,
                  -1.2526, 3.025, 2.5114, 7.0162])

# level 1: one FP32 scale for the whole tensor
global_amax = x.abs().max()
s_enc = (E4M3_MAX * E2M1_MAX) / global_amax
s_dec = 1.0 / s_enc

# level 2: one E4M3 scale for each block of 16
block_amax = x.abs().max()                    # one block here
dec_scale = (block_amax / E2M1_MAX) * s_enc
dec_scale_e4m3 = dec_scale.to(torch.float8_e4m3fn)
enc_scale = 1.0 / (dec_scale_e4m3.float() * s_dec)
print(f"level 1, the tensor's float32 scale: 448 x 6 / "
      f"{global_amax:.4f} = {s_enc:.2f}")
print(f"level 2, this block's E4M3 scale: "
      f"{dec_scale_e4m3.float():g}")
print(f"together: {s_enc:.2f} / {dec_scale_e4m3.float():g} = "
      f"{enc_scale:.4f}, the factor every value is multiplied by")

# quantize: scale, clamp to E2M1's range, round onto its 16 values
scaled = (x * enc_scale).clamp(-E2M1_MAX, E2M1_MAX)
q = rne_e2m1(scaled)

# dequantize: multiply back by both scales
dq = q * dec_scale_e4m3.float() * s_dec

print(f"global amax {global_amax:.4f}   block encode scale "
      f"{enc_scale:.4f}\n")
print(f"{'input':>9s} {'scaled':>8s} {'E2M1':>6s} {'back':>9s}")
for a, b, c, d in zip(x, scaled, q, dq):
    print(f"{a:>9.4f} {b:>8.4f} {c:>6.1f} {d:>9.4f}")

print("\nread the two ends: 15.011 comes back as 15.011 exactly")
print("(it set the scales), 0.25 and 0.5 come back as 0 (they")
print("fell below half of E2M1's smallest step at this scale),")
print("3.2002 comes back as 3.7528: about 17% off. one 4-bit")
print("number is coarse; the training recipe works because these")
print("errors average out over billions of them.")

One subtlety earned a hardware rule. The E8M0 scale can only be a power of 2, so a block whose largest value is 5.9 must round its scale one way or the other, and the 2 ways are not close:

NVIDIA measured the difference as the gap between divergence and parity at trillion-token scale [11]; proof shows the mechanism in 2 lines.

The whole scaling story now fits on one ladder, and every recipe of the last 5 sections is 1 rung of it:

Training at 4 bits

The destination format, E2M1, was drawn whole 2 sections back: 15 values, 12x wide. Training inside those 16 codes needs 3 rescues beyond blocks, and each is an idea from earlier on this page pushed one step:

The first is stochastic rounding, the lost-update repair now doing the main work. At 4 bits nearly every update is below half the local step, so deterministic rounding freezes everything; rounding up with probability proportional to progress is unbiased in expectation. Our proof runs 10,000 micro-updates: nearest rounds to a frozen 1.0, stochastic reaches a mean of 3.0113 against a true 3.0, noisy per run and honest on average (proof).

"""Proof: round-to-nearest is biased when every step is smaller than
half the local spacing; stochastic rounding is unbiased in
expectation. The same fact that freezes bf16 weights, isolated."""
import torch
N, u, TRIALS = 10000, 2e-4, 50   # each step ~1/39 of bf16's step at 1.0

def sr_walk(seed):
    g = torch.Generator().manual_seed(seed)
    val = 1.0
    for _ in range(N):
        x = val + u
        lo = torch.tensor(x, dtype=torch.bfloat16)
        lo_f = lo.float().item()
        hi_f = torch.nextafter(lo, torch.tensor(float("inf"), dtype=torch.bfloat16)).float().item()
        if lo_f > x:
            hi_f, lo_f = lo_f, torch.nextafter(lo, torch.tensor(float("-inf"), dtype=torch.bfloat16)).float().item()
        p = 0.0 if hi_f == lo_f else (x - lo_f) / (hi_f - lo_f)
        val = hi_f if torch.rand((), generator=g).item() < p else lo_f
    return val

rne = torch.tensor(1.0, dtype=torch.bfloat16)
for _ in range(N):
    rne = rne + torch.tensor(u, dtype=torch.bfloat16)
finals = torch.tensor([sr_walk(s) for s in range(TRIALS)])
true = 1.0 + N * u
print(f"true value                     : {true:.4f}")
print(f"round-to-nearest, {N} steps  : {rne.item():.4f}  (frozen)")
print(f"stochastic rounding, {TRIALS} runs  : "
      f"mean {finals.mean().item():.4f}, spread +-{finals.std().item():.4f}")
print("unbiased in expectation, noisy per run: that is the trade.")

The second is the random Hadamard transform: rotate the tensor by an orthogonal matrix before quantizing and undo it after, which changes no matmul (the rotation cancels) but spreads an outlier's energy across every coordinate:

Our 64-value proof takes the max-to-mean ratio from 11.8 to 2.6 (proof). The third is selective precision: the first and last layers, where quantization error is measured to hurt most, stay in higher precision. With all three plus NVFP4, a 12-billion-parameter model has been pretrained on 10 trillion tokens at nearly the same loss as its FP8 twin [10]; MXFP4 needs more tokens for the same loss [12]. Where each rescue is applied is itself measured engineering [10][24]: stochastic rounding only on the gradients (on the forward pass it adds noise for nothing, so weights and activations round to nearest); the Hadamard rotation only on the inputs of the weight-gradient matmul; and weights are scaled in 16 x 16 squares rather than 1 x 16 rows, for a reason worth drawing:

A row-scaled weight matrix and its column-scaled transpose are 2 different quantized matrices, and the forward and backward passes must see the same weights. The last roughly 15% of layers stay in higher precision, and if a gap to the wide baseline remains at the end, switching the forward pass to higher precision for the final phase of training closes most of it [10][24]. 4-bit training is real and it is young; treat recipes as versioned, not settled.

"""Proof: an orthogonal rotation spreads one outlier's energy across
every coordinate, without changing the vector's length or the matmul
it feeds (H is orthogonal). The max shrinks; the format's points get
used again."""
import torch

def hadamard(n):
    H = torch.tensor([[1.0]])
    while H.shape[0] < n:
        H = torch.cat([torch.cat([H, H], 1), torch.cat([H, -H], 1)], 0) / (2 ** 0.5)
    return H

torch.manual_seed(3)
n = 64
x = torch.randn(n) * 0.5
x[7], x[23], x[51] = 8.0, -7.0, 6.0
H = hadamard(n)
y = x @ H
print(f"norm preserved: {torch.allclose(x.norm(), y.norm())}")
for name, v in (("before", x), ("after H", y)):
    print(f"{name:8s} max {v.abs().max():.3f}   mean {v.abs().mean():.3f}   "
          f"max/mean {(v.abs().max()/v.abs().mean()).item():.2f}")

The other grids

Inference relaxed one constraint: if only the weights are quantized and arithmetic runs wider, the 16 points need not be a float at all. 3 designs compete on the same bell-shaped distribution of trained weights:

INT4 spaces its points equally and wastes them where weights are rare. FP4 doubles its steps and crowds zero. NF4, from the QLoRA work, places its 16 values at the quantiles of a gaussian, which is approximately what trained weights are, and wins the tail (proof, [13]). Around these grids grew the post-training toolkit: GPTQ chooses each weight's rounding to compensate the error of the previous ones [14]; AWQ rescales channels so the few weights that matter most land on finer points [15]; and the same how-many-bits question now runs through the KV cache and even optimizer states, where 8-bit Adam keeps its moments in blocks with per-block scales [16], the MX idea arriving from the other direction.

"""Proof: three ways to spend sixteen points. INT4's uniform grid,
FP4 E2M1's doubling grid, and NF4's quantile grid (the sixteen
values from the QLoRA paper), each quantizing the same gaussian
weights with one absmax scale."""
import torch
torch.manual_seed(0)
w = torch.randn(100_000)

INT4 = torch.linspace(-7, 7, 15) / 7.0            # symmetric int4
E2M1 = torch.tensor([0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 6.0]) / 6.0
E2M1 = torch.cat([-E2M1.flip(0), E2M1]).unique()
NF4 = torch.tensor([-1.0, -0.6961928009986877, -0.5250730514526367,
    -0.39491748809814453, -0.28444138169288635, -0.18477343022823334,
    -0.09105003625154495, 0.0, 0.07958029955625534, 0.16093020141124725,
    0.24611230194568634, 0.33791524171829224, 0.44070982933044434,
    0.5626170039176941, 0.7229568362236023, 1.0])  # Dettmers et al. 2023

s = w.abs().max()
for name, grid in (("INT4 uniform", INT4), ("FP4 E2M1", E2M1),
                   ("NF4 quantile", NF4)):
    q = grid[( (w / s).unsqueeze(-1) - grid ).abs().argmin(-1)] * s
    err = (q - w).abs()
    print(f"{name:14s} mean abs err {err.mean():.4f}   "
          f"p99 {err.quantile(0.99):.4f}")
print("\nthe bell wants points where the bell is; NF4 puts them there.")

The whole family on one sheet

Every format on this page has now been met. Gather them:

"""Proof: the whole-family sheet, derived from each format's bit
counts and checked against torch wherever torch has the type.
For every format: ceiling, smallest positive value, the number of
doublings between them (log2 of their ratio), step at 1.0, and
decimal digits. kind says how the top exponent codes are used:
  ieee  top exponent code reserved for inf and NaN
  fn    no inf; only the top mantissa pattern of the top code is NaN
  all   no inf and no NaN; every code is a number (FP4 E2M1)
"""
import math
import torch

def family(E, M, bias, kind):
    if kind == "ieee":
        emax = 2 ** E - 2 - bias
        ceiling = (2 - 2.0 ** -M) * 2.0 ** emax
    elif kind == "fn":
        emax = 2 ** E - 1 - bias
        ceiling = (2 - 2.0 ** (1 - M)) * 2.0 ** emax
    else:  # all
        emax = 2 ** E - 1 - bias
        ceiling = (2 - 2.0 ** -M) * 2.0 ** emax
    smallest = 2.0 ** (1 - bias - M)          # smallest subnormal
    return {
        "ceiling": ceiling,
        "smallest": smallest,
        "doublings": math.log2(ceiling) - math.log2(smallest),
        "step1": 2.0 ** -M,
        "digits": (M + 1) * math.log10(2),
    }

ROWS = [
    ("float64", 11, 52, 1023, "ieee", torch.float64),
    ("float32", 8, 23, 127, "ieee", torch.float32),
    ("TF32", 8, 10, 127, "ieee", None),
    ("bfloat16", 8, 7, 127, "ieee", torch.bfloat16),
    ("float16", 5, 10, 15, "ieee", torch.float16),
    ("FP8 E5M2", 5, 2, 15, "ieee", torch.float8_e5m2),
    ("FP8 E4M3", 4, 3, 7, "fn", torch.float8_e4m3fn),
    ("FP6 E3M2", 3, 2, 3, "all", None),
    ("FP6 E2M3", 2, 3, 1, "all", None),
    ("FP4 E2M1", 2, 1, 1, "all", None),
]

print(f"{'format':10s} {'ceiling':>10s} {'smallest':>10s} "
      f"{'doublings':>9s} {'step at 1':>10s} {'digits':>6s}")
for name, E, M, bias, kind, dt in ROWS:
    f = family(E, M, bias, kind)
    print(f"{name:10s} {f['ceiling']:>10.4g} {f['smallest']:>10.3g} "
          f"{f['doublings']:>9.1f} {f['step1']:>10.4g} "
          f"{f['digits']:>6.1f}")
    if dt is not None:
        fi = torch.finfo(dt)
        assert f["ceiling"] == fi.max, (name, f["ceiling"], fi.max)
        assert f["step1"] == fi.eps, (name, f["step1"], fi.eps)
print("\ntorch.finfo agrees on ceiling and step for every format "
      "torch carries\n(float64/32/16, bfloat16, both FP8 types).")
print("FP6 sheets from the same formulas; Blackwell tensor cores "
      "run them,\ntorch has no storage type for them yet.")

Read the sheet by columns. The bit bars share one scale: you can see float64 dwarf everything and the narrow rows shrink to almost nothing. The reach bars collapse from float64's 2,098 doublings to E2M1's 3.6; every format from float16 down fits its whole reach on the drawing. The digits column falls from 16 to 0.6. The block rows at the bottom are the repair: the same narrow elements, plus a shared scale whose dashed arrows say what it does, sliding each block's small reach to wherever its values live. And the sheet is the page in one drawing: everything before the field guide taught you what the columns mean, and everything after it was the story of how the narrow rows are made usable.

Where the bytes live

All of it, on one shelf:

These panels are the honest reason this whole family of formats exists. A 7-billion-parameter model is 28 GB of float32 or 3.7 GB of MXFP4 with its scales; a training run holds 8x its bf16 weights; and in serving, the KV cache decides how many conversations one GPU carries. gpt-oss ships its 120-billion-parameter mixture-of-experts in MXFP4 so it fits a single 80 GB GPU [9].

Serving adds a loop of its own. A chat model generates 1 token at a time, and every token's step reads all the weights and the whole KV cache, the stored attention keys and values of every token so far; serving engines keep that cache in fixed-size pages so many conversations can grow side by side without fragmenting memory [29]. That loop, end to end:

Hardware truth

Throughput follows the same halvings on hardware that speaks the format natively, and only there: on a GPU that must dequantize in software, a narrow format saves memory but not time. That warning is measurable. One published comparison ran the same image-generation model on a GPU with no MXFP8 hardware: the MXFP8 file was smaller, and generation was slower than bfloat16 (112.8 seconds against 96.0), because every block had to be unpacked in software [25]. The same tests ranked output quality FP32, then FP16 and BF16 about equal, then MXFP8, then per-tensor-scaled FP8, then NVFP4, then plain FP8 last [25]: size and quality move together, and a scale at any granularity beats no scale. Who speaks what natively today: NVIDIA's Blackwell chips run MXFP8/6/4 and NVFP4 in their tensor cores [24]; AMD's newest Instinct chips run the MX family [26]; Hopper GPUs and Trainium2 can store MX blocks but unpack them for the math [26]. The same facts, drawn so a spec sheet can be checked against them:

And what a native format buys, next to the measured price of one the hardware cannot speak:

Read a spec sheet's format row before believing its speedup column.

The contract

Two runs of the same training script, same seeds, same data, on two different GPUs, will not match bit for bit, and now every reason is on this page: sums land in different orders across different core counts; tensor cores round differently than the unfused loop; TF32 trims mantissas silently on one machine and not another. And there is a quieter actor: the compiler. IEEE 754 fixes each instruction's bits, but no language promises which instructions your line becomes. A compiler may legally fuse a * b + c into 1 fused multiply-add (1 rounding) or keep it as 2 instructions (2 roundings), and the choice can flip with an optimization flag or with something as small as inlining a function [27]; the rounding section's proof showed the 2 paths disagreeing. torch.compile is a compiler too: its fused kernels are allowed to round differently than eager mode [28]. And rounding in 2 hops is not rounding once: take an exact value to float64 first and then to float32, and you can land on a different number than going straight to float32, because the first rounding can create a tie the second one breaks the other way (proof builds one). This double rounding is why the old x87 unit, which computed in 80-bit registers and rounded again on every store to memory, made a program's bits depend on register allocation [4]. None of it is a bug; all of it is the map.

"""Proof: rounding twice can differ from rounding once. Take the
exact value 1 + 2^-24 + 2^-60. Rounded straight to float32 it goes
UP to 1 + 2^-23, because it sits just above the halfway point.
Rounded first to float64, the tiny 2^-60 tail is lost and the
float64 result lands exactly ON the halfway point; rounding that
to float32 then goes DOWN to 1.0 by ties-to-even. Two legal
paths, 2 different answers: double rounding, the failure mode of
every system that computes wider first and stores narrower.
"""
from fractions import Fraction as F
import numpy as np

exact = F(1) + F(1, 2 ** 24) + F(1, 2 ** 60)

# path 1: round once, straight to float32 (decided exactly)
lo, hi = F(1), F(1) + F(1, 2 ** 23)          # float32 neighbors
assert lo < exact < hi
once = np.float32(1.0) if (exact - lo) <= (hi - exact) else \
       np.float32(1.0 + 2.0 ** -23)

# path 2: round to float64 first, then to float32
via64 = float(exact)                          # correctly rounded
twice = np.float32(via64)

print(f"exact value          : 1 + 2^-24 + 2^-60")
print(f"rounded once to fp32 : {once!r}")
print(f"fp64 first           : {via64!r}")
print(f"then to fp32         : {twice!r}")
assert once == np.float32(1.0 + 2.0 ** -23)
assert twice == np.float32(1.0)
print("\nthe 2 paths disagree. the old x87 unit computed in 80-bit")
print("registers and rounded again on every spill to memory, so a")
print("program's bits depended on register allocation; SSE and its")
print("one-width successors retired the problem, and this proof is")
print("why narrow-then-narrower casts still deserve suspicion.")

Where this came from

Konrad Zuse's Z3 computed in 22-bit floating point in 1941, special values included. 44 years later IEEE 754, largely William Kahan's design, ended an era in which every vendor's arithmetic disagreed [3]. The formats since bfloat16 are the same standard's ideas at new bit counts, and the full table of every named format that ever shipped, from the Z3 to the G.711 telephone float of 1972 (an 8-bit float in every phone call for 50 years) to NVFP4, is longer than this page and belongs to a follow-up table of its own. The roads not taken, posits and logarithmic number systems, are worth reading about [17].

Build it yourself

You already did, in the proofs. The 15-line decoder in p2_e4m3_map.py is a complete FP8 implementation that torch agrees with on all 256 patterns; the block quantizer in p11_mxblock.py is MXFP4 minus the packing; the stochastic rounder lives in p12_sr_drift.py. Change E4M3's constants to E5M2's and check yourself against torch again: that is the exercise that makes every format in this family yours.