Collect value statistics for formats
This notebook presents various statistics for a variety of float formats.
Some of these are present on the FormatInfo class, and are presented for all formats.
Others are obtained by enumerating all values (only for the =16 bit formats).
Statistics obtained from FormatInfo
name: Format
B: Bits in the format
P: Precision in bits
E: Exponent field width in bits
T: Trailing significand field width in bits
max: Largest finite value
min: Most negative value (typically the same, unless twos complement)
smallest: Smallest positive value
smallest_normal: Smallest positive normal value, NaN if all finite values are subnormal
%run utils.py
D = pandas_render # from utils
import pandas
from functools import partial
from gfloat import *
from gfloat.formats import *
import numpy as np
# Special rendering for float values -
# if they don't round-trip in 8.5g, prepend with "≈", or render as rational
def render_float(approx: bool, v):
if not isinstance(v, float):
return str(v)
if np.isnan(v):
return "n/a"
s = f"{v:8.5g}"
if float(s) == v:
return s
if approx:
return "≈" + s
else:
return float_pow2str(v)
def collect_stats(fi: FormatInfo):
return dict(
name=fi.name,
B=fi.bits,
P=fi.precision,
E=fi.expBits,
smallest=fi.smallest,
smallest_normal=fi.smallest_normal if not fi.is_all_subnormal else np.nan,
max=fi.max,
num_nans=float(fi.num_nans),
infs=2 if fi.has_infs else 0,
)
stats = [collect_stats(fi) for fi in all_formats]
df = pandas.DataFrame(stats)
D(df, format=partial(render_float, True))
| name | B | P | E | smallest | smallest_normal | max | num_nans | infs |
|---|---|---|---|---|---|---|---|---|
| ocp_e2m1 | 4 | 2 | 2 | 0.5 | 1 | 6 | 0 | 0 |
| ocp_e2m3 | 6 | 4 | 2 | 0.125 | 1 | 7.5 | 0 | 0 |
| ocp_e3m2 | 6 | 3 | 3 | 0.0625 | 0.25 | 28 | 0 | 0 |
| ocp_e4m3 | 8 | 4 | 4 | ≈0.0019531 | 0.015625 | 448 | 2 | 0 |
| ocp_e5m2 | 8 | 3 | 5 | ≈1.5259e-05 | ≈6.1035e-05 | 57344 | 6 | 2 |
| p3109_p1 | 8 | 1 | 7 | ≈2.1684e-19 | ≈2.1684e-19 | ≈9.2234e+18 | 1 | 2 |
| p3109_p2 | 8 | 2 | 6 | ≈2.3283e-10 | ≈4.6566e-10 | ≈2.1475e+09 | 1 | 2 |
| p3109_p3 | 8 | 3 | 5 | ≈7.6294e-06 | ≈3.0518e-05 | 49152 | 1 | 2 |
| p3109_p4 | 8 | 4 | 4 | ≈0.00097656 | 0.0078125 | 224 | 1 | 2 |
| p3109_p5 | 8 | 5 | 3 | 0.0078125 | 0.125 | 15 | 1 | 2 |
| p3109_p6 | 8 | 6 | 2 | 0.015625 | 0.5 | 3.875 | 1 | 2 |
| binary16 | 16 | 11 | 5 | ≈5.9605e-08 | ≈6.1035e-05 | 65504 | 2046 | 2 |
| bfloat16 | 16 | 8 | 8 | ≈9.1835e-41 | ≈1.1755e-38 | ≈3.3895e+38 | 254 | 2 |
| binary32 | 32 | 24 | 8 | ≈1.4013e-45 | ≈1.1755e-38 | ≈3.4028e+38 | ≈1.6777e+07 | 2 |
| binary64 | 64 | 53 | 11 | 4.9407e-324 | ≈2.2251e-308 | ≈1.7977e+308 | ≈9.0072e+15 | 2 |
| ocp_e8m0 | 8 | 1 | 8 | ≈5.8775e-39 | ≈5.8775e-39 | ≈1.7014e+38 | 1 | 0 |
| ocp_int8 | 8 | 8 | 0 | 0.015625 | n/a | ≈ 1.9844 | 0 | 0 |
Statistics computed by exhaustive inspection
lt1: Number of values x such that
0 < x < 1gt1: Number of values x such that
1 < x < Infrt16: True if all values are exactly representable in IEEE binary16
min/maxSubnormal: Smallest/largest subnormal value, “n/a” if no values are subnormal
min/maxNormal: Smallest/largest normal value, “n/a” if no values are normal
def compute_stats(fi: FormatInfo):
# Generate all values
values = [decode_float(fi, i) for i in range(2**fi.bits)]
df = pandas.DataFrame(values)
# Compute statistics: lt1,gt1
fval = df["fval"]
total_01 = fval.between(0, 1, inclusive="neither").sum()
total_1Inf = fval.between(1, np.inf, inclusive="neither").sum()
# Compute statistics: maxFinite,minFinite
finite_vals = fval[np.isfinite(fval)]
maxFinite = finite_vals.loc[finite_vals.idxmax()]
minFinite = finite_vals.loc[finite_vals.idxmin()]
assert maxFinite == fi.max
assert minFinite == fi.min
# Compute statistics: maxNormal,minNormal
normal_vals = fval[(df["fclass"] == FloatClass.NORMAL) & (fval > 0)]
maxNormal = normal_vals.loc[normal_vals.idxmax()] if normal_vals.any() else np.nan
minNormal = normal_vals.loc[normal_vals.idxmin()] if normal_vals.any() else np.nan
assert np.isnan(maxNormal) or maxNormal == fi.max
assert np.isnan(minNormal) or minNormal == fi.smallest_normal
# Compute statistics: minSubnormal
pos_subnormal = fval[(df["fclass"] == FloatClass.SUBNORMAL) & (fval > 0)]
maxSubnormal = (
pos_subnormal.loc[pos_subnormal.idxmax()] if pos_subnormal.any() else np.nan
)
minSubnormal = (
pos_subnormal.loc[pos_subnormal.idxmin()] if pos_subnormal.any() else np.nan
)
assert np.isnan(minSubnormal) or minSubnormal == fi.smallest_subnormal
assert np.nanmin([minSubnormal, minNormal]) == fi.smallest
# Compute roundtrips: rt16, rt32
with np.errstate(over="ignore"):
rt16 = (np.float64(np.float16(fval)) == np.float64(fval)) | ~np.isfinite(fval)
rt32 = (np.float64(np.float32(fval)) == np.float64(fval)) | ~np.isfinite(fval)
rt16 = rt16.all()
rt32 = rt32.all()
assert rt32 # If not, we should include rt32 in the table
# Assemble tuple
return dict(
name=fi.name,
B=fi.bits,
P=fi.precision,
E=fi.expBits,
rt16=rt16,
lt1=total_01,
gt1=total_1Inf,
minSubnormal=minSubnormal,
maxSubnormal=maxSubnormal,
minNormal=minNormal,
maxNormal=maxNormal,
)
stats = [compute_stats(fi) for fi in all_formats if fi.bits <= 16]
df2 = pandas.DataFrame(stats)
D(df2, format=partial(render_float, True))
| name | B | P | E | rt16 | lt1 | gt1 | minSubnormal | maxSubnormal | minNormal | maxNormal |
|---|---|---|---|---|---|---|---|---|---|---|
| ocp_e2m1 | 4 | 2 | 2 | True | 1 | 5 | 0.5 | 0.5 | 1 | 6 |
| ocp_e2m3 | 6 | 4 | 2 | True | 7 | 23 | 0.125 | 0.875 | 1 | 7.5 |
| ocp_e3m2 | 6 | 3 | 3 | True | 11 | 19 | 0.0625 | 0.1875 | 0.25 | 28 |
| ocp_e4m3 | 8 | 4 | 4 | True | 55 | 70 | ≈0.0019531 | ≈0.013672 | 0.015625 | 448 |
| ocp_e5m2 | 8 | 3 | 5 | True | 59 | 63 | ≈1.5259e-05 | ≈4.5776e-05 | ≈6.1035e-05 | 57344 |
| p3109_p1 | 8 | 1 | 7 | False | 62 | 63 | n/a | n/a | ≈2.1684e-19 | ≈9.2234e+18 |
| p3109_p2 | 8 | 2 | 6 | False | 63 | 62 | ≈2.3283e-10 | ≈2.3283e-10 | ≈4.6566e-10 | ≈2.1475e+09 |
| p3109_p3 | 8 | 3 | 5 | True | 63 | 62 | ≈7.6294e-06 | ≈2.2888e-05 | ≈3.0518e-05 | 49152 |
| p3109_p4 | 8 | 4 | 4 | True | 63 | 62 | ≈0.00097656 | ≈0.0068359 | 0.0078125 | 224 |
| p3109_p5 | 8 | 5 | 3 | True | 63 | 62 | 0.0078125 | ≈ 0.11719 | 0.125 | 15 |
| p3109_p6 | 8 | 6 | 2 | True | 63 | 62 | 0.015625 | ≈ 0.48438 | 0.5 | 3.875 |
| binary16 | 16 | 11 | 5 | True | 15359 | 16383 | ≈5.9605e-08 | ≈6.0976e-05 | ≈6.1035e-05 | 65504 |
| bfloat16 | 16 | 8 | 8 | False | 16255 | 16383 | ≈9.1835e-41 | ≈1.1663e-38 | ≈1.1755e-38 | ≈3.3895e+38 |
| ocp_e8m0 | 8 | 1 | 8 | False | 127 | 127 | n/a | n/a | ≈5.8775e-39 | ≈1.7014e+38 |
| ocp_int8 | 8 | 8 | 0 | True | 63 | 63 | 0.015625 | ≈ 1.9844 | n/a | n/a |
Emit the same table, but with exact values
In this table, float values are printed as decimals, unless the decimals are not an exact representation of the value, in which case, they are printed as rationals (between 1 and 2) times 2^E.
D(df2, format=partial(render_float, False))
| name | B | P | E | rt16 | lt1 | gt1 | minSubnormal | maxSubnormal | minNormal | maxNormal |
|---|---|---|---|---|---|---|---|---|---|---|
| ocp_e2m1 | 4 | 2 | 2 | True | 1 | 5 | 0.5 | 0.5 | 1 | 6 |
| ocp_e2m3 | 6 | 4 | 2 | True | 7 | 23 | 0.125 | 0.875 | 1 | 7.5 |
| ocp_e3m2 | 6 | 3 | 3 | True | 11 | 19 | 0.0625 | 0.1875 | 0.25 | 28 |
| ocp_e4m3 | 8 | 4 | 4 | True | 55 | 70 | 2^-9 | 7/4*2^-7 | 0.015625 | 448 |
| ocp_e5m2 | 8 | 3 | 5 | True | 59 | 63 | 2^-16 | 3/2*2^-15 | 2^-14 | 57344 |
| p3109_p1 | 8 | 1 | 7 | False | 62 | 63 | n/a | n/a | 2^-62 | 2^63 |
| p3109_p2 | 8 | 2 | 6 | False | 63 | 62 | 2^-32 | 2^-32 | 2^-31 | 2^31 |
| p3109_p3 | 8 | 3 | 5 | True | 63 | 62 | 2^-17 | 3/2*2^-16 | 2^-15 | 49152 |
| p3109_p4 | 8 | 4 | 4 | True | 63 | 62 | 2^-10 | 7/4*2^-8 | 0.0078125 | 224 |
| p3109_p5 | 8 | 5 | 3 | True | 63 | 62 | 0.0078125 | 15/8*2^-4 | 0.125 | 15 |
| p3109_p6 | 8 | 6 | 2 | True | 63 | 62 | 0.015625 | 31/16*2^-2 | 0.5 | 3.875 |
| binary16 | 16 | 11 | 5 | True | 15359 | 16383 | 2^-24 | 1023/512*2^-15 | 2^-14 | 65504 |
| bfloat16 | 16 | 8 | 8 | False | 16255 | 16383 | 2^-133 | 127/64*2^-127 | 2^-126 | 255/128*2^127 |
| ocp_e8m0 | 8 | 1 | 8 | False | 127 | 127 | n/a | n/a | 2^-127 | 2^127 |
| ocp_int8 | 8 | 8 | 0 | True | 63 | 63 | 0.015625 | 127/64*2^0 | n/a | n/a |
Tables in RST/Markdown
These are used to generate gfloat documentation, but may be of use in other contexts so left here.
from tabulate import tabulate
dfstr = df.map(lambda x: render_float(True, x))
print(
tabulate(dfstr, df.columns, tablefmt="rst", showindex=False).replace(" nan", " n/a")
)
======== === === === =========== ================= ============ =========== ======
name B P E smallest smallest_normal max num_nans infs
======== === === === =========== ================= ============ =========== ======
ocp_e2m1 4 2 2 0.5 1 6 0 0
ocp_e2m3 6 4 2 0.125 1 7.5 0 0
ocp_e3m2 6 3 3 0.0625 0.25 28 0 0
ocp_e4m3 8 4 4 ≈0.0019531 0.015625 448 2 0
ocp_e5m2 8 3 5 ≈1.5259e-05 ≈6.1035e-05 57344 6 2
p3109_p1 8 1 7 ≈2.1684e-19 ≈2.1684e-19 ≈9.2234e+18 1 2
p3109_p2 8 2 6 ≈2.3283e-10 ≈4.6566e-10 ≈2.1475e+09 1 2
p3109_p3 8 3 5 ≈7.6294e-06 ≈3.0518e-05 49152 1 2
p3109_p4 8 4 4 ≈0.00097656 0.0078125 224 1 2
p3109_p5 8 5 3 0.0078125 0.125 15 1 2
p3109_p6 8 6 2 0.015625 0.5 3.875 1 2
binary16 16 11 5 ≈5.9605e-08 ≈6.1035e-05 65504 2046 2
bfloat16 16 8 8 ≈9.1835e-41 ≈1.1755e-38 ≈3.3895e+38 254 2
binary32 32 24 8 ≈1.4013e-45 ≈1.1755e-38 ≈3.4028e+38 ≈1.6777e+07 2
binary64 64 53 11 4.9407e-324 ≈2.2251e-308 ≈1.7977e+308 ≈9.0072e+15 2
ocp_e8m0 8 1 8 ≈5.8775e-39 ≈5.8775e-39 ≈1.7014e+38 1 0
ocp_int8 8 8 0 0.015625 n/a ≈ 1.9844 0 0
======== === === === =========== ================= ============ =========== ======
from tabulate import tabulate
dfstr = df.map(lambda x: render_float(False, x))
print(tabulate(dfstr, df.columns, tablefmt="rst", showindex=False))
======== === === === =========== ================= ======================================== ====================================== ======
name B P E smallest smallest_normal max num_nans infs
======== === === === =========== ================= ======================================== ====================================== ======
ocp_e2m1 4 2 2 0.5 1 6 0 0
ocp_e2m3 6 4 2 0.125 1 7.5 0 0
ocp_e3m2 6 3 3 0.0625 0.25 28 0 0
ocp_e4m3 8 4 4 2^-9 0.015625 448 2 0
ocp_e5m2 8 3 5 2^-16 2^-14 57344 6 2
p3109_p1 8 1 7 2^-62 2^-62 2^63 1 2
p3109_p2 8 2 6 2^-32 2^-31 2^31 1 2
p3109_p3 8 3 5 2^-17 2^-15 49152 1 2
p3109_p4 8 4 4 2^-10 0.0078125 224 1 2
p3109_p5 8 5 3 0.0078125 0.125 15 1 2
p3109_p6 8 6 2 0.015625 0.5 3.875 1 2
binary16 16 11 5 2^-24 2^-14 65504 2046 2
bfloat16 16 8 8 2^-133 2^-126 255/128*2^127 254 2
binary32 32 24 8 2^-149 2^-126 16777215/8388608*2^127 8388607/4194304*2^23 2
binary64 64 53 11 4.9407e-324 2^-1022 9007199254740991/9007199254740992*2^1024 4503599627370495/4503599627370496*2^53 2
ocp_e8m0 8 1 8 2^-127 2^-127 2^127 1 0
ocp_int8 8 8 0 0.015625 n/a 127/64*2^0 0 0
======== === === === =========== ================= ======================================== ====================================== ======
print(df.to_markdown())
| | name | B | P | E | smallest | smallest_normal | max | num_nans | infs |
|---:|:---------|----:|----:|----:|-------------:|------------------:|-----------------:|---------------:|-------:|
| 0 | ocp_e2m1 | 4 | 2 | 2 | 0.5 | 1 | 6 | 0 | 0 |
| 1 | ocp_e2m3 | 6 | 4 | 2 | 0.125 | 1 | 7.5 | 0 | 0 |
| 2 | ocp_e3m2 | 6 | 3 | 3 | 0.0625 | 0.25 | 28 | 0 | 0 |
| 3 | ocp_e4m3 | 8 | 4 | 4 | 0.00195312 | 0.015625 | 448 | 2 | 0 |
| 4 | ocp_e5m2 | 8 | 3 | 5 | 1.52588e-05 | 6.10352e-05 | 57344 | 6 | 2 |
| 5 | p3109_p1 | 8 | 1 | 7 | 2.1684e-19 | 2.1684e-19 | 9.22337e+18 | 1 | 2 |
| 6 | p3109_p2 | 8 | 2 | 6 | 2.32831e-10 | 4.65661e-10 | 2.14748e+09 | 1 | 2 |
| 7 | p3109_p3 | 8 | 3 | 5 | 7.62939e-06 | 3.05176e-05 | 49152 | 1 | 2 |
| 8 | p3109_p4 | 8 | 4 | 4 | 0.000976562 | 0.0078125 | 224 | 1 | 2 |
| 9 | p3109_p5 | 8 | 5 | 3 | 0.0078125 | 0.125 | 15 | 1 | 2 |
| 10 | p3109_p6 | 8 | 6 | 2 | 0.015625 | 0.5 | 3.875 | 1 | 2 |
| 11 | binary16 | 16 | 11 | 5 | 5.96046e-08 | 6.10352e-05 | 65504 | 2046 | 2 |
| 12 | bfloat16 | 16 | 8 | 8 | 9.18355e-41 | 1.17549e-38 | 3.38953e+38 | 254 | 2 |
| 13 | binary32 | 32 | 24 | 8 | 1.4013e-45 | 1.17549e-38 | 3.40282e+38 | 1.67772e+07 | 2 |
| 14 | binary64 | 64 | 53 | 11 | 4.94066e-324 | 2.22507e-308 | 1.79769e+308 | 9.0072e+15 | 2 |
| 15 | ocp_e8m0 | 8 | 1 | 8 | 5.87747e-39 | 5.87747e-39 | 1.70141e+38 | 1 | 0 |
| 16 | ocp_int8 | 8 | 8 | 0 | 0.015625 | nan | 1.98438 | 0 | 0 |