crfix: correctly rounded fixed-point functions for Python

crfix.sigmoid(x, fin, fout, bits): for every integer x, the exact sigmoid of x·2^−fin rounded once to an integer with fout fractional bits, in the rounding mode you ask for, then saturated to bits. The same goes for exp, exp2, log, log2, tanh, sqrt, rsqrt, the reciprocal, division, SiLU and GELU (the exact one, v·Φ(v)). A container is 8, 16 or 32 bits, or "u8" and "u16", unsigned (TFLite's uint8). Every result is the one an exact computation followed by one rounding would give, so a lookup table or a fixed-point kernel built from it has no error beyond its format.

import numpy as np, crfix
x = np.arange(-128, 128, dtype=np.int8)
y = crfix.sigmoid(x, fin=4, fout=7, bits=8)              # sigmoid(x / 16) as round(sigmoid * 128), int8
t = crfix.table("tanh", fin=5, fout=7, bits=8)            # an int8 kernel's lookup table: t[x + 128]
q = crfix.div(a, 8, b, 8, fout=12, bits=16, mode="down")  # (a / 256) / (b / 256), rounded down

Layer normalization over the last axis, exact, with γ, β and ε in fixed point:

y = crfix.layernorm(x, fin=3, fout=5, eps=1, fe=10, gamma=g, fg=7, beta=b, fb=4, bits=8)

TFLite's affine formats, where a value is scale·(q − zero_point), for any scales:

inp, out = crfix.Affine(0.0478, 3), crfix.Affine(1 / 256, -128)
t = crfix.table_affine("sigmoid", inp, out)               # a LOGISTIC kernel's table, every entry correctly rounded
p = crfix.softmax(logits, c=beta * 0.0478, out=out)       # the exact softmax of c·logits, rounded into out

A lookup table as C source, CSV or a list, from the command line (--f32 rounds decimal scales to float32 first, as a TFLite model stores them; hex floats such as 0x1p-8 are taken exactly):

python -m crfix table tanh --fin 5 --fout 7 --bits 8 --format c
python -m crfix table-affine sigmoid --in 0.0478,3 --out 0x1p-8,-128 --f32 --format c --name logistic

• Modes: "nearest" (ties to even), "up", "down" and "zero". In an affine format, "zero" rounds the value itself toward 0, before the zero point is added.
• Domain edges: log of a value ≤ 0 gives the most negative result. sqrt of a negative value gives 0. rsqrt of a value ≤ 0, and the reciprocal of 0, give the largest. a/0 goes by a's sign, and 0/0 gives 0.
• Integer arithmetic only: the library has no floating point, and gives the same bits on x86-64, ARM64, 32-bit ARM and a Cortex-M0.
• How it is proven: every 16-bit input, in every format, mode and container, is checked against MPFR. Results that the fast 64-bit enclosures cannot decide go to interval arithmetic that is correct by construction. This package's own tests use a second, independent reference: Python's decimal and exact fractions.

pip install crfix brings a wheel with the library, libcrfix.so.0 (or libcrfix.0.dylib on macOS); CRFIX_LIBRARY names another. The methods and their checks are in crnumerics' fix/README.md.