g_math 0.6.7

Multi-domain fixed-point arithmetic with geometric extension: Lie groups, manifolds, ODE solvers, tensors, fiber bundles — pure Rust, zero-float, deterministic
Documentation 
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#!/usr/bin/env python3
"""References for the 0.6.7 wide-tier gate functions (tests/data/gate_refs.rs).

Q64.64 functions (`g_math::wide`): sigmoid, softplus, silu, sqrt. Inputs are
Q.40 raws (the value k / 2^40), given to the function as `k << 24`. For each,
the reference is the exact value rounded to nearest (ties toward +infinity)
at 64, 40 and 20 fractional bits, from mpmath at 140 digits (sqrt: exact
integer arithmetic), so the test can measure the Q64.64 error in units and
check the narrowed results for exact equality.

Storage functions (`fused::sigmoid_mul`, `fused::entropy`): dyadic inputs
with at most 8 fraction bits, stored exactly at every gated split; the
reference is a 120-digit literal the test parses with the exact literal
parser, which rounds it to nearest at the build's split.

Integer functions (`wide::mul_div_*`, `wide::try_ratio_from_str`): exact
Python integers and `Fraction`.

Usage: python3 scripts/generate_gate_refs.py > tests/data/gate_refs.rs
"""
import random
from fractions import Fraction
from math import isqrt

from mpmath import mp, mpf, exp, log, floor

mp.dps = 140
SEED = 20261005
DIGITS = 120


def nearest(v, frac_bits):
    """round(v * 2^frac_bits), ties toward +infinity."""
    return int(floor(v * mpf(2) ** frac_bits + mpf(1) / 2))


def sigmoid(x):
    return 1 / (1 + exp(-x))


def softplus(x):
    return log(1 + exp(x))


def lit(v):
    return mp.nstr(v, DIGITS, strip_zeros=False, min_fixed=-10**9, max_fixed=10**9)


def dy_lit(fr):
    """Exact decimal literal of a dyadic Fraction."""
    sign = "-" if fr < 0 else ""
    fr = abs(fr)
    d = 0
    while (fr * 10 ** d).denominator != 1:
        d += 1
    n = int(fr * 10 ** d)
    if d == 0:
        return f"{sign}{n}"
    s = str(n).rjust(d + 1, "0")
    return f"{sign}{s[:-d]}.{s[-d:]}"


out = ["// Generated by scripts/generate_gate_refs.py. Do not edit.", ""]
rng = random.Random(SEED)

# ---- Q64.64 functions on Q.40 inputs -----------------------------------------
def q40_inputs(n, lo, hi):
    xs = [0, 1, -1, 1 << 40, -(1 << 40), 12 << 40, -(12 << 40), 39 << 40, -(39 << 40)]
    while len(xs) < n:
        mag = rng.choice([1, 4, 12, 40])
        xs.append(rng.randrange(int(lo * mag / 40 * 2 ** 40), int(hi * mag / 40 * 2 ** 40)))
    return [x for x in xs if lo * 2 ** 40 <= x <= hi * 2 ** 40]

for name, f in (("SIGMOID", sigmoid), ("SOFTPLUS", softplus), ("SILU", lambda x: x * sigmoid(x))):
    rows = []
    for k in q40_inputs(400, -40, 40):
        v = f(mpf(k) / mpf(2) ** 40)
        rows.append(f"({k}, {nearest(v, 64)}, {nearest(v, 40)}, {nearest(v, 20)})")
    out.append(f"/// (x at Q.40, f(x) at Q64.64, at Q.40, at Q.20), each the nearest value.")
    out.append(f"pub const {name}: &[(i128, i128, i128, i128)] = &[{', '.join(rows)}];")

# sqrt: exact. round(sqrt(x) * 2^f) for x = k / 2^40 is the nearest integer to sqrt(k * 2^(2f - 40)).
def sqrt_nearest(k, f):
    assert 2 * f >= 40
    n = k << (2 * f - 40)
    r = isqrt(n)
    return r + 1 if n > r * r + r else r

rows = []
ks = [0, 1, 2, 3, 1 << 40, 1 << 41, (1 << 40) + 1, 9 << 38, 1 << 86]
while len(ks) < 400:
    ks.append(rng.randrange(0, 1 << rng.choice([10, 30, 41, 50, 70, 86])))
for k in ks:
    rows.append(f"({k}, {sqrt_nearest(k, 64)}, {sqrt_nearest(k, 40)}, {sqrt_nearest(k, 20)})")
out.append("/// (x at Q.40, sqrt(x) at Q64.64, at Q.40, at Q.20), each the nearest value (exact integer arithmetic).")
out.append(f"pub const SQRT: &[(i128, i128, i128, i128)] = &[{', '.join(rows)}];")

# ---- fused::sigmoid_mul ---------------------------------------------------------
def dy(lo, hi, bits=8):
    return Fraction(rng.randrange(int(lo * 2 ** bits), int(hi * 2 ** bits)), 1 << bits)

def m_of(fr):
    return mpf(fr.numerator) / fr.denominator

rows = []
cases = [(Fraction(1), Fraction(0)), (Fraction(-1), Fraction(0)), (Fraction(3), Fraction(-100)), (Fraction(3), Fraction(100)),
         (Fraction(127), Fraction(1, 256)), (Fraction(-127), Fraction(-1, 256)), (Fraction(0), Fraction(5))]
while len(cases) < 240:
    cases.append((dy(-127, 127), dy(*rng.choice([(-2, 2), (-12, 12), (-60, 60)]))))
for x, g in cases:
    rows.append(f'("{dy_lit(x)}", "{dy_lit(g)}", "{lit(m_of(x) * sigmoid(m_of(g)))}")')
out.append("/// (x, gate, x * sigmoid(gate)); |x| < 128 so every gated split holds it.")
out.append(f"pub const SIGMOID_MUL: &[(&str, &str, &str)] = &[{', '.join(rows)}];")

# ---- fused::entropy ---------------------------------------------------------------
rows = []
for _ in range(60):
    n = rng.choice([1, 2, 5, 16, 64, 300])
    kind = rng.randrange(3)
    if kind == 0:      # a distribution on a 1/256 grid (sums to 1), zeros included
        cuts = sorted(rng.randrange(0, 257) for _ in range(n - 1))
        parts = [b - a for a, b in zip([0] + cuts, cuts + [256])]
        ws = [Fraction(p, 256) for p in parts]
    elif kind == 1:    # arbitrary small weights
        ws = [dy(0, 1) for _ in range(n)]
    else:              # weights above one (negative terms)
        ws = [dy(0, 4) for _ in range(min(n, 16))]
    h = -sum((m_of(w) * log(m_of(w)) for w in ws if w > 0), mpf(0))
    rows.append("(&[" + ", ".join(f'"{dy_lit(w)}"' for w in ws) + f'], "{lit(h)}")')
out.append("/// (weights, -sum(w ln w)).")
out.append(f"pub const ENTROPY: &[(&[&str], &str)] = &[{', '.join(rows)}];")

# ---- integer mul_div ---------------------------------------------------------------
I128_MIN, I128_MAX = -(1 << 127), (1 << 127) - 1
def r128():
    bits = rng.randrange(1, 128)
    v = rng.randrange(0, 1 << bits)
    return -v if rng.randrange(2) else v
rows = []
trip = [(I128_MAX, I128_MAX, I128_MAX), (I128_MIN, I128_MAX, I128_MIN), (I128_MIN, I128_MIN, I128_MAX), (7, -3, 2), (-7, 3, -2),
        (7, 3, 2), (-7, -3, -2), (1, 1, 2), (-1, 1, 2), (3, 1, 2), (-3, 1, 2), (0, 5, -9), (I128_MAX, 2, 1)]
while len(trip) < 600:
    trip.append((r128(), r128(), r128() or 1))
for a, b, d in trip:
    fl = (a * b) // d
    ne = (2 * a * b + d) // (2 * d)            # floor(p/d + 1/2): ties toward +infinity
    enc = lambda v: f"Some({v})" if I128_MIN <= v <= I128_MAX else "None"
    rows.append(f"({a}, {b}, {d}, {enc(fl)}, {enc(ne)})")
out.append("/// (a, b, d, floor(a*b/d), nearest(a*b/d) ties toward +infinity); None when it does not fit i128.")
out.append(f"pub const MUL_DIV: &[(i128, i128, i128, Option<i128>, Option<i128>)] = &[{', '.join(rows)}];")
rows = []
U = (1 << 128) - 1
ut = [(U, U, U), (U, U, 1), (U, 2, 3), (0, U, 7), (1 << 127, 2, 1), (1 << 127, 2, 2)]
while len(ut) < 300:
    ut.append((rng.randrange(0, 1 << rng.randrange(1, 129)), rng.randrange(0, 1 << rng.randrange(1, 129)), rng.randrange(1, 1 << rng.randrange(1, 129))))
for a, b, d in ut:
    q = (a * b) // d
    rows.append(f"({a}, {b}, {d}, {'Some(%d)' % q if q <= U else 'None'})")
out.append("/// (a, b, d, floor(a*b/d)); None when it does not fit u128.")
out.append(f"pub const MUL_DIV_U128: &[(u128, u128, u128, Option<u128>)] = &[{', '.join(rows)}];")

# ---- exact ratios -------------------------------------------------------------------
rows = []
lits = ["0.25", "10000000.0", "-1.5e1", "1e-5", ".5", "5.", "-0", "0.000", "+3", "1e3", "12.3400", "  7.5  ", "0.1", "1e38", "1e-38",
        "170141183460469231731687303715884105727", "0.5000000000000000000000000000000000000000000000000000", "1E2", "2.5e-1", "-0.125"]
for _ in range(180):
    s = rng.choice(["", "-", "+"]) + str(rng.randrange(0, 10 ** rng.randrange(1, 12)))
    if rng.randrange(3):
        s += "." + "".join(rng.choice("0123456789") for _ in range(rng.randrange(0, 14)))
    if rng.randrange(4) == 0:
        s += "e" + str(rng.randrange(-12, 13))
    lits.append(s)
for s in lits:
    body = s.strip().lstrip("+")
    fr = Fraction(body if body not in ("5.",) and not body.endswith(".") else body + "0") if not body.startswith(".") and not body.startswith("-.") else Fraction("0" + body)
    ok = I128_MIN <= fr.numerator <= I128_MAX and fr.denominator <= I128_MAX
    rows.append(f'("{s}", {"Some((%d, %d))" % (fr.numerator, fr.denominator) if ok else "None"})')
out.append("/// (literal, the exact fraction in lowest terms); None when it does not fit.")
out.append(f"pub const RATIO: &[(&str, Option<(i128, i128)>)] = &[{', '.join(rows)}];")
out.append('/// Strings outside the grammar.')
out.append('pub const RATIO_BAD: &[&str] = &["", " ", ".", "-", "e5", "1e", "1e+", "1.2.3", "0x10", "1/3", "pi", "1 2", "--1", "1e1.5", "٣"];')
print("\n".join(out))