math: the remaining missing names, and degrees() associates the wrong way

• Type: bug (missing surface + last-ulp divergence) — Track N
• Found: 2026-08-12, sweeping the whole math surface against CPython.
• Companion to [[bug-n-math-trunc-and-log-need-frontend-intercepts]], which covers trunc / log / pow / copysign / atan2. This is what a full sweep found besides those.

1. degrees() — a real value divergence, one line to fix

math.degrees(3.14)   pxx: 179.90874767107852   CPython: 179.9087476710785

Measured directly: pxx's answer is exactly 3.14 * 180 / math.pi and CPython's is exactly 3.14 * (180 / math.pi). The conversion factor must be formed FIRST and then multiplied — the same constant CPython uses — rather than multiplying by 180 and dividing by pi. radians() agrees already, so only degrees is wrong.

2. Names that do not resolve at all

Each is error: undefined variable:

sin/cos/tan/hypot/comb/gcd/factorial/isclose/fmod/copysign /log2/log10/exp/sqrt/floor/ceil/inf/nan/pi/e are all present and agree, so this is a short tail rather than a hole.

fsum is worth doing together with [[bug-nilpy-sum-of-floats-has-no-compensated-summation]] — same algorithm, two callers.

3. import random does not exist

random.random() / random.randint() / random.seed() are all "undefined variable". Not a divergence so much as an absent module, but it is one of the first imports a script reaches for, and seeding makes it testable against CPython only for the shape (a float in [0,1), an int in range) — the sequence itself is deliberately implementation-defined, so the gate must assert the contract, not the values.

Gate

A .npy diffed against CPython: degrees over several angles (including the row above, which is what shows the association), every name in the table with a value CPython pins exactly, and — if random lands here rather than in its own ticket — range/type assertions plus seed() making two runs of the same program agree with each other.

Resolution (2026-08-15)

All three sections done, with one deliberate exclusion.

1. degrees() — lib/rtl/math.pas now forms the conversion FACTOR first (r * (180/pi), and d * (pi/180) for the sibling), which is how both CPython and FPC associate it. math.degrees(3.14) is 179.9087476710785. The fix lands in a Track B file because that is where RadToDeg lives; it is two lines, additive to nothing, and it corrects the PASCAL frontend's FPC parity by the same edit.

2. The eight missing names — seven landed, one refused for a stated reason.

• asin/acos/atan were a pure NAME difference: the RTL spells them ArcSin/ArcCos/ArcTan, so three rows in the intercept table is the whole change — exactly what math.log -> Ln already does.
• modf, prod, fsum, perm are new pylib routines. None needs a transcendental, which is precisely what lets them live in a BUILTIN unit (the wall math.log/math.pow hit). fsum is Neumaier compensated summation and should be SHARED with sum() when [[bug-nilpy-sum-of-floats-has-no-compensated-summation]] lands, not copied.
• dist is NOT done, and this is the interesting one. It needs Sqrt, which lives in lib/rtl/math.pas and a builtin unit cannot reach, while its arguments are pylib CONTAINERS the RTL cannot name. It is the one entry that needs a composed lowering rather than a table row, so it is left out with a comment at the table saying why, rather than half-built.

Two things the CPython diff caught that reasoning would not have:

• math.modf(-0.25) is (-0.25, -0.0) — the integral part keeps the sign even when it is ZERO, and Int() hands back a positive zero. Read from the sign BIT, like copysign does.
• asin/acos are 1-2 ulps off libm for mid-range arguments while atan and the endpoints are exact — an accuracy defect in the RTL, not in the wiring. Filed as [[bug-b-arcsin-arccos-lose-2-ulps-vs-libm]] (Track B, p20); this test asserts only the exact cases until it lands.

3. random exists now — seed, random, randint, randrange, uniform, choice, shuffle, on pylib's own SplitMix64. The sequence is deliberately not CPython's (its Mersenne stream is an implementation detail there too), so the test asserts the CONTRACT: random() in [0, 1), randint(a, b) inclusive at both ends, a repeated seed repeating, choice() an element of the sequence, shuffle() a permutation that mutates in place and returns None. The state starts at a FIXED value rather than a clock reading, so a program that never seeds still reproduces — which is what makes a failure reportable.

Gate: test/test_nilpy_math_surface_and_random.npy (+.expected, wired into the Makefile). The math half is byte-identical to CPython; the random half asserts the contract. tools/gate.sh quick GREEN, self-host byte-identical.

Log

• 2026-08-15 — resolved, commit 3799e1d77.