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proofs/math/typed/f64mulp.bend

bend-collections-laws-math@1.0.0.0 · 76 lines · 9 definitions

1import Base
2import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64light.bend as FL
3import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
4import bend-collections-laws-math@1.0.0.0/spec/math/f64.bend as SF
5import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
6import bend-collections-laws-math@1.0.0.0/src/math/f64.bend as F
7import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
8import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
9import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
10import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
11import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
12import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD
13import bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend as NR
14import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW
15import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA
16import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64sh.bend as SH
17import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bits.bend as FB
18import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64round.bend as FR
19import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64rtools.bend as RT
20
21# Pieces of SoftFloat's f64_mul and roundPackToF64 used by the arithmetic
22# proofs: the sticky OR into bit 0, the one-bit renormalization of rp62, and
23# exact products of 53-bit significands.
24
25def v(+x: U32) -> Nat:
26  U32.to_nat(x)
27
28# or_bit(a, b) is jam(a, b)
29def orv_c(+l: U32, +h: U32, +b: Bool) -> {SW.value(X.or_bit(WU.U64{l, h}, b)) == SW.jam(SW.value(WU.U64{l, h}), SF.b2n(b)) : Nat}:
30  FL.orv_c(l, h, b)
31
32def orv(+a: WU.U64, +b: Bool) -> {SW.value(X.or_bit(a, b)) == SW.jam(SW.value(a), SF.b2n(b)) : Nat}:
33  FL.orv(a, b)
34
35def minz(+r: Nat) -> {Nat.min(r, 0n) == 0n : Nat}:
36  FL.minz(r)
37
38# jam reads only whether r is zero
39def jmin(+q: Nat, +r: Nat) -> {SW.jam(q, SF.b2n(Bool.not(Nat.is_eq(r, 0n)))) == SW.jam(q, r) : Nat}:
40  FL.jmin(q, r)
41
42# ---- rp62: a significand in [2^61, 2^63) renormalized to bit 62 ----
43
44def lt62(+one: Nat, +h1: {one == 1n : Nat}, +sig: WU.U64, +c: U32, +hc: {c == U32{WD.pw(32n, 30n)} : U32}) -> {X.lt(sig, WU.U64{0, c}) == C.fits(62n, SW.value(sig)) : Bool}:
45  l1 = WA.lt_value(sig, WU.U64{0, c})
46  +ec = Equal.trans(Nat, C.shift(32n, v(c)), C.shift(32n, C.shift(30n, one)), C.shift(62n, one), Equal.cong(Nat, Nat, z => C.shift(32n, z), v(c), C.shift(30n, one), FB.pwv(30n, {==}, one, h1, c, hc)), Equal.sym(Nat, C.shift(62n, one), C.shift(32n, C.shift(30n, one)), WW.shift_comp(32n, 30n, one)))
47  Equal.trans(Bool, X.lt(sig, WU.U64{0, c}), Nat.is_lt(SW.value(sig), C.shift(32n, v(c))), C.fits(62n, SW.value(sig)), l1, Equal.trans(Bool, Nat.is_lt(SW.value(sig), C.shift(32n, v(c))), Nat.is_lt(SW.value(sig), C.shift(62n, one)), C.fits(62n, SW.value(sig)), Equal.cong(Nat, Bool, z => Nat.is_lt(SW.value(sig), z), C.shift(32n, v(c)), C.shift(62n, one), ec), FR.lt_fit(62n, one, h1, SW.value(sig))))
48
49def rp62_c(+s: Bool, +e: Nat, +sig: WU.U64, +x: Nat, +hx: {Nat.add(x, 2180n) == e : Nat}, +hx1: {Nat.is_le(1n, x) == True{} : Bool}, +h61: {C.fits(61n, SW.value(sig)) == False{} : Bool}, +h63: {C.fits(63n, SW.value(sig)) == True{} : Bool}, +f: Bool, +hf: {C.fits(62n, SW.value(sig)) == f : Bool}) -> {F.rp62_pick(s, e, sig, f) == SF.round(s, SW.value(sig), x) : F.F64}:
50  match f:
51    case True{}:
52      +Vs = SW.value(sig)
53      +d = X.add(sig, sig)
54      a1 = WA.add_value(sig, sig)
55      +f63 = L.subst(Nat, z => {C.fits(63n, z) == True{} : Bool}, Nat.add(0n, C.shift(1n, Vs)), Nat.double(Vs), {==}, WW.limbs_fit(1n, 62n, 0n, Vs, FR.fz(1n), hf))
56      +eVV = Equal.sym(Nat, Nat.double(Vs), Nat.add(Vs, Vs), NA.double_self(Vs))
57      +ed = Equal.trans(Nat, SW.value(d), C.low(64n, Nat.add(Vs, Vs)), Nat.double(Vs), a1, Equal.trans(Nat, C.low(64n, Nat.add(Vs, Vs)), C.low(64n, Nat.double(Vs)), Nat.double(Vs), Equal.cong(Nat, Nat, z => C.low(64n, z), Nat.add(Vs, Vs), Nat.double(Vs), eVV), WW.low_fit(64n, Nat.double(Vs), SH.fits_mono(63n, 64n, Nat.double(Vs), {==}, f63))))
58      +f62 = Equal.trans(Bool, C.fits(62n, Nat.double(Vs)), C.fits(61n, Vs), False{}, RT.fit_td(61n, 0n, Vs, {==}), h61)
59      +h62d = L.subst(Nat, z => {C.fits(62n, z) == False{} : Bool}, Nat.double(Vs), SW.value(d), Equal.sym(Nat, SW.value(d), Nat.double(Vs), ed), f62)
60      +h63d = L.subst(Nat, z => {C.fits(63n, z) == True{} : Bool}, Nat.double(Vs), SW.value(d), Equal.sym(Nat, SW.value(d), Nat.double(Vs), ed), f63)
61      +ex = Equal.trans(Nat, Nat.add(Nat.sub(x, 1n), 2180n), Nat.sub(Nat.add(x, 2180n), 1n), Nat.sub(e, 1n), Equal.sym(Nat, Nat.sub(Nat.add(x, 2180n), 1n), Nat.add(Nat.sub(x, 1n), 2180n), FR.sub_add_r(x, 2180n, 1n, hx1)), Equal.cong(Nat, Nat, z => Nat.sub(z, 1n), Nat.add(x, 2180n), e, hx))
62      +r1 = FR.round_pack(s, Nat.sub(e, 1n), d, Nat.sub(x, 1n), ex, h62d, h63d)
63      +r2 = Equal.cong(Nat, F.F64, z => SF.round(s, z, Nat.sub(x, 1n)), SW.value(d), C.shift(1n, Vs), ed)
64      +r3 = RT.round_shift(s, 1n, Vs, Nat.sub(x, 1n))
65      +r4 = Equal.cong(Nat, F.F64, z => SF.round(s, Vs, z), Nat.add(Nat.sub(x, 1n), 1n), x, Equal.trans(Nat, Nat.add(Nat.sub(x, 1n), 1n), Nat.add(1n, Nat.sub(x, 1n)), x, NA.add_comm(Nat.sub(x, 1n), 1n), N.sub_add(x, 1n, hx1)))
66      Equal.trans(F.F64, F.round_pack(s, Nat.sub(e, 1n), d), SF.round(s, SW.value(d), Nat.sub(x, 1n)), SF.round(s, Vs, x), r1, Equal.trans(F.F64, SF.round(s, SW.value(d), Nat.sub(x, 1n)), SF.round(s, C.shift(1n, Vs), Nat.sub(x, 1n)), SF.round(s, Vs, x), r2, Equal.trans(F.F64, SF.round(s, C.shift(1n, Vs), Nat.sub(x, 1n)), SF.round(s, Vs, Nat.add(Nat.sub(x, 1n), 1n)), SF.round(s, Vs, x), r3, r4)))
67    case False{}:
68      FR.round_pack(s, e, sig, x, hx, hf, h63)
69
70def rp62_g(+one: Nat, +h1: {one == 1n : Nat}, +s: Bool, +e: Nat, +sig: WU.U64, +x: Nat, +hx: {Nat.add(x, 2180n) == e : Nat}, +hx1: {Nat.is_le(1n, x) == True{} : Bool}, +h61: {C.fits(61n, SW.value(sig)) == False{} : Bool}, +h63: {C.fits(63n, SW.value(sig)) == True{} : Bool}, +c: U32, +hc: {c == U32{WD.pw(32n, 30n)} : U32}) -> {F.rp62_pick(s, e, sig, X.lt(sig, WU.U64{0, c})) == SF.round(s, SW.value(sig), x) : F.F64}:
71  +el = lt62(one, h1, sig, c, hc)
72  Equal.trans(F.F64, F.rp62_pick(s, e, sig, X.lt(sig, WU.U64{0, c})), F.rp62_pick(s, e, sig, C.fits(62n, SW.value(sig))), SF.round(s, SW.value(sig), x), Equal.cong(Bool, F.F64, t => F.rp62_pick(s, e, sig, t), X.lt(sig, WU.U64{0, c}), C.fits(62n, SW.value(sig)), el), rp62_c(s, e, sig, x, hx, hx1, h61, h63, C.fits(62n, SW.value(sig)), {==}))
73
74# SoftFloat's rp62(s, e, sig) for sig in [2^61, 2^63) is round(s, sig, e - 2180)
75def rp62(+s: Bool, +e: Nat, +sig: WU.U64, +x: Nat, +hx: {Nat.add(x, 2180n) == e : Nat}, +hx1: {Nat.is_le(1n, x) == True{} : Bool}, +h61: {C.fits(61n, SW.value(sig)) == False{} : Bool}, +h63: {C.fits(63n, SW.value(sig)) == True{} : Bool}) -> {F.rp62(s, e, sig) == SF.round(s, SW.value(sig), x) : F.F64}:
76  rp62_g(1n, {==}, s, e, sig, x, hx, hx1, h61, h63, 1073741824, {==})