def sqr_root_v

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqr.bend:388

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/f64.bend as SF
import bend-collections-laws-math@1.0.0.0/spec/math/w64.bend as SW
import bend-collections-laws-math@1.0.0.0/src/math/f64.bend as F
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64div.bend as W64D
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64sqv.bend as QV

388# the F64 root: roundPackToF64 of isqrt(nh 2^64) with its sticky bit
389def sqr_root_v(+one: Nat, +h1: {one == 1n : Nat}, +nh: WU.U64, +h62: {C.fits(62n, SW.value(nh)) == True{} : Bool}, +h60: {C.fits(60n, SW.value(nh)) == False{} : Bool}, +e: Nat, +x: Nat, +hx: {Nat.add(x, 2180n) == e : Nat}) -> {F.sq_root(e, nh) == SF.round(False{}, SW.jam(M.isqrt(C.shift(64n, SW.value(nh))), SF.b2n(Bool.not(Nat.is_eq(Nat.mul(M.isqrt(C.shift(64n, SW.value(nh))), M.isqrt(C.shift(64n, SW.value(nh)))), C.shift(64n, SW.value(nh)))))), x) : F.F64}:
390  +hu = Equal.trans(Nat, v(X.snd_r(X.div32(WU.U64{0, X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))}, U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))), Nat.mod(SW.value(WU.U64{0, X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))}), v(U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), Nat.mod(C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), W64D.div32_rem(WU.U64{0, X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))}, U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))), QV.dw_nz(one, h1, nh, h62, h60)), Equal.trans(Nat, Nat.mod(SW.value(WU.U64{0, X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))}), v(U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), Nat.mod(C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))), v(U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), Nat.mod(C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), Equal.cong(Nat, Nat, z => Nat.mod(C.shift(32n, z), v(U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh))))), v(X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))), Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), QV.lo_w(one, h1, nh, h62, h60)), Equal.cong(Nat, Nat, z => Nat.mod(C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))), z), v(U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))), QV.dw_v(one, h1, nh, h62, h60))))
391  div_v(one, h1, nh, h62, h60, e, x, hx, X.div32(WU.U64{0, X.lo(X.sub(nh, X.mul32(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))))}, U32.add(X.lo(X.isqrt(nh)), X.lo(X.isqrt(nh)))), QV.q_v(one, h1, nh, h62, h60), hu)