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Added solution "Day 24: Never Tell Me The Odds", part 2

This commit is contained in:
Stefan Müller 2024-05-27 02:29:49 +02:00
parent 44caf3e21c
commit 3e3e1d45d3
2 changed files with 332 additions and 443 deletions
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773
solvers/UNeverTellMeTheOdds.pas
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@ -22,7 +22,7 @@ unit UNeverTellMeTheOdds;
interface
uses
Classes, SysUtils, Generics.Collections, Math, matrix, USolver, UNumberTheory, UBigInt;
Classes, SysUtils, Generics.Collections, Math, USolver, UNumberTheory, UBigInt, UPolynomial, UPolynomialRoots;
type
@ -30,26 +30,15 @@ type
THailstone = class
public
Position, Velocity: Tvector3_extended;
P0, P1, P2: Int64;
V0, V1, V2: Integer;
constructor Create(const ALine: string);
constructor Create(const APosition, AVelocity: Tvector3_extended);
constructor Create;
end;
THailstones = specialize TObjectList<THailstone>;
{ TFirstCollisionPolynomial }
TFirstCollisionPolynomial = class
private
FA: array[0..10] of TBigInt;
FH: array[0..6] of TBigInt;
procedure NormalizeCoefficients;
public
procedure Init(constref AHailstone1, AHailstone2, AHailstone3: THailstone; const t_0, t_1, t_2: Int64);
function EvaluateAt(const AT0: Int64): TBigInt;
function CalcPositiveIntegerRoot: Int64;
function CalcT1(const AT0: Int64): Int64;
end;
TInt64Array = array of Int64;
{ TNeverTellMeTheOdds }
@ -57,10 +46,15 @@ type
private
FMin, FMax: Int64;
FHailstones: THailstones;
FA: array[0..10] of Int64;
FH: array[0..6] of Int64;
FA: array[0..10] of TBigInt;
FH: array[0..6] of TBigInt;
function AreIntersecting(constref AHailstone1, AHailstone2: THailstone): Boolean;
procedure FindRockThrow(const AIndex1, AIndex2, AIndex3: Integer);
function FindRockThrow(const AIndex0, AIndex1, AIndex2: Integer): Int64;
procedure CalcCollisionPolynomials(constref AHailstone0, AHailstone1, AHailstone2: THailstone; out OPolynomial0,
OPolynomial1: TBigIntPolynomial);
function CalcRockThrowCollisionOptions(constref AHailstone0, AHailstone1, AHailstone2: THailstone): TInt64Array;
function ValidateRockThrow(constref AHailstone0, AHailstone1, AHailstone2: THailstone; const AT0, AT1: Int64):
Int64;
public
constructor Create(const AMin: Int64 = 200000000000000; const AMax: Int64 = 400000000000000);
destructor Destroy; override;
@ -70,10 +64,6 @@ type
function GetPuzzleName: string; override;
end;
const
CIterationThreshold = 0.00001;
CEpsilon = 0.0000000001;
implementation
{ THailstone }
@ -83,69 +73,71 @@ var
split: TStringArray;
begin
split := ALine.Split([',', '@']);
Position.init(
StrToFloat(Trim(split[0])),
StrToFloat(Trim(split[1])),
StrToFloat(Trim(split[2])));
Velocity.init(
StrToFloat(Trim(split[3])),
StrToFloat(Trim(split[4])),
StrToFloat(Trim(split[5])));
P0 := StrToInt64(Trim(split[0]));
P1 := StrToInt64(Trim(split[1]));
P2 := StrToInt64(Trim(split[2]));
V0 := StrToInt(Trim(split[3]));
V1 := StrToInt(Trim(split[4]));
V2 := StrToInt(Trim(split[5]));
end;
constructor THailstone.Create(const APosition, AVelocity: Tvector3_extended);
constructor THailstone.Create;
begin
Position := APosition;
Velocity := AVelocity;
end;
{ TFirstCollisionPolynomial }
{ TNeverTellMeTheOdds }
procedure TFirstCollisionPolynomial.NormalizeCoefficients;
function TNeverTellMeTheOdds.AreIntersecting(constref AHailstone1, AHailstone2: THailstone): Boolean;
var
shift: Integer;
i: Low(FA)..High(FA);
//gcd: TBigInt;
m1, m2, x, y: Double;
begin
// Eliminates zero constant term.
shift := 0;
while (shift <= High(FA)) and (FA[shift] = 0) do
Inc(shift);
if shift <= High(FA) then
Result := False;
m1 := AHailstone1.V1 / AHailstone1.V0;
m2 := AHailstone2.V1 / AHailstone2.V0;
if m1 <> m2 then
begin
if shift > 0 then
x := (AHailstone2.P1 - m2 * AHailstone2.P0
- AHailstone1.P1 + m1 * AHailstone1.P0)
/ (m1 - m2);
if (FMin <= x) and (x <= FMax)
and (x * Sign(AHailstone1.V0) >= AHailstone1.P0 * Sign(AHailstone1.V0))
and (x * Sign(AHailstone2.V0) >= AHailstone2.P0 * Sign(AHailstone2.V0))
then
begin
for i := Low(FA) to High(FA) - shift do
FA[i] := FA[i + shift];
for i := High(FA) - shift + 1 to High(FA) do
FA[i] := 0;
y := m1 * (x - AHailstone1.P0) + AHailstone1.P1;
if (FMin <= y) and (y <= FMax) then
Result := True
end;
//// Finds GCD of all coefficients.
//gcd := FA[Low(FA)];
//for i := Low(FA) + 1 to High(FA) do
// if FA[i] <> 0 then
// gcd := TNumberTheory.GreatestCommonDivisor(gcd, FA[i]);
//WriteLn('GCD: ', gcd);
//
//for i := Low(FA) to High(FA) do
// FA[i] := FA[i] div gcd;
end;
//WriteLn('(', FA[10], ') * x^10 + (', FA[9], ') * x^9 + (', FA[8], ') * x^8 + (', FA[7], ') * x^7 + (',
// FA[6], ') * x^6 + (', FA[5], ') * x^5 + (', FA[4], ') * x^4 + (', FA[3], ') * x^3 + (', FA[2], ') * x^2 + (',
// FA[1], ') * x + (', FA[0], ')');
end;
procedure TFirstCollisionPolynomial.Init(constref AHailstone1, AHailstone2, AHailstone3: THailstone; const t_0, t_1,
t_2: Int64);
function TNeverTellMeTheOdds.FindRockThrow(const AIndex0, AIndex1, AIndex2: Integer): Int64;
var
t0, t1: TInt64Array;
i, j: Int64;
begin
t0 := CalcRockThrowCollisionOptions(FHailstones[AIndex0], FHailstones[AIndex1], FHailstones[AIndex2]);
t1 := CalcRockThrowCollisionOptions(FHailstones[AIndex1], FHailstones[AIndex0], FHailstones[AIndex2]);
Result := 0;
for i in t0 do
begin
for j in t1 do
begin
Result := ValidateRockThrow(FHailstones[AIndex0], FHailstones[AIndex1], FHailstones[AIndex2], i, j);
if Result > 0 then
Break;
end;
if Result > 0 then
Break;
end;
end;
procedure TNeverTellMeTheOdds.CalcCollisionPolynomials(constref AHailstone0, AHailstone1, AHailstone2: THailstone; out
OPolynomial0, OPolynomial1: TBigIntPolynomial);
var
P_00, P_01, P_02, P_10, P_11, P_12, P_20, P_21, P_22,
V_00, V_01, V_02, V_10, V_11, V_12, V_20, V_21, V_22: Int64;
k: array[0..139] of TBigInt;
// For debug calculations
act, a_1, a_2, b_0, b_1, c_0, c_1, d_0, d_1, e_0, e_1, f_0, f_1, f_2: Int64;
begin
// Solving this non-linear equation system, with velocities V_i and start positions P_i:
// V_0 * t_0 + P_0 = V_x * t_0 + P_x
@ -155,55 +147,88 @@ begin
// P_x = (V_0 - V_x) * t_0 + P_0
// V_x = (V_0 * t_0 - V_1 * t_1 + P_0 - P_1) / (t_0 - t_1)
// And with vertex components:
// 1: 0 = (t_1 - t_0) * (V_00 * t_0 - V_20 * t_2 + P_00 - P_20) - (t_2 - t_0) * (V_00 * t_0 - V_10 * t_1 + P_00 - P_10)
// 2: t_1 = (((V_01 - V_21) * t_2 + P_11 - P_21) * t_0 + (P_01 - P_11) * t_2)
// 1: 0 = (t_1 - t_0) * (V_00 * t_0 - V_20 * t_2 + P_00 - P_20)
// - (t_2 - t_0) * (V_00 * t_0 - V_10 * t_1 + P_00 - P_10)
// 2: t_1 = (((V_01 - V_21) * t_2 + P_11 - P_21) * t_0 + (P_01 - P_11) * t_2)
// / ((V_01 - V_11) * t_0 + (V_11 - V_21) * t_2 + P_01 - P_21)
// 3: t_2 = (((V_02 - V_12) * t_1 + P_22 - P_12) * t_0 + (P_02 - P_22) * t_1)
// 3: t_2 = (((V_02 - V_12) * t_1 + P_22 - P_12) * t_0 + (P_02 - P_22) * t_1)
// / ((V_02 - V_22) * t_0 + (V_22 - V_12) * t_1 + P_02 - P_12)
// for t_0, t_1, t_2 not pairwise equal.
// With some substitutions depending only on t_0 this gives
// 1: 0 = (t_1 - t_0) * (f_2 - V_20 * t_2) - (t_2 - t_0) * (f_1 - V_10 * t_1)
// 2: t_1 = (b_0 + b_1 * t_2) / (c_0 + c_1 * t_2)
// 3: t_2 = (d_0 + d_1 * t_1) / (e_0 + e_1 * t_1)
// 1: 0 = (t_1 - t_0) * (a_1 - V_20 * t_2) - (t_2 - t_0) * (a_2 - V_10 * t_1)
// 2: t_1 = (b_0 + b_1 * t_2) / (c_0 + c_1 * t_2)
// 3: t_2 = (d_0 + d_1 * t_1) / (e_0 + e_1 * t_1)
// And 3 in 2 gives:
// 4: g_2 * t_1^2 - g_1 * t_1 - g_0 = 0
// Then, with 4 and 3 in 1 and lengthy calculations with many substitutions (see constants k below, now independent of
// t_0), the following polynomial can be constructed, with t_0 being a positive integer root of this polynomial.
// y = a_10 * x^10 + a_9 * x^9 + ... + a_0
// 4: f_2 * t_1^2 + f_1 * t_1 - f_0 = 0
// Then, with 4 and 3 in 1 and many substitutions (see constants k below, now independent of t_0), the equation
// 5: 0 = p_0(t_0) + p_1(t_0) * sqrt(p_2(t_0))
// can be constructed, where p_0, p_1, and p_2 are polynomials in t_0. Since we are searching for an integer solution,
// we assume that there is an integer t_0 that is a root of both p_0 and p_1, which would solve the equation.
P_00 := Round(AHailstone1.Position.data[0]);
P_01 := Round(AHailstone1.Position.data[1]);
P_02 := Round(AHailstone1.Position.data[2]);
P_10 := Round(AHailstone2.Position.data[0]);
P_11 := Round(AHailstone2.Position.data[1]);
P_12 := Round(AHailstone2.Position.data[2]);
P_20 := Round(AHailstone3.Position.data[0]);
P_21 := Round(AHailstone3.Position.data[1]);
P_22 := Round(AHailstone3.Position.data[2]);
V_00 := Round(AHailstone1.Velocity.data[0]);
V_01 := Round(AHailstone1.Velocity.data[1]);
V_02 := Round(AHailstone1.Velocity.data[2]);
V_10 := Round(AHailstone2.Velocity.data[0]);
V_11 := Round(AHailstone2.Velocity.data[1]);
V_12 := Round(AHailstone2.Velocity.data[2]);
V_20 := Round(AHailstone3.Velocity.data[0]);
V_21 := Round(AHailstone3.Velocity.data[1]);
V_22 := Round(AHailstone3.Velocity.data[2]);
// Subsitutions depending on t_0:
// a_1 = V_00 * t_0 + P_00 - P_20
// a_2 = V_00 * t_0 + P_00 - P_10
// b_0 = (P_11 - P_21) * t_0
// b_1 = (V_01 - V_21) * t_0 + P_01 - P_11
// c_0 = (V_01 - V_11) * t_0 + P_01 - P_21
// c_1 = V_11 - V_21
// d_0 = (P_22 - P_12) * t_0
// d_1 = (V_02 - V_12) * t_0 + P_02 - P_22
// e_0 = (V_02 - V_22) * t_0 + P_02 - P_12
// e_1 = V_22 - V_12
// f_0 = b_1 * d_0 + b_0 * e_0
// f_1 = c_0 * e_0 + c_1 * d_0 - b_0 * e_1 - b_1 * d_1
// f_2 = c_0 * e_1 + c_1 * d_1
k[0] := P_00 - P_20;
k[1] := P_00 - P_10;
k[2] := P_11 - P_21;
k[3] := P_01 - P_11;
k[4] := P_01 - P_21;
k[5] := P_22 - P_12;
k[6] := P_02 - P_22;
k[7] := P_02 - P_12;
k[8] := V_11 - V_21;
k[9] := V_22 - V_12;
k[10] := V_01 - V_21;
k[11] := V_01 - V_11;
k[12] := V_02 - V_12;
k[13] := V_02 - V_22;
// Calculations for equation 5 (4 and 3 in 1).
// 1: 0 = (t_1 - t_0) * (a_1 - V_20 * t_2) - (t_2 - t_0) * (a_2 - V_10 * t_1)
// 3: (e_0 + e_1 * t_1) * t_2 = (d_0 + d_1 * t_1)
// 0 = (t_1 - t_0) * (a_1 - V_20 * t_2) - (t_2 - t_0) * (a_2 - V_10 * t_1)
// = (t_1 - t_0) * (a_1 * (e_0 + e_1 * t_1) - V_20 * (e_0 + e_1 * t_1) * t_2) - ((e_0 + e_1 * t_1) * t_2 - (e_0 + e_1 * t_1) * t_0) * (a_2 - V_10 * t_1)
// = (t_1 - t_0) * (a_1 * (e_0 + e_1 * t_1) - V_20 * (d_0 + d_1 * t_1)) - ((d_0 + d_1 * t_1) - (e_0 + e_1 * t_1) * t_0) * (a_2 - V_10 * t_1)
// = (t_1 - t_0) * (a_1 * e_0 + a_1 * e_1 * t_1 - V_20 * d_0 - V_20 * d_1 * t_1) - (d_0 + d_1 * t_1 - e_0 * t_0 - e_1 * t_1 * t_0) * (a_2 - V_10 * t_1)
// = (a_1 * e_1 - V_20 * d_1) * t_1^2 + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1 - V_20 * d_1)) * t_1 - t_0 * (a_1 * e_0 - V_20 * d_0)
// - ( - V_10 * (d_1 - e_1 * t_0) * t_1^2 + ((d_1 - e_1 * t_0) * a_2 - V_10 * (d_0 - e_0 * t_0)) * t_1 + (d_0 - e_0 * t_0) * a_2)
// = (a_1 * e_1 - V_20 * d_1 + V_10 * (d_1 - e_1 * t_0)) * t_1^2
// + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1 - V_20 * d_1) - (d_1 - e_1 * t_0) * a_2 + V_10 * (d_0 - e_0 * t_0)) * t_1
// + t_0 * (V_20 * d_0 - a_1 * e_0) + (e_0 * t_0 - d_0) * a_2
// Inserting 4, solved for t_0: t_1 = - f_1 / (2 * f_2) + sqrt((f_1 / (2 * f_2))^2 + f_0 / f_2)
// = (a_1 * e_1 - V_20 * d_1 + V_10 * (d_1 - e_1 * t_0)) * (f_1^2 + 2 * f_0 * f_2 - f_1 * sqrt(f_1^2 + 4 * f_0 * f_2))
// + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1 - V_20 * d_1) - (d_1 - e_1 * t_0) * a_2 + V_10 * (d_0 - e_0 * t_0)) * (- f_1 * f_2 + f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// + t_0 * (V_20 * d_0 - a_1 * e_0) * 2 * f_2^2 + (e_0 * t_0 - d_0) * a_2 * 2 * f_2^2
// a_1 = V_00 * t_0 + k_0
// a_2 = V_00 * t_0 + k_1
// b_0 = k_2 * t_0
// b_1 = k_10 * t_0 + k_3
// c_0 = k_11 * t_0 + k_4
// d_0 = k_5 * t_0
// d_1 = k_12 * t_0 + k_6
// e_0 = k_13 * t_0 + k_7
// f_2 = (k_11 * t_0 + k_4) * k_9 + k_8 * (k_12 * t_0 + k_6)
// = (k_11 * k_9 + k_8 * k_12) * t_0 + k_4 * k_9 + k_8 * k_6
// = FH_0 * t_0 + FH_1
// f_1 = (k_11 * t_0 + k_4) * (k_13 * t_0 + k_7) + k_8 * k_5 * t_0 - k_2 * t_0 * k_9 - (k_10 * t_0 + k_3) * (k_12 * t_0 + k_6)
// = (k_11 * k_13 - k_10 * k_12) * t_0^2 + (k_11 * k_7 + k_4 * k_12 + k_8 * k_5 - k_2 * k_9 - k_10 * k_6 - k_3 * k_12) * t_0 + k_4 * k_7 - k_3 * k_6
// = FH_2 * t_0^2 + FH_3 * t_0 + FH_4
// f_0 = (k_10 * t_0 + k_3) * k_5 * t_0 + k_2 * t_0 * (k_13 * t_0 + k_7)
// = (k_10 * k_5 + k_2 * k_13) * t_0^2 + (k_3 * k_5 + k_2 * k_7) * t_0
// = FH_5 * t_0^2 + FH_6 * t_0
k[0] := AHailstone0.P0 - AHailstone2.P0;
k[1] := AHailstone0.P0 - AHailstone1.P0;
k[2] := AHailstone1.P1 - AHailstone2.P1;
k[3] := AHailstone0.P1 - AHailstone1.P1;
k[4] := AHailstone0.P1 - AHailstone2.P1;
k[5] := AHailstone2.P2 - AHailstone1.P2;
k[6] := AHailstone0.P2 - AHailstone2.P2;
k[7] := AHailstone0.P2 - AHailstone1.P2;
k[8] := AHailstone1.V1 - AHailstone2.V1;
k[9] := AHailstone2.V2 - AHailstone1.V2;
k[10] := AHailstone0.V1 - AHailstone2.V1;
k[11] := AHailstone0.V1 - AHailstone1.V1;
k[12] := AHailstone0.V2 - AHailstone1.V2;
k[13] := AHailstone0.V2 - AHailstone2.V2;
FH[0] := k[11] * k[9] + k[8] * k[12];
FH[1] := k[4] * k[9] + k[8] * k[6];
@ -213,10 +238,45 @@ begin
FH[5] := k[10] * k[5] + k[2] * k[13];
FH[6] := k[3] * k[5] + k[2] * k[7];
k[14] := V_00 * k[9] - V_20 * k[12];
k[15] := k[0] * k[9] - V_20 * k[6];
k[16] := V_00 * k[13];
k[17] := V_00 * k[7] + k[0] * k[13] - V_20 * k[5];
// Additional substitutions.
// a_1 * k_9 - V_20 * d_1
// = (V_00 * t_0 + k_0) * k_9 - V_20 * (k_12 * t_0 + k_6)
// = (V_00 * k_9 - V_20 * k_12) * t_0 + k_0 * k_9 - V_20 * k_6
// = k_14 * t_0 + k_15
// d_1 - k_9 * t_0
// = k_12 * t_0 + k_6 - k_9 * t_0
// = (k_12 - k_9) * t_0 + k_6
// a_1 * e_0 - V_20 * d_0
// = (V_00 * t_0 + k_0) * (k_13 * t_0 + k_7) - V_20 * k_5 * t_0
// = V_00 * k_13 * t_0^2 + (V_00 * k_7 + k_0 * k_13 - V_20 * k_5) * t_0 + k_0 * k_7
// = k_16 * t_0^2 + k_17 * t_0 + k_18
// d_0 - e_0 * t_0
// = k_5 * t_0 - k_13 * t_0^2 - k_7 * t_0
// = - k_13 * t_0^2 + k_19 * t_0
// f_1^2
// = (FH_2 * t_0^2 + FH_3 * t_0 + FH_4)^2
// = FH_2^2 * t_0^4 + FH_3^2 * t_0^2 + FH_4^2 + 2 * FH_2 * t_0^2 * FH_3 * t_0 + 2 * FH_2 * t_0^2 * FH_4 + 2 * FH_3 * t_0 * FH_4
// = FH_2^2 * t_0^4 + 2 * FH_2 * FH_3 * t_0^3 + (FH_3^2 + 2 * FH_2 * FH_4) * t_0^2 + 2 * FH_3 * FH_4 * t_0 + FH_4^2
// = FH_2^2 * t_0^4 + k_20 * t_0^3 + k_22 * t_0^2 + k_23 * t_0 + FH_4^2
// f_2^2
// = (FH_0 * t_0 + FH_1)^2
// = FH_0^2 * t_0^2 + 2 * FH_0 * FH_1 * t_0 + FH_1^2
// = FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2
// f_0 * f_2
// = (FH_5 * t_0^2 + FH_6 * t_0) * (FH_0 * t_0 + FH_1)
// = FH_5 * FH_0 * t_0^3 + (FH_5 * FH_1 + FH_6 * FH_0) * t_0^2 + FH_6 * FH_1 * t_0
// = k_126 * t_0^3 + k_127 * t_0^2 + k_128 * t_0
// f_1^2 + 4 * f_0 * f_2
// = FH_2^2 * t_0^4 + k_20 * t_0^3 + k_22 * t_0^2 + k_23 * t_0 + FH_4^2 + 4 * (k_126 * t_0^3 + k_127 * t_0^2 + k_128 * t_0)
// = k_31 * t_0^4 + k_132 * t_0^3 + k_133 * t_0^2 + k_134 * t_0 + k_58
// f_1^2 + 2 * f_0 * f_2
// = FH_2^2 * t_0^4 + k_20 * t_0^3 + k_22 * t_0^2 + k_23 * t_0 + FH_4^2 + 2 * (k_126 * t_0^3 + k_127 * t_0^2 + k_128 * t_0)
// = k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58
k[14] := AHailstone0.V0 * k[9] - AHailstone2.V0 * k[12];
k[15] := k[0] * k[9] - AHailstone2.V0 * k[6];
k[16] := AHailstone0.V0 * k[13];
k[17] := AHailstone0.V0 * k[7] + k[0] * k[13] - AHailstone2.V0 * k[5];
k[18] := k[0] * k[7];
k[19] := k[5] - k[7];
k[20] := 2 * FH[2] * FH[3];
@ -224,34 +284,19 @@ begin
k[22] := k[21] + 2 * FH[2] * FH[4];
k[23] := 2 * FH[3] * FH[4];
k[24] := 2 * FH[0] * FH[1];
k[25] := FH[0] * FH[0]; // KILL?
k[26] := FH[5] * k[25]; // KILL?
k[25] := FH[0] * FH[0];
k[126] := FH[5] * FH[0];
k[127] := FH[5] * FH[1] + FH[6] * FH[0];
k[128] := FH[6] * FH[1];
k[27] := FH[5] * k[24] + FH[6] * k[25]; // KILL?
k[28] := FH[1] * FH[1]; // KILL?
k[29] := FH[5] * k[28] + FH[6] * k[24]; // KILL?
k[30] := FH[6] * k[28]; // KILL?
k[28] := FH[1] * FH[1];
k[31] := FH[2] * FH[2];
k[132] := k[20] + 4 * k[126];
k[133] := k[22] + 4 * k[127];
k[134] := k[23] + 4 * k[128];
k[32] := k[31] + 4 * k[26]; // KILL?
k[33] := k[20] + 4 * k[27]; // KILL?
k[34] := k[22] + 4 * k[29]; // KILL?
k[35] := k[23] + 4 * k[30]; // KILL?
k[36] := k[31] + 2 * k[26]; // KILL?
k[37] := k[20] + 2 * k[27]; // KILL?
k[38] := k[22] + 2 * k[29]; // KILL?
k[39] := k[23] + 2 * k[30]; // KILL?
k[137] := k[20] + 2 * k[126];
k[138] := k[22] + 2 * k[127];
k[139] := k[23] + 2 * k[128];
k[40] := k[14] + V_10 * (k[12] - k[9]);
k[41] := k[15] + V_10 * k[6];
k[42] := k[16] - k[14] - V_10 * k[13] - (k[12] - k[9]) * V_00;
k[43] := k[17] - k[15] + V_10 * k[19] - (k[12] - k[9]) * k[1] - k[6] * V_00;
k[40] := k[14] + AHailstone1.V0 * (k[12] - k[9]);
k[41] := k[15] + AHailstone1.V0 * k[6];
k[42] := k[16] - k[14] - AHailstone1.V0 * k[13] - (k[12] - k[9]) * AHailstone0.V0;
k[43] := k[17] - k[15] + AHailstone1.V0 * k[19] - (k[12] - k[9]) * k[1] - k[6] * AHailstone0.V0;
k[44] := k[18] - k[6] * k[1];
k[45] := k[42] * FH[0] - k[40] * FH[2];
k[46] := k[42] * FH[1] + k[43] * FH[0] - k[41] * FH[2] - k[40] * FH[3];
@ -269,9 +314,9 @@ begin
k[58] := FH[4] * FH[4];
k[59] := k[40] * k[58] + k[41] * k[139] - k[57] * FH[0] - k[55] * FH[1];
k[60] := k[41] * k[58] - k[57] * FH[1];
k[61] := k[13] * V_00 - k[16];
k[61] := k[13] * AHailstone0.V0 - k[16];
k[62] := 2 * k[25] * k[61];
k[63] := k[13] * k[1] - k[19] * V_00 - k[17];
k[63] := k[13] * k[1] - k[19] * AHailstone0.V0 - k[17];
k[64] := 2 * (k[24] * k[61] + k[25] * k[63]);
k[65] := - k[19] * k[1] - k[18];
k[66] := 2 * (k[28] * k[61] + k[24] * k[63] + k[25] * k[65]);
@ -282,319 +327,167 @@ begin
k[71] := k[54] + k[66];
k[72] := k[56] + k[67];
k[73] := k[59] + k[68];
k[74] := k[45] * k[45];
k[75] := 2 * k[45] * k[46];
k[76] := k[46] * k[46] + 2 * k[45] * k[47];
k[77] := 2 * (k[45] * k[48] + k[46] * k[47]);
k[78] := k[47] * k[47] + 2 * k[46] * k[48];
k[79] := 2 * k[47] * k[48];
k[80] := k[48] * k[48];
// Unused, they are part of the polynomial inside the square root.
//k[132] := k[20] + 4 * k[126];
//k[133] := k[22] + 4 * k[127];
//k[134] := k[23] + 4 * k[128];
FA[0] := k[58] * k[80] - k[60] * k[60];
FA[1] := k[134] * k[80] + k[58] * k[79] - 2 * k[73] * k[60];
FA[2] := k[133] * k[80] + k[134] * k[79] + k[58] * k[78] - k[73] * k[73] - 2 * k[72] * k[60];
FA[3] := k[133] * k[79] + k[134] * k[78] + k[58] * k[77] + k[132] * k[80]
- 2 * (k[71] * k[60] + k[72] * k[73]);
FA[4] := k[31] * k[80] + k[133] * k[78] + k[134] * k[77] + k[58] * k[76] + k[132] * k[79] - k[72] * k[72]
- 2 * (k[70] * k[60] + k[71] * k[73]);
FA[5] := k[31] * k[79] + k[133] * k[77] + k[134] * k[76] + k[58] * k[75] + k[132] * k[78]
- 2 * (k[69] * k[60] + k[70] * k[73] + k[71] * k[72]);
FA[6] := k[31] * k[78] + k[133] * k[76] + k[134] * k[75] + k[58] * k[74] + k[132] * k[77] - k[71] * k[71]
- 2 * (k[69] * k[73] + k[70] * k[72]);
FA[7] := k[31] * k[77] + k[133] * k[75] + k[134] * k[74] + k[132] * k[76] - 2 * (k[69] * k[72] + k[70] * k[71]);
FA[8] := k[31] * k[76] + k[132] * k[75] + k[133] * k[74] - k[70] * k[70] - 2 * k[69] * k[71];
FA[9] := k[31] * k[75] + k[132] * k[74] - 2 * k[69] * k[70];
FA[10] := k[31] * k[74] - k[69] * k[69];
// Continuing calculations for equation 5.
// 0 = (k_14 * t_0 + k_15 + V_10 * ((k_12 - k_9) * t_0 + k_6)) * (k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58 -+ f_1 * sqrt(f_1^2 + 4 * f_0 * f_2))
// + (k_16 * t_0^2 + k_17 * t_0 + k_18 - t_0 * (k_14 * t_0 + k_15) - ((k_12 - k_9) * t_0 + k_6) * a_2 - V_10 * (k_13 * t_0^2 - k_19 * t_0)) * (- f_1 * f_2 +- f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// - 2 * t_0 * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2) + 2 * (k_13 * t_0^2 - k_19 * t_0) * a_2 * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2)
// 0 = (k_40 * t_0 + k_41) * (k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58 -+ f_1 * sqrt(f_1^2 + 4 * f_0 * f_2))
// + ((k_16 - k_14 - V_10 * k_13 - (k_12 - k_9) * V_00) * t_0^2 + (k_17 - k_15 + V_10 * k_19 - (k_12 - k_9) * k_1 - k_6 * V_00) * t_0 + k_18 - k_6 * k_1) * (- f_1 * f_2 +- f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// - 2 * t_0 * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2) + 2 * (k_13 * t_0^2 - k_19 * t_0) * a_2 * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2)
// 0 = (k_40 * t_0 + k_41) * (k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58)
// -+ (k_40 * t_0 + k_41) * f_1 * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_42 * t_0^2 + k_43 * t_0 + k_44) * (- f_1 * f_2 +- f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// - 2 * t_0 * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2) + 2 * (k_13 * t_0^2 - k_19 * t_0) * a_2 * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2)
// 0 = (k_40 * t_0 + k_41) * (k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58)
// -+ (k_40 * t_0 + k_41) * f_1 * sqrt(f_1^2 + 4 * f_0 * f_2)
// - (k_42 * t_0^2 + k_43 * t_0 + k_44) * f_1 * f_2
// +- (k_42 * t_0^2 + k_43 * t_0 + k_44) * f_2 * sqrt(f_1^2 + 4 * f_0 * f_2)
// - 2 * t_0 * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2) + 2 * (k_13 * t_0^2 - k_19 * t_0) * a_2 * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2)
// 0 = +- ((k_42 * t_0^2 + k_43 * t_0 + k_44) * f_2 - (k_40 * t_0 + k_41) * f_1) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_40 * t_0 + k_41) * (k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58)
// - (k_42 * t_0^2 + k_43 * t_0 + k_44) * f_1 * f_2
// - 2 * t_0 * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2) + 2 * (k_13 * t_0^2 - k_19 * t_0) * a_2 * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2)
// 0 = +- ((k_42 * t_0^2 + k_43 * t_0 + k_44) * (FH_0 * t_0 + FH_1) - (k_40 * t_0 + k_41) * (FH_2 * t_0^2 + FH_3 * t_0 + FH_4)) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_40 * t_0 + k_41) * (k_31 * t_0^4 + k_137 * t_0^3 + k_138 * t_0^2 + k_139 * t_0 + k_58)
// - (k_42 * t_0^2 + k_43 * t_0 + k_44) * (FH_2 * t_0^2 + FH_3 * t_0 + FH_4) * (FH_0 * t_0 + FH_1)
// - 2 * t_0 * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2) + 2 * (k_13 * t_0^2 - k_19 * t_0) * (V_00 * t_0 + k_1) * (FH_0^2 * t_0^2 + k_24 * t_0 + FH_1^2)
// 0 = +- (
// (k_42 * FH_0 - k_40 * FH_2) * t_0^3
// + (k_42 * FH_1 + k_43 * FH_0 - k_41 * FH_2 - k_40 * FH_3) * t_0^2
// + (k_43 * FH_1 + k_44 * FH_0 - k_41 * FH_3 - k_40 * FH_4) * t_0
// + k_44 * FH_1 - k_41 * FH_4
// ) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_40 * k_31 - k_42 * FH_2 * FH_0) * t_0^5
// + (k_40 * k_137 + k_41 * k_31 - k_42 * FH_3 * FH_0 - k_43 * FH_2 * FH_0 - k_42 * FH_2 * FH_1) * t_0^4
// + (k_40 * k_138 + k_41 * k_137 - k_42 * FH_4 * FH_0 - k_43 * FH_3 * FH_0 - k_44 * FH_2 * FH_0 - k_42 * FH_3 * FH_1 - k_43 * FH_2 * FH_1) * t_0^3
// + (k_40 * k_139 + k_41 * k_138 - k_43 * FH_4 * FH_0 - k_44 * FH_3 * FH_0 - k_42 * FH_4 * FH_1 - k_43 * FH_3 * FH_1 - k_44 * FH_2 * FH_1) * t_0^2
// + (k_40 * k_58 + k_41 * k_139 - k_44 * FH_4 * FH_0 - k_43 * FH_4 * FH_1 - k_44 * FH_3 * FH_1) * t_0
// + k_41 * k_58 - k_44 * FH_4 * FH_1
// + 2 * (k_13 * V_00 * FH_0^2 - k_16 * FH_0^2) * t_0^5
// + 2 * (k_13 * V_00 * k_24 + k_13 * k_1 * FH_0^2 - k_19 * V_00 * FH_0^2 - k_16 * k_24 - k_17 * FH_0^2) * t_0^4
// + 2 * (k_13 * V_00 * FH_1^2 + k_13 * k_1 * k_24 - k_19 * V_00 * k_24 - k_19 * k_1 * FH_0^2 - k_16 * FH_1^2 - k_17 * k_24 - k_18 * FH_0^2) * t_0^3
// + 2 * (k_13 * k_1 * FH_1^2 - k_19 * V_00 * FH_1^2 - k_19 * k_1 * k_24 - k_17 * FH_1^2 - k_18 * k_24) * t_0^2
// + 2 * (- k_19 * k_1 * FH_1^2 - k_18 * FH_1^2) * t_0
// 0 = +- (k_45 * t_0^3 + k_46 * t_0^2 + k_47 * t_0 + k_48) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_50 + k_62) * t_0^5 + (k_52 + k_64) * t_0^4 + (k_54 + k_66) * t_0^3 + (k_56 + k_67) * t_0^2 + (k_59 + k_68) * t_0 + k_60
// 0 = +- (k_45 * t_0^3 + k_46 * t_0^2 + k_47 * t_0 + k_48) * sqrt(k_31 * t_0^4 + k_132 * t_0^3 + k_133 * t_0^2 + k_134 * t_0 + k_58)
// + k_69 * t_0^5 + k_70 * t_0^4 + k_71 * t_0^3 + k_72 * t_0^2 + k_73 * t_0 + k_60
// Debug calculations
//a_1 := V_00 * t_0 + P_00 - P_20;
//a_2 := V_00 * t_0 + P_00 - P_10;
//b_0 := (P_11 - P_21) * t_0;
//b_1 := (V_01 - V_21) * t_0 + P_01 - P_11;
//c_0 := (V_01 - V_11) * t_0 + P_01 - P_21;
//c_1 := V_11 - V_21;
//d_0 := (P_22 - P_12) * t_0;
//d_1 := (V_02 - V_12) * t_0 + P_02 - P_22;
//e_0 := (V_02 - V_22) * t_0 + P_02 - P_12;
//e_1 := V_22 - V_12;
//f_2 := c_0 * e_1 + c_1 * d_1;
//f_1 := c_0 * e_0 + c_1 * d_0 - b_0 * e_1 - b_1 * d_1;
//f_0 := b_1 * d_0 + b_0 * e_0;
//
//act := f_2 * t_1 * t_1 + f_1 * t_1 - f_0;
//Write('debug10: ', 0 = act, ' ');
//
//if f_2 <> 0 then
//begin
// act := Round(- f_1 / (2 * f_2) + Sqrt((f_1 / (2 * f_2)) * (f_1 / (2 * f_2)) + f_0 / f_2));
// Write('debug15: ', t_1 = act);
// act := Round(- f_1 / (2 * f_2) - Sqrt((f_1 / (2 * f_2)) * (f_1 / (2 * f_2)) + f_0 / f_2));
// Write(' OR ', t_1 = act, ' ');
//end;
//
//act := (e_0 + e_1 * t_1) * t_2 - (d_0 + d_1 * t_1);
//Write('debug20: ', 0 = act, ' ');
//
//act := (a_1 * e_1 - V_20 * d_1 + V_10 * (d_1 - e_1 * t_0)) * t_1 * t_1
// + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1 - V_20 * d_1) - (d_1 - e_1 * t_0) * a_2 + V_10 * (d_0 - e_0 * t_0)) * t_1
// + t_0 * (V_20 * d_0 - a_1 * e_0) + (e_0 * t_0 - d_0) * a_2;
//Write('debug30: ', 0 = act, ' ');
//
//act := Round((a_1 * e_1 - V_20 * d_1 + V_10 * (d_1 - e_1 * t_0)) * (f_1 * f_1 + 2 * f_0 * f_2 - f_1 * Sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + (a_1 * e_0 - V_20 * d_0 - t_0 * (a_1 * e_1 - V_20 * d_1) - (d_1 - e_1 * t_0) * a_2 + V_10 * (d_0 - e_0 * t_0)) * (- f_1 * f_2 + f_2 * Sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + t_0 * (V_20 * d_0 - a_1 * e_0) * 2 * f_2 * f_2 + (e_0 * t_0 - d_0) * a_2 * 2 * f_2 * f_2);
//Write('debug40: ', 0 = act, ' ');
//
//Write('debug41: ',
// a_1 * k[9] - V_20 * d_1
// = k[14] * t_0 + k[15], ' ');
//Write('debug42: ',
// d_1 - k[9] * t_0
// = (k[12] - k[9]) * t_0 + k[6], ' ');
//Write('debug43: ',
// a_1 * e_0 - V_20 * d_0
// = k[16] * t_0 * t_0 + k[17] * t_0 + k[18], ' ');
//Write('debug44: ',
// d_0 - e_0 * t_0
// = - k[13] * t_0 * t_0 + k[19] * t_0, ' ');
//Write('debug45: ',
// f_1 * f_1
// = FH[2] * FH[2] * t_0 * t_0 * t_0 * t_0 + k[20] * t_0 * t_0 * t_0 + k[22] * t_0 * t_0 + k[23] * t_0 + FH[4] * FH[4], ' ');
//Write('debug46: ',
// f_2 * f_2
// = FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1], ' ');
//Write('debug47: ',
// f_0 * f_2
// = k[126] * t_0 * t_0 * t_0 + k[127] * t_0 * t_0 + k[128] * t_0, ' ');
//Write('debug48: ',
// f_1 * f_1 + 4 * f_0 * f_2
// = k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58], ' ');
//Write('debug49: ',
// f_1 * f_1 + 2 * f_0 * f_2
// = k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58], ' ');
//
//act := Round((k[14] * t_0 + k[15] + V_10 * ((k[12] - k[9]) * t_0 + k[6])) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58] - f_1 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + (k[16] * t_0 * t_0 + k[17] * t_0 + k[18] - t_0 * (k[14] * t_0 + k[15]) - ((k[12] - k[9]) * t_0 + k[6]) * a_2 - V_10 * (k[13] * t_0 * t_0 - k[19] * t_0)) * (- f_1 * f_2 + f_2 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * a_2 * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]));
//Write('debug50: ', 0 = act, ' ');
//
//Write('debug53: ',
// 0 = Round((k[40] * t_0 + k[41]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58] - f_1 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + ((k[16] - k[14] - V_10 * k[13] - (k[12] - k[9]) * V_00) * t_0 * t_0 + (k[17] - k[15] + V_10 * k[19] - (k[12] - k[9]) * k[1] - k[6] * V_00) * t_0 + k[18] - k[6] * k[1]) * (- f_1 * f_2 + f_2 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * a_2 * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1])),
// ' ');
//
//Write('debug55: ',
// 0 = Round((k[40] * t_0 + k[41]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58])
// - (k[40] * t_0 + k[41]) * f_1 * sqrt(f_1 * f_1 + 4 * f_0 * f_2)
// + (k[42] * t_0 * t_0 + k[43] * t_0 + k[44]) * (- f_1 * f_2 + f_2 * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * a_2 * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1])),
// ' ');
//
//Write('debug70: ',
// 0 = Round(((k[42] * t_0 * t_0 + k[43] * t_0 + k[44]) * (FH[0] * t_0 + FH[1]) - (k[40] * t_0 + k[41]) * (FH[2] * t_0 * t_0 + FH[3] * t_0 + FH[4])) * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + (k[40] * t_0 + k[41]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[137] * t_0 * t_0 * t_0 + k[138] * t_0 * t_0 + k[139] * t_0 + k[58])
// - (k[42] * t_0 * t_0 + k[43] * t_0 + k[44]) * (FH[2] * t_0 * t_0 + FH[3] * t_0 + FH[4]) * (FH[0] * t_0 + FH[1])
// - 2 * t_0 * (k[16] * t_0 * t_0 + k[17] * t_0 + k[18]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]) + 2 * (k[13] * t_0 * t_0 - k[19] * t_0) * (V_00 * t_0 + k[1]) * (FH[0] * FH[0] * t_0 * t_0 + k[24] * t_0 + FH[1] * FH[1]),
// ' ');
//
// Write('debug73: ',
// 0 = Round((
// (k[42] * FH[0] - k[40] * FH[2]) * t_0 * t_0 * t_0
// + (k[42] * FH[1] + k[43] * FH[0] - k[41] * FH[2] - k[40] * FH[3]) * t_0 * t_0
// + (k[43] * FH[1] + k[44] * FH[0] - k[41] * FH[3] - k[40] * FH[4]) * t_0
// + k[44] * FH[1] - k[41] * FH[4]
// ) * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + (k[40] * k[31] - k[42] * FH[2] * FH[0]) * t_0 * t_0 * t_0 * t_0 * t_0
// + (k[40] * k[137] + k[41] * k[31] - k[42] * FH[3] * FH[0] - k[43] * FH[2] * FH[0] - k[42] * FH[2] * FH[1]) * t_0 * t_0 * t_0 * t_0
// + (k[40] * k[138] + k[41] * k[137] - k[42] * FH[4] * FH[0] - k[43] * FH[3] * FH[0] - k[44] * FH[2] * FH[0] - k[42] * FH[3] * FH[1] - k[43] * FH[2] * FH[1]) * t_0 * t_0 * t_0
// + (k[40] * k[139] + k[41] * k[138] - k[43] * FH[4] * FH[0] - k[44] * FH[3] * FH[0] - k[42] * FH[4] * FH[1] - k[43] * FH[3] * FH[1] - k[44] * FH[2] * FH[1]) * t_0 * t_0
// + (k[40] * k[58] + k[41] * k[139] - k[44] * FH[4] * FH[0] - k[43] * FH[4] * FH[1] - k[44] * FH[3] * FH[1]) * t_0
// + k[41] * k[58] - k[44] * FH[4] * FH[1]
// + 2 * (k[13] * V_00 * FH[0] * FH[0] - k[16] * FH[0] * FH[0]) * t_0 * t_0 * t_0 * t_0 * t_0
// + 2 * (k[13] * V_00 * k[24] + k[13] * k[1] * FH[0] * FH[0] - k[19] * V_00 * FH[0] * FH[0] - k[16] * k[24] - k[17] * FH[0] * FH[0]) * t_0 * t_0 * t_0 * t_0
// + 2 * (k[13] * V_00 * FH[1] * FH[1] + k[13] * k[1] * k[24] - k[19] * V_00 * k[24] - k[19] * k[1] * FH[0] * FH[0] - k[16] * FH[1] * FH[1] - k[17] * k[24] - k[18] * FH[0] * FH[0]) * t_0 * t_0 * t_0
// + 2 * (k[13] * k[1] * FH[1] * FH[1] - k[19] * V_00 * FH[1] * FH[1] - k[19] * k[1] * k[24] - k[17] * FH[1] * FH[1] - k[18] * k[24]) * t_0 * t_0
// + 2 * (- k[19] * k[1] * FH[1] * FH[1] - k[18] * FH[1] * FH[1]) * t_0,
// ' ');
//
// Write('debug78: ',
// 0 = Round((k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * sqrt(f_1 * f_1 + 4 * f_0 * f_2))
// + (k[50] + k[62]) * t_0 * t_0 * t_0 * t_0 * t_0 + (k[52] + k[64]) * t_0 * t_0 * t_0 * t_0 + (k[54] + k[66]) * t_0 * t_0 * t_0 + (k[56] + k[67]) * t_0 * t_0 + (k[59] + k[68]) * t_0 + k[60],
// ' ');
//
// Write('debug80: ',
// 0 = Round((k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * sqrt(k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58])
// + k[69] * t_0 * t_0 * t_0 * t_0 * t_0 + k[70] * t_0 * t_0 * t_0 * t_0 + k[71] * t_0 * t_0 * t_0 + k[72] * t_0 * t_0 + k[73] * t_0 + k[60]),
// ' ');
// WriteLn;
// WriteLn(' 0 = ((', k[45], ') * x^3 + (', k[46], ') * x^2 + (', k[47], ') * x + (', k[48], ')) * sqrt((', k[31], ') * x^4 + (', k[132], ') * x^3 + (', k[133], ') * x^2 + (', k[134], ') * x + (', k[58], ')) + (',
// k[69], ') * x^5 + (', k[70], ') * x^4 + (', k[71], ') * x^3 + (', k[72], ') * x^2 + (', k[73], ') * x + (', k[60], ')');
OPolynomial0 := TBigIntPolynomial.Create([k[60], k[73], k[72], k[71], k[70], k[69]]);
OPolynomial1 := TBigIntPolynomial.Create([k[48], k[47], k[46], k[45]]);
Write('debug83: ',
(k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * (k[45] * t_0 * t_0 * t_0 + k[46] * t_0 * t_0 + k[47] * t_0 + k[48]) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58]) =
(k[69] * t_0 * t_0 * t_0 * t_0 * t_0 + k[70] * t_0 * t_0 * t_0 * t_0 + k[71] * t_0 * t_0 * t_0 + k[72] * t_0 * t_0 + k[73] * t_0 + k[60]) * (k[69] * t_0 * t_0 * t_0 * t_0 * t_0 + k[70] * t_0 * t_0 * t_0 * t_0 + k[71] * t_0 * t_0 * t_0 + k[72] * t_0 * t_0 + k[73] * t_0 + k[60]),
' ');
Write('debug85: ',
0 =
(
k[45] * k[45] * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
+ 2 * k[45] * k[46] * t_0 * t_0 * t_0 * t_0 * t_0
+ k[46] * k[46] * t_0 * t_0 * t_0 * t_0
+ 2 * k[45] * k[47] * t_0 * t_0 * t_0 * t_0
+ 2 * k[45] * k[48] * t_0 * t_0 * t_0
+ 2 * k[46] * k[47] * t_0 * t_0 * t_0
+ k[47] * k[47] * t_0 * t_0
+ 2 * k[46] * k[48] * t_0 * t_0
+ 2 * k[47] * k[48] * t_0
+ k[48] * k[48]
) * (k[31] * t_0 * t_0 * t_0 * t_0 + k[132] * t_0 * t_0 * t_0 + k[133] * t_0 * t_0 + k[134] * t_0 + k[58])
- k[69] * k[69] * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
- 2 * k[69] * k[70] * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
- (k[70] * k[70] + 2 * k[69] * k[71]) * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
- 2 * (k[69] * k[72] + k[70] * k[71]) * t_0 * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
- (k[71] * k[71] + 2 * k[69] * k[73] + 2 * k[70] * k[72]) * t_0 * t_0 * t_0 * t_0 * t_0 * t_0
- 2 * (k[69] * k[60] + k[70] * k[73] + k[71] * k[72]) * t_0 * t_0 * t_0 * t_0 * t_0
- (k[72] * k[72] + 2 * k[70] * k[60] + 2 * k[71] * k[73]) * t_0 * t_0 * t_0 * t_0
- 2 * (k[71] * k[60] + k[72] * k[73]) * t_0 * t_0 * t_0
- (k[73] * k[73] + 2 * k[72] * k[60]) * t_0 * t_0
- 2 * k[73] * k[60] * t_0
- k[60] * k[60],
' ');
// Squaring that formula eliminates the square root, but may lead to a polynomial with all coefficients zero in some
// cases. Therefore this part is merely included for the interested reader.
// -+ (k_45 * t_0^3 + k_46 * t_0^2 + k_47 * t_0 + k_48) * sqrt(k_31 * t_0^4 + k_132 * t_0^3 + k_133 * t_0^2 + k_134 * t_0 + k_58) =
// k_69 * t_0^5 + k_70 * t_0^4 + k_71 * t_0^3 + k_72 * t_0^2 + k_73 * t_0 + k_60
// (k_45 * t_0^3 + k_46 * t_0^2 + k_47 * t_0 + k_48)^2 * (k_31 * t_0^4 + k_132 * t_0^3 + k_133 * t_0^2 + k_134 * t_0 + k_58) =
// (k_69 * t_0^5 + k_70 * t_0^4 + k_71 * t_0^3 + k_72 * t_0^2 + k_73 * t_0 + k_60)^2
// 0 =
// (k_45^2 * t_0^6
// + 2 * k_45 * k_46 * t_0^5
// + k_46^2 * t_0^4 + 2 * k_45 * k_47 * t_0^4
// + 2 * k_45 * k_48 * t_0^3 + 2 * k_46 * k_47 * t_0^3
// + k_47^2 * t_0^2 + 2 * k_46 * k_48 * t_0^2
// + 2 * k_47 * k_48 * t_0
// + k_48^2
// ) * (k_31 * t_0^4 + k_132 * t_0^3 + k_133 * t_0^2 + k_134 * t_0 + k_58)
// - k_69^2 * t_0^10
// - 2 * k_69 * k_70 * t_0^9
// - (k_70^2 + 2 * k_69 * k_71) * t_0^8
// - 2 * (k_69 * k_72 + k_70 * k_71) * t_0^7
// - (k_71^2 + 2 * k_69 * k_73 + 2 * k_70 * k_72) * t_0^6
// - 2 * (k_69 * k_60 + k_70 * k_73 + k_71 * k_72) * t_0^5
// - (k_72^2 + 2 * k_70 * k_60 + 2 * k_71 * k_73) * t_0^4
// - 2 * (k_71 * k_60 + k_72 * k_73) * t_0^3
// - (k_73^2 + 2 * k_72 * k_60) * t_0^2
// - 2 * k_73 * k_60 * t_0
// - k_60^2
// 0 = ak_10 * t_0^10 + ak_9 * t_0^9 + ak_8 * t_0^8 + ak_7 * t_0^7 + ak_6 * t_0^6 + ak_5 * t_0^5 + ak_4 * t_0^4 + ak_3 * t_0^3 + ak_2 * t_0^2 + ak_1 * t_0 + ak_0
WriteLn('debug96: ', EvaluateAt(t_0) = 0);
NormalizeCoefficients;
WriteLn('debug99: ', EvaluateAt(t_0) = 0, ' ');
//k[74] := k[45] * k[45];
//k[75] := 2 * k[45] * k[46];
//k[76] := k[46] * k[46] + 2 * k[45] * k[47];
//k[77] := 2 * (k[45] * k[48] + k[46] * k[47]);
//k[78] := k[47] * k[47] + 2 * k[46] * k[48];
//k[79] := 2 * k[47] * k[48];
//k[80] := k[48] * k[48];
//ak[0] := k[58] * k[80] - k[60] * k[60];
//ak[1] := k[134] * k[80] + k[58] * k[79] - 2 * k[73] * k[60];
//ak[2] := k[133] * k[80] + k[134] * k[79] + k[58] * k[78] - k[73] * k[73] - 2 * k[72] * k[60];
//ak[3] := k[133] * k[79] + k[134] * k[78] + k[58] * k[77] + k[132] * k[80]
// - 2 * (k[71] * k[60] + k[72] * k[73]);
//ak[4] := k[31] * k[80] + k[133] * k[78] + k[134] * k[77] + k[58] * k[76] + k[132] * k[79] - k[72] * k[72]
// - 2 * (k[70] * k[60] + k[71] * k[73]);
//ak[5] := k[31] * k[79] + k[133] * k[77] + k[134] * k[76] + k[58] * k[75] + k[132] * k[78]
// - 2 * (k[69] * k[60] + k[70] * k[73] + k[71] * k[72]);
//ak[6] := k[31] * k[78] + k[133] * k[76] + k[134] * k[75] + k[58] * k[74] + k[132] * k[77] - k[71] * k[71]
// - 2 * (k[69] * k[73] + k[70] * k[72]);
//ak[7] := k[31] * k[77] + k[133] * k[75] + k[134] * k[74] + k[132] * k[76] - 2 * (k[69] * k[72] + k[70] * k[71]);
//ak[8] := k[31] * k[76] + k[132] * k[75] + k[133] * k[74] - k[70] * k[70] - 2 * k[69] * k[71];
//ak[9] := k[31] * k[75] + k[132] * k[74] - 2 * k[69] * k[70];
//ak[10] := k[31] * k[74] - k[69] * k[69];
end;
function TFirstCollisionPolynomial.EvaluateAt(const AT0: Int64): TBigInt;
function TNeverTellMeTheOdds.CalcRockThrowCollisionOptions(constref AHailstone0, AHailstone1, AHailstone2: THailstone):
TInt64Array;
var
i: Low(FA)..High(FA);
a0, a1: TBigIntPolynomial;
a0Roots, a1Roots: TBigIntArray;
options: specialize TList<Int64>;
i, j: TBigInt;
val: Int64;
begin
Result := TBigInt.Zero;
for i := High(FA) downto Low(FA) do
Result := Result * AT0 + FA[i];
CalcCollisionPolynomials(AHailstone0, AHailstone1, AHailstone2, a0, a1);
a0Roots := TPolynomialRoots.BisectInteger(a0, 64);
a1Roots := TPolynomialRoots.BisectInteger(a1, 64);
options := specialize TList<Int64>.Create;
for i in a0Roots do
for j in a1Roots do
if (i = j) and i.TryToInt64(val) then
options.Add(val);
Result := options.ToArray;
options.Free;
end;
function TFirstCollisionPolynomial.CalcPositiveIntegerRoot: Int64;
function TNeverTellMeTheOdds.ValidateRockThrow(constref AHailstone0, AHailstone1, AHailstone2: THailstone; const AT0,
AT1: Int64): Int64;
var
dividers: TDividers;
factors: TInt64Array;
divider: Int64;
divisor, t: Int64;
rock: THailstone;
begin
Result := 0;
//factors := TIntegerFactorization.PollardsRhoAlgorithm(FA[0]);
//dividers := TDividers.Create(factors);
//
//try
//for divider in dividers do
//begin
// //WriteLn('Check if ', divider, ' is a root...');
// if EvaluateAt(divider) = 0 then
// begin
// Result := divider;
// Break;
// end;
//end;
//
//finally
// dividers.Free;
//end;
end;
// V_x = (V_0 * t_0 - V_1 * t_1 + P_0 - P_1) / (t_0 - t_1)
divisor := AT0 - AT1;
rock := THailstone.Create;
rock.V0 := (AHailstone0.V0 * AT0 - AHailstone1.V0 * AT1 + AHailstone0.P0 - AHailstone1.P0) div divisor;
rock.V1 := (AHailstone0.V1 * AT0 - AHailstone1.V1 * AT1 + AHailstone0.P1 - AHailstone1.P1) div divisor;
rock.V2 := (AHailstone0.V2 * AT0 - AHailstone1.V2 * AT1 + AHailstone0.P2 - AHailstone1.P2) div divisor;
function TFirstCollisionPolynomial.CalcT1(const AT0: Int64): Int64;
var
g_0, g_1, g_2: Int64;
g: Extended;
begin
//g_2 := FH[0] * AT0 + FH[1];
//g_1 := FH[2] * AT0 * AT0 + FH[3] * AT0 + FH[4];
//g_0 := FH[5] * AT0 * AT0 + FH[6] * AT0;
//g := - g_1 / (2 * g_2);
//Result := Round(g + sqrt(g * g + g_0));
end;
// P_x = (V_0 - V_x) * t_0 + P_0
rock.P0 := (AHailstone0.V0 - rock.V0) * AT0 + AHailstone0.P0;
rock.P1 := (AHailstone0.V1 - rock.V1) * AT0 + AHailstone0.P1;
rock.P2 := (AHailstone0.V2 - rock.V2) * AT0 + AHailstone0.P2;
{ TNeverTellMeTheOdds }
Result := rock.P0 + rock.P1 + rock.P2;
function TNeverTellMeTheOdds.AreIntersecting(constref AHailstone1, AHailstone2: THailstone): Boolean;
var
m1, m2, x, y: Double;
begin
Result := False;
m1 := AHailstone1.Velocity.data[1] / AHailstone1.Velocity.data[0];
m2 := AHailstone2.Velocity.data[1] / AHailstone2.Velocity.data[0];
if m1 <> m2 then
begin
x := (AHailstone2.Position.data[1] - m2 * AHailstone2.Position.data[0]
- AHailstone1.Position.data[1] + m1 * AHailstone1.Position.data[0])
/ (m1 - m2);
if (FMin <= x) and (x <= FMax)
and (x * Sign(AHailstone1.Velocity.data[0]) >= AHailstone1.Position.data[0] * Sign(AHailstone1.Velocity.data[0]))
and (x * Sign(AHailstone2.Velocity.data[0]) >= AHailstone2.Position.data[0] * Sign(AHailstone2.Velocity.data[0]))
then
begin
y := m1 * (x - AHailstone1.Position.data[0]) + AHailstone1.Position.data[1];
if (FMin <= y) and (y <= FMax) then
Result := True
end;
// Checks collision with the third hailstone.
if ((AHailstone2.V0 = rock.V0) and (AHailstone2.P0 <> rock.P0))
or ((AHailstone2.V1 = rock.V1) and (AHailstone2.P1 <> rock.P1))
or ((AHailstone2.V2 = rock.V2) and (AHailstone2.P2 <> rock.P2)) then
Result := 0
else begin
t := (AHailstone2.P0 - rock.P0) div (rock.V0 - AHailstone2.V0);
if (t <> (AHailstone2.P1 - rock.P1) div (rock.V1 - AHailstone2.V1))
or (t <> (AHailstone2.P2 - rock.P2) div (rock.V2 - AHailstone2.V2)) then
Result := 0;
end;
end;
// For debug calculations:
Const
T : array[0..4] of Byte = (5, 3, 4, 6, 1);
procedure TNeverTellMeTheOdds.FindRockThrow(const AIndex1, AIndex2, AIndex3: Integer);
var
//i, j, k: Integer;
//x0, x1, x2: Extended;
f: TFirstCollisionPolynomial;
t0, t1: Int64;
p, v: Tvector3_extended;
test: TBigInt;
begin
WriteLn;
WriteLn(AIndex1, ' ', AIndex2, ' ', AIndex3);
f := TFirstCollisionPolynomial.Create;
f.Init(FHailstones[AIndex1], FHailstones[AIndex2], FHailstones[AIndex3], T[AIndex1], T[AIndex2], T[AIndex3]);
//t0 := f.CalcPositiveIntegerRoot;
//WriteLn('t0: ', t0, ' ', t0 = T[AIndex1]);
//t1 := f.CalcT1(t0);
//WriteLn(', t1: ', t1);
f.Free;
//// V_x = (V_0 * t_0 - V_1 * t_1 + P_0 - P_1) / (t_0 - t_1)
//v := (FHailstones[AIndex1].Velocity * t0 - FHailstones[AIndex2].Velocity * t1
// + FHailstones[AIndex1].Position - FHailstones[AIndex2].Position) / (t0 - t1);
//// P_x = (V_0 - V_x) * t_0 + P_0
//p := (FHailstones[AIndex1].Velocity - v) * t0 + FHailstones[AIndex1].Position;
//FPart2 := Round(p.data[0]) + Round(p.data[1]) + Round(p.data[2]);
//for i := 0 to FHailstones.Count - 3 do
// for j := i + 1 to FHailstones.Count - 2 do
// for k:= j + 1 to FHailstones.Count - 1 do
// begin
// WriteLn(i, j, k);
// solver := TRockThrowSolver.Create(FHailstones[i], FHailstones[j], FHailstones[k], 0);
// case i of
// 0: x0 := 5;
// 1: x0 := 3;
// 2: x0 := 4;
// end;
// f := solver.CalcValue(x0);
// solver.Free;
// end;
//for i := 80 to 120 do
//begin
// solver := TRockThrowSolver.Create(FHailstones[0], FHailstones[1], FHailstones[2], 0);
// x0 := i / 20;
// f := solver.CalcValue(x0);
// WriteLn(x0, ' ', f.Valid, ' ', f.Value);
// solver.Free;
//end;
rock.Free;
end;
constructor TNeverTellMeTheOdds.Create(const AMin: Int64; const AMax: Int64);
@ -617,19 +510,15 @@ end;
procedure TNeverTellMeTheOdds.Finish;
var
i, j, k: Integer;
i, j: Integer;
begin
for i := 0 to FHailstones.Count - 2 do
for j := i + 1 to FHailstones.Count - 1 do
if AreIntersecting(FHailstones[i], FHailstones[j]) then
Inc(FPart1);
for i := 0 to FHailstones.Count - 1 do
for j := 0 to FHailstones.Count - 1 do
for k := 0 to FHailstones.Count - 1 do
if (i <> j) and (i <> k) and (j <> k) then
FindRockThrow(i, j, k);
//FindRockThrow(0, 1, 2);
if FHailstones.Count >= 3 then
FPart2 := FindRockThrow(0, 1, 2);
end;
function TNeverTellMeTheOdds.GetDataFileName: string;

2
tests/UNeverTellMeTheOddsTestCases.pas
View File

@ -81,7 +81,7 @@ end;
procedure TNeverTellMeTheOddsFullDataTestCase.TestPart2;
begin
AssertEquals(-1, FSolver.GetResultPart2);
AssertEquals(856642398547748, FSolver.GetResultPart2);
end;
{ TNeverTellMeTheOddsExampleTestCase }