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Fixed day 24 helper variable indices

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This commit is contained in:
Stefan Müller 2024-05-27 02:52:22 +02:00
parent 3e3e1d45d3
commit b27b14a153
1 changed files with 174 additions and 176 deletions
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350
solvers/UNeverTellMeTheOdds.pas
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@ -46,8 +46,6 @@ type
private
FMin, FMax: Int64;
FHailstones: THailstones;
FA: array[0..10] of TBigInt;
FH: array[0..6] of TBigInt;
function AreIntersecting(constref AHailstone1, AHailstone2: THailstone): Boolean;
function FindRockThrow(const AIndex0, AIndex1, AIndex2: Integer): Int64;
procedure CalcCollisionPolynomials(constref AHailstone0, AHailstone1, AHailstone2: THailstone; out OPolynomial0,
@ -137,7 +135,7 @@ end;
procedure TNeverTellMeTheOdds.CalcCollisionPolynomials(constref AHailstone0, AHailstone1, AHailstone2: THailstone; out
OPolynomial0, OPolynomial1: TBigIntPolynomial);
var
k: array[0..139] of TBigInt;
k: array[0..74] of TBigInt;
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
@ -207,13 +205,13 @@ begin
// 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
// = k_14 * t_0 + k_15
// 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
// = k_16 * t_0^2 + k_17 * t_0 + k_18
// 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_19 * t_0^2 + k_20 * t_0
k[0] := AHailstone0.P0 - AHailstone2.P0;
k[1] := AHailstone0.P0 - AHailstone1.P0;
@ -230,207 +228,207 @@ begin
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];
FH[2] := k[11] * k[13] - k[10] * k[12];
FH[3] := k[11] * k[7] + k[4] * k[13] + k[8] * k[5] - k[2] * k[9] - k[10] * k[6] - k[3] * k[12];
FH[4] := k[4] * k[7] - k[3] * k[6];
FH[5] := k[10] * k[5] + k[2] * k[13];
FH[6] := k[3] * k[5] + k[2] * k[7];
k[14] := k[11] * k[9] + k[8] * k[12];
k[15] := k[4] * k[9] + k[8] * k[6];
k[16] := k[11] * k[13] - k[10] * k[12];
k[17] := k[11] * k[7] + k[4] * k[13] + k[8] * k[5] - k[2] * k[9] - k[10] * k[6] - k[3] * k[12];
k[18] := k[4] * k[7] - k[3] * k[6];
k[19] := k[10] * k[5] + k[2] * k[13];
k[20] := k[3] * k[5] + k[2] * k[7];
// 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
// = k_21 * t_0 + k_22
// 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
// = k_23 * t_0^2 + k_24 * t_0 + k_25
// 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
// = - k_13 * t_0^2 + k_26 * 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
// = (k_16 * t_0^2 + k_17 * t_0 + k_18)^2
// = k_16^2 * t_0^4 + k_17^2 * t_0^2 + k_18^2 + 2 * k_16 * t_0^2 * k_17 * t_0 + 2 * k_16 * t_0^2 * k_18 + 2 * k_17 * t_0 * k_18
// = k_16^2 * t_0^4 + 2 * k_16 * k_17 * t_0^3 + (k_17^2 + 2 * k_16 * k_18) * t_0^2 + 2 * k_17 * k_18 * t_0 + k_18^2
// = k_16^2 * t_0^4 + k_27 * t_0^3 + k_29 * t_0^2 + k_30 * t_0 + k_18^2
// 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
// = (k_14 * t_0 + k_15)^2
// = k_14^2 * t_0^2 + 2 * k_14 * k_15 * t_0 + k_15^2
// = k_14^2 * t_0^2 + k_31 * t_0 + k_15^2
// 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
// = (k_19 * t_0^2 + k_20 * t_0) * (k_14 * t_0 + k_15)
// = k_19 * k_14 * t_0^3 + (k_19 * k_15 + k_20 * k_14) * t_0^2 + k_20 * k_15 * t_0
// = k_33 * t_0^3 + k_34 * t_0^2 + k_35 * 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
// = k_16^2 * t_0^4 + k_27 * t_0^3 + k_29 * t_0^2 + k_30 * t_0 + k_18^2 + 4 * (k_33 * t_0^3 + k_34 * t_0^2 + k_35 * t_0)
// = k_37 * t_0^4 + k_75 * t_0^3 + k_76 * t_0^2 + k_77 * t_0 + k_59
// 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_16^2 * t_0^4 + k_27 * t_0^3 + k_29 * t_0^2 + k_30 * t_0 + k_18^2 + 2 * (k_33 * t_0^3 + k_34 * t_0^2 + k_35 * t_0)
// = k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59
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];
k[21] := FH[3] * FH[3];
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];
k[126] := FH[5] * FH[0];
k[127] := FH[5] * FH[1] + FH[6] * FH[0];
k[128] := FH[6] * FH[1];
k[28] := FH[1] * FH[1];
k[31] := FH[2] * FH[2];
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] + 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];
k[47] := k[43] * FH[1] + k[44] * FH[0] - k[41] * FH[3] - k[40] * FH[4];
k[48] := k[44] * FH[1] - k[41] * FH[4];
k[49] := k[42] * FH[2];
k[50] := k[40] * k[31] - k[49] * FH[0];
k[51] := k[42] * FH[3] + k[43] * FH[2];
k[52] := k[40] * k[137] + k[41] * k[31] - k[51] * FH[0] - k[49] * FH[1];
k[53] := k[42] * FH[4] + k[43] * FH[3] + k[44] * FH[2];
k[54] := k[40] * k[138] + k[41] * k[137] - k[53] * FH[0] - k[51] * FH[1];
k[55] := k[43] * FH[4] + k[44] * FH[3];
k[56] := k[40] * k[139] + k[41] * k[138] - k[55] * FH[0] - k[53] * FH[1];
k[57] := k[44] * FH[4];
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] * AHailstone0.V0 - k[16];
k[62] := 2 * k[25] * k[61];
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]);
k[67] := 2 * (k[28] * k[63] + k[24] * k[65]);
k[68] := 2 * k[28] * k[65];
k[69] := k[50] + k[62];
k[70] := k[52] + k[64];
k[71] := k[54] + k[66];
k[72] := k[56] + k[67];
k[73] := k[59] + k[68];
k[21] := AHailstone0.V0 * k[9] - AHailstone2.V0 * k[12];
k[22] := k[0] * k[9] - AHailstone2.V0 * k[6];
k[23] := AHailstone0.V0 * k[13];
k[24] := AHailstone0.V0 * k[7] + k[0] * k[13] - AHailstone2.V0 * k[5];
k[25] := k[0] * k[7];
k[26] := k[5] - k[7];
k[27] := 2 * k[16] * k[17];
k[28] := k[17] * k[17];
k[29] := k[28] + 2 * k[16] * k[18];
k[30] := 2 * k[17] * k[18];
k[31] := 2 * k[14] * k[15];
k[32] := k[14] * k[14];
k[33] := k[19] * k[14];
k[34] := k[19] * k[15] + k[20] * k[14];
k[35] := k[20] * k[15];
k[36] := k[15] * k[15];
k[37] := k[16] * k[16];
k[38] := k[27] + 2 * k[33];
k[39] := k[29] + 2 * k[34];
k[40] := k[30] + 2 * k[35];
k[41] := k[21] + AHailstone1.V0 * (k[12] - k[9]);
k[42] := k[22] + AHailstone1.V0 * k[6];
k[43] := k[23] - k[21] - AHailstone1.V0 * k[13] - (k[12] - k[9]) * AHailstone0.V0;
k[44] := k[24] - k[22] + AHailstone1.V0 * k[26] - (k[12] - k[9]) * k[1] - k[6] * AHailstone0.V0;
k[45] := k[25] - k[6] * k[1];
k[46] := k[43] * k[14] - k[41] * k[16];
k[47] := k[43] * k[15] + k[44] * k[14] - k[42] * k[16] - k[41] * k[17];
k[48] := k[44] * k[15] + k[45] * k[14] - k[42] * k[17] - k[41] * k[18];
k[49] := k[45] * k[15] - k[42] * k[18];
k[50] := k[43] * k[16];
k[51] := k[41] * k[37] - k[50] * k[14];
k[52] := k[43] * k[17] + k[44] * k[16];
k[53] := k[41] * k[38] + k[42] * k[37] - k[52] * k[14] - k[50] * k[15];
k[54] := k[43] * k[18] + k[44] * k[17] + k[45] * k[16];
k[55] := k[41] * k[39] + k[42] * k[38] - k[54] * k[14] - k[52] * k[15];
k[56] := k[44] * k[18] + k[45] * k[17];
k[57] := k[41] * k[40] + k[42] * k[39] - k[56] * k[14] - k[54] * k[15];
k[58] := k[45] * k[18];
k[59] := k[18] * k[18];
k[60] := k[41] * k[59] + k[42] * k[40] - k[58] * k[14] - k[56] * k[15];
k[61] := k[42] * k[59] - k[58] * k[15];
k[62] := k[13] * AHailstone0.V0 - k[23];
k[63] := 2 * k[32] * k[62];
k[64] := k[13] * k[1] - k[26] * AHailstone0.V0 - k[24];
k[65] := 2 * (k[31] * k[62] + k[32] * k[64]);
k[66] := - k[26] * k[1] - k[25];
k[67] := 2 * (k[36] * k[62] + k[31] * k[64] + k[32] * k[66]);
k[68] := 2 * (k[36] * k[64] + k[31] * k[66]);
k[69] := 2 * k[36] * k[66];
k[70] := k[51] + k[63];
k[71] := k[53] + k[65];
k[72] := k[55] + k[67];
k[73] := k[57] + k[68];
k[74] := k[60] + k[69];
// 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];
//k[75] := k[27] + 4 * k[33];
//k[76] := k[29] + 4 * k[34];
//k[77] := k[30] + 4 * k[35];
// 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_21 * t_0 + k_22 + V_10 * ((k_12 - k_9) * t_0 + k_6)) * (k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59 -+ f_1 * sqrt(f_1^2 + 4 * f_0 * f_2))
// + (k_23 * t_0^2 + k_24 * t_0 + k_25 - t_0 * (k_21 * t_0 + k_22) - ((k_12 - k_9) * t_0 + k_6) * a_2 - V_10 * (k_13 * t_0^2 - k_26 * t_0)) * (- f_1 * f_2 +- f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// - 2 * t_0 * (k_23 * t_0^2 + k_24 * t_0 + k_25) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2) + 2 * (k_13 * t_0^2 - k_26 * t_0) * a_2 * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2)
// 0 = (k_41 * t_0 + k_42) * (k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59 -+ f_1 * sqrt(f_1^2 + 4 * f_0 * f_2))
// + ((k_23 - k_21 - V_10 * k_13 - (k_12 - k_9) * V_00) * t_0^2 + (k_24 - k_22 + V_10 * k_26 - (k_12 - k_9) * k_1 - k_6 * V_00) * t_0 + k_25 - k_6 * k_1) * (- f_1 * f_2 +- f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// - 2 * t_0 * (k_23 * t_0^2 + k_24 * t_0 + k_25) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2) + 2 * (k_13 * t_0^2 - k_26 * t_0) * a_2 * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2)
// 0 = (k_41 * t_0 + k_42) * (k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59)
// -+ (k_41 * t_0 + k_42) * f_1 * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_43 * t_0^2 + k_44 * t_0 + k_45) * (- f_1 * f_2 +- f_2 * sqrt(f_1^2 + 4 * f_0 * f_2))
// - 2 * t_0 * (k_23 * t_0^2 + k_24 * t_0 + k_25) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2) + 2 * (k_13 * t_0^2 - k_26 * t_0) * a_2 * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2)
// 0 = (k_41 * t_0 + k_42) * (k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59)
// -+ (k_41 * t_0 + k_42) * f_1 * sqrt(f_1^2 + 4 * f_0 * f_2)
// - (k_43 * t_0^2 + k_44 * t_0 + k_45) * f_1 * f_2
// +- (k_43 * t_0^2 + k_44 * t_0 + k_45) * f_2 * sqrt(f_1^2 + 4 * f_0 * f_2)
// - 2 * t_0 * (k_23 * t_0^2 + k_24 * t_0 + k_25) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2) + 2 * (k_13 * t_0^2 - k_26 * t_0) * a_2 * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2)
// 0 = +- ((k_43 * t_0^2 + k_44 * t_0 + k_45) * f_2 - (k_41 * t_0 + k_42) * f_1) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_41 * t_0 + k_42) * (k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59)
// - (k_43 * t_0^2 + k_44 * t_0 + k_45) * f_1 * f_2
// - 2 * t_0 * (k_23 * t_0^2 + k_24 * t_0 + k_25) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2) + 2 * (k_13 * t_0^2 - k_26 * t_0) * a_2 * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2)
// 0 = +- ((k_43 * t_0^2 + k_44 * t_0 + k_45) * (k_14 * t_0 + k_15) - (k_41 * t_0 + k_42) * (k_16 * t_0^2 + k_17 * t_0 + k_18)) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_41 * t_0 + k_42) * (k_37 * t_0^4 + k_38 * t_0^3 + k_39 * t_0^2 + k_40 * t_0 + k_59)
// - (k_43 * t_0^2 + k_44 * t_0 + k_45) * (k_16 * t_0^2 + k_17 * t_0 + k_18) * (k_14 * t_0 + k_15)
// - 2 * t_0 * (k_23 * t_0^2 + k_24 * t_0 + k_25) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2) + 2 * (k_13 * t_0^2 - k_26 * t_0) * (V_00 * t_0 + k_1) * (k_14^2 * t_0^2 + k_31 * t_0 + k_15^2)
// 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
// (k_43 * k_14 - k_41 * k_16) * t_0^3
// + (k_43 * k_15 + k_44 * k_14 - k_42 * k_16 - k_41 * k_17) * t_0^2
// + (k_44 * k_15 + k_45 * k_14 - k_42 * k_17 - k_41 * k_18) * t_0
// + k_45 * k_15 - k_42 * k_18
// ) * 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
// + (k_41 * k_37 - k_43 * k_16 * k_14) * t_0^5
// + (k_41 * k_38 + k_42 * k_37 - k_43 * k_17 * k_14 - k_44 * k_16 * k_14 - k_43 * k_16 * k_15) * t_0^4
// + (k_41 * k_39 + k_42 * k_38 - k_43 * k_18 * k_14 - k_44 * k_17 * k_14 - k_45 * k_16 * k_14 - k_43 * k_17 * k_15 - k_44 * k_16 * k_15) * t_0^3
// + (k_41 * k_40 + k_42 * k_39 - k_44 * k_18 * k_14 - k_45 * k_17 * k_14 - k_43 * k_18 * k_15 - k_44 * k_17 * k_15 - k_45 * k_16 * k_15) * t_0^2
// + (k_41 * k_59 + k_42 * k_40 - k_45 * k_18 * k_14 - k_44 * k_18 * k_15 - k_45 * k_17 * k_15) * t_0
// + k_42 * k_59 - k_45 * k_18 * k_15
// + 2 * (k_13 * V_00 * k_14^2 - k_23 * k_14^2) * t_0^5
// + 2 * (k_13 * V_00 * k_31 + k_13 * k_1 * k_14^2 - k_26 * V_00 * k_14^2 - k_23 * k_31 - k_24 * k_14^2) * t_0^4
// + 2 * (k_13 * V_00 * k_15^2 + k_13 * k_1 * k_31 - k_26 * V_00 * k_31 - k_26 * k_1 * k_14^2 - k_23 * k_15^2 - k_24 * k_31 - k_25 * k_14^2) * t_0^3
// + 2 * (k_13 * k_1 * k_15^2 - k_26 * V_00 * k_15^2 - k_26 * k_1 * k_31 - k_24 * k_15^2 - k_25 * k_31) * t_0^2
// + 2 * (- k_26 * k_1 * k_15^2 - k_25 * k_15^2) * t_0
// 0 = +- (k_46 * t_0^3 + k_47 * t_0^2 + k_48 * t_0 + k_49) * sqrt(f_1^2 + 4 * f_0 * f_2)
// + (k_51 + k_63) * t_0^5 + (k_53 + k_65) * t_0^4 + (k_55 + k_67) * t_0^3 + (k_57 + k_68) * t_0^2 + (k_60 + k_69) * t_0 + k_61
// 0 = +- (k_46 * t_0^3 + k_47 * t_0^2 + k_48 * t_0 + k_49) * sqrt(k_37 * t_0^4 + k_75 * t_0^3 + k_76 * t_0^2 + k_77 * t_0 + k_59)
// + k_70 * t_0^5 + k_71 * t_0^4 + k_72 * t_0^3 + k_73 * t_0^2 + k_74 * t_0 + k_61
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]]);
OPolynomial0 := TBigIntPolynomial.Create([k[61], k[74], k[73], k[72], k[71], k[70]]);
OPolynomial1 := TBigIntPolynomial.Create([k[49], k[48], k[47], k[46]]);
// 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
// -+ (k_46 * t_0^3 + k_47 * t_0^2 + k_48 * t_0 + k_49) * sqrt(k_37 * t_0^4 + k_75 * t_0^3 + k_76 * t_0^2 + k_77 * t_0 + k_59) =
// k_70 * t_0^5 + k_71 * t_0^4 + k_72 * t_0^3 + k_73 * t_0^2 + k_74 * t_0 + k_61
// (k_46 * t_0^3 + k_47 * t_0^2 + k_48 * t_0 + k_49)^2 * (k_37 * t_0^4 + k_75 * t_0^3 + k_76 * t_0^2 + k_77 * t_0 + k_59) =
// (k_70 * t_0^5 + k_71 * t_0^4 + k_72 * t_0^3 + k_73 * t_0^2 + k_74 * t_0 + k_61)^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
// (k_46^2 * t_0^6
// + 2 * k_46 * k_47 * t_0^5
// + k_47^2 * t_0^4 + 2 * k_46 * k_48 * t_0^4
// + 2 * k_46 * k_49 * t_0^3 + 2 * k_47 * k_48 * t_0^3
// + k_48^2 * t_0^2 + 2 * k_47 * k_49 * t_0^2
// + 2 * k_48 * k_49 * t_0
// + k_49^2
// ) * (k_37 * t_0^4 + k_75 * t_0^3 + k_76 * t_0^2 + k_77 * t_0 + k_59)
// - k_70^2 * t_0^10
// - 2 * k_70 * k_71 * t_0^9
// - (k_71^2 + 2 * k_70 * k_72) * t_0^8
// - 2 * (k_70 * k_73 + k_71 * k_72) * t_0^7
// - (k_72^2 + 2 * k_70 * k_74 + 2 * k_71 * k_73) * t_0^6
// - 2 * (k_70 * k_61 + k_71 * k_74 + k_72 * k_73) * t_0^5
// - (k_73^2 + 2 * k_71 * k_61 + 2 * k_72 * k_74) * t_0^4
// - 2 * (k_72 * k_61 + k_73 * k_74) * t_0^3
// - (k_74^2 + 2 * k_73 * k_61) * t_0^2
// - 2 * k_74 * k_61 * t_0
// - k_61^2
// 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
//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];
//k[78] := k[46] * k[46];
//k[79] := 2 * k[46] * k[47];
//k[80] := k[47] * k[47] + 2 * k[46] * k[48];
//k[81] := 2 * (k[46] * k[49] + k[47] * k[48]);
//k[82] := k[48] * k[48] + 2 * k[47] * k[49];
//k[83] := 2 * k[48] * k[49];
//k[84] := k[49] * k[49];
//ak[0] := k[59] * k[84] - k[61] * k[61];
//ak[1] := k[77] * k[84] + k[59] * k[83] - 2 * k[74] * k[61];
//ak[2] := k[76] * k[84] + k[77] * k[83] + k[59] * k[82] - k[74] * k[74] - 2 * k[73] * k[61];
//ak[3] := k[76] * k[83] + k[77] * k[82] + k[59] * k[81] + k[75] * k[84]
// - 2 * (k[72] * k[61] + k[73] * k[74]);
//ak[4] := k[37] * k[84] + k[76] * k[82] + k[77] * k[81] + k[59] * k[80] + k[75] * k[83] - k[73] * k[73]
// - 2 * (k[71] * k[61] + k[72] * k[74]);
//ak[5] := k[37] * k[83] + k[76] * k[81] + k[77] * k[80] + k[59] * k[79] + k[75] * k[82]
// - 2 * (k[70] * k[61] + k[71] * k[74] + k[72] * k[73]);
//ak[6] := k[37] * k[82] + k[76] * k[80] + k[77] * k[79] + k[59] * k[78] + k[75] * k[81] - k[72] * k[72]
// - 2 * (k[70] * k[74] + k[71] * k[73]);
//ak[7] := k[37] * k[81] + k[76] * k[79] + k[77] * k[78] + k[75] * k[80] - 2 * (k[70] * k[73] + k[71] * k[72]);
//ak[8] := k[37] * k[80] + k[75] * k[79] + k[76] * k[78] - k[71] * k[71] - 2 * k[70] * k[72];
//ak[9] := k[37] * k[79] + k[75] * k[78] - 2 * k[70] * k[71];
//ak[10] := k[37] * k[78] - k[70] * k[70];
end;
function TNeverTellMeTheOdds.CalcRockThrowCollisionOptions(constref AHailstone0, AHailstone1, AHailstone2: THailstone):