**G:**Reduction Based Telescoping with Lazy Hermite Reduction (LHR) as described in the paper. In the*i*th iteration, LHR is applied to the remainder of the previous iteration*after*polynomial reduction.**H:**Reduction Based Telescoping with Lazy Hermite Reduction (LHR) as described in the paper. In the*i*th iteration, LHR is applied to the output of the LHR call of the previous iteration, i.e., the form of the remainder*before*polynomial reduction was applied to it.**I:**Reduction Based Telescoping with Hermite Reduction applied to an integral basis that is normal at infinity. The reported computation time includes the time for computing the integral bases, which is typically a dominant portion. In the*i*th iteration, LHR is applied to the output of the LHR call of the previous iteration, i.e., the form of the remainder*before*polynomial reduction was applied to it.

in | out | A | A' | B | C | D | E | F | G | H | I | ord | deg | bytecount | meaning |

in-1.m | out-1.m | 0.71 | 7.45 | 1.41 | 16.11 | 6.56 | 5.88 | 4.98 | 6.66 | 4.47 | 392.61 | 3 | 11 | 3824 | aztec diamond edge gfun, [x^0 y^0] |

in-2.m | out-2.m | 5.90 | 11.03 | 7.89 | 51.75 | 11.06 | 10.90 | 9.30 | 10.93 | 6.36 | 957.27 | 4 | 13 | 4576 | aztec diamond edge gfun, [x^1 y^0] |

in-3.m | out-3.m | 1.16 | 34.96 | 7.30 | 61.89 | 12.21 | 12.48 | 11.42 | 22.64 | 15.54 | 990.96 | 4 | 15 | 5632 | aztec diamond edge gfun, [x^0 y^1] |

in-4.m | out-4.m | 5.96 | 11.87 | 7.82 | 52.97 | 11.38 | 11.15 | 9.09 | 12.47 | 6.79 | 652.16 | 4 | 13 | 4576 | aztec diamond edge gfun, [x^(-1) y^0] |

in-5.m | out-5.m | 0.73 | 6.32 | 1.66 | 21.57 | 6.19 | 5.63 | 4.79 | 12.58 | 6.44 | 457.73 | 3 | 10 | 3600 | aztec diamond edge gfun, [x^0 y^(-1)] |

in-6.m | out-6.m | 6.04 | 11.47 | 11.32 | 77.44 | 11.54 | 11.28 | 9.27 | 16.52 | 9.70 | 365.82 | 4 | 14 | 6104 | aztec diamond edge gfun, [x^(-1) y^(-1)] |

in-7.m | out-7.m | 6.04 | 32.20 | 11.24 | 52.98 | 11.90 | 11.59 | 11.49 | 29.96 | 18.55 | 583.24 | 4 | 14 | 5432 | aztec diamond edge gfun, [x^(-1) y^1] |

in-8.m | out-8.m | 5.98 | 11.06 | 11.54 | 60.53 | 11.11 | 11.04 | 9.39 | 14.96 | 8.37 | 596.67 | 4 | 14 | 6104 | aztec diamond edge gfun, [x^1 y^(-1)] |

in-9.m | out-9.m | 5.94 | 31.59 | 11.50 | 47.14 | 12.09 | 12.54 | 11.71 | 23.98 | 20.67 | 1309.75 | 4 | 14 | 5432 | aztec diamond edge gfun, [x^1 y^1] |

in-10.m | out-10.m | 34.02 | 495.75 | 23.72 | 465.36 | 69.87 | 87.31 | 60.35 | 63.51 | 87.62 | 8227.50 | 7 | 11 | 10368 | cores of a planar graph, [x^(2n)y^n] |

in-11.m | out-11.m | 2.89 | 147.87 | 2.33 | 32.67 | 1.63 | 1.68 | 1.76 | 1.31 | 1.22 | 2.57 | 4 | 6 | 4144 | cores of a planar graph, [x^n y^(2n)] |

in-12.m | out-12.m | 18.82 | 409.63 | 20.45 | 199.60 | 54.72 | 63.09 | 44.66 | 182.94 | 225.07 | 8441.65 | 6 | 9 | 7056 | cores of a planar graph, [x^(2n)y^(2n)] |

in-13.m | out-13.m | 314.34 | 2079.81 | 565.10 | 6022.78 | 6098.31 | 630.98 | 287.44 | 325.08 | 411.58 | >30h | 11 | 17 | 20928 | cores of a planar graph, [x^(3n)y^n] |

in-14.m | out-14.m | 1.82 | 55.74 | 1.57 | 16.63 | 17.88 | 1.57 | 1.84 | 0.94 | 0.94 | 2.32 | 3 | 4 | 2192 | cores of a planar graph, [x^n y^(3n)] |

in-15.m | out-15.m | 1385.74 | 5198.98 | 2434.02 | 44390.07 | 848.92 | 2491.66 | 1332.63 | 589.46 | 759.66 | >30h | 13 | 21 | 32392 | cores of a planar graph, [x^(4n)y^n] |

in-16.m | out-16.m | 252.61 | 2052.27 | 454.48 | 3623.02 | 337.07 | 414.81 | 298.00 | 774.64 | 1465.19 | >30h | 10 | 16 | 19768 | cores of a planar graph, [x^(2n) y^(3n)] |

in-17.m | out-17.m | 13.47 | 194.59 | 9.54 | 98.62 | 29.84 | 33.49 | 25.17 | 102.89 | 79.84 | 6620.72 | 5 | 8 | 5904 | cores of a planar graph, [x^(3n) y^(2n)] |

in-18.m | out-18.m | 3.12 | 1751.68 | 2.38 | 56.55 | 1.10 | 1.17 | 1.69 | 2.00 | 1.99 | 3.00 | 4 | 6 | 4144 | cores of a planar graph, [x^n y^(4n)] |

in-19.m | out-19.m | >30h | 2.17 | 4431.06 | 399.11 | 21.43 | 22.10 | 23.40 | 56.27 | 55.93 | 155.31 | 3 | 6 | 4216 | artificial example from the paper |

in-20.m | out-20.m | 19.79 | 19.98 | 9.50 | 104.79 | 6.93 | 8.77 | 6.85 | 11.42 | 8.71 | 1841.74 | 4 | 9 | 6144 | random groves, [x^n y^n] |

in-21.m | out-21.m | 10712.13 | 125.60 | 421.60 | 5062.14 | 37.60 | 52.26 | 33.18 | 45.41 | 43.41 | 2739.10 | 7 | 20 | 19480 | random groves, [x^(2n) y^n] |

in-22.m | out-22.m | 117.96 | 11099.43 | 87.98 | 6060.53 | 37.11 | 54.08 | 33.48 | 45.87 | 43.24 | 2756.84 | 7 | 20 | 19480 | random groves, [x^n y^(2n)] |

in-23.m | out-23.m | >30h | 529.80 | 10946.79 | >30h | 166.86 | 403.41 | 173.14 | 170.91 | 202.49 | 7004.39 | 10 | 32 | 51904 | random groves, [x^(3n) y^n] |

in-24.m | out-24.m | 16773.93 | 14784.08 | 765.81 | 15679.77 | 759.26 | 758.02 | 1997.19 | 78485.99 | >30h | >30h | 7 | 27 | 31440 | random groves, [x^(2n) y^(2n)] |

in-25.m | out-25.m | 873.93 | >33h | 1125.10 | >30h | 168.78 | 244.90 | 158.93 | 167.70 | 193.23 | 6996.08 | 10 | 32 | 51904 | random groves, [x^n y^(3n)] |

in-26.m | out-26.m | 126.55 | 125.61 | 275.13 | 2655.64 | 2666.71 | 280.68 | 127.92 | >30h | >30h | >30h | 6 | 23 | 13224 | knight walks ending at (0,0) |

in-27.m | out-27.m | 4294.85 | 2227.09 | 31336.71 | >30h | 2178.55 | 3116.89 | 4378.15 | >30h | >30h | >30h | 12 | 76 | 89360 | knight walks ending at (1,0) |

in-28.m | out-28.m | 2164.31 | 4283.18 | 21465.62 | >30h | 4264.47 | 5166.89 | 65365.06 | >30h | >30h | >30h | 12 | 76 | 89752 | knight walks ending at (0,1) |

in-29.m | out-29.m | 4271.47 | 2303.72 | 37185.64 | >30h | 2287.83 | 3273.19 | 2376.63 | >30h | >30h | >30h | 12 | 76 | 90272 | knight walks ending at (-1,0) |

in-30.m | out-30.m | 2441.47 | 4247.27 | 43244.79 | >30h | 5386.17 | 6190.89 | 64508.09 | >30h | >30h | >30h | 12 | 76 | 89752 | knight walks ending at (0,-1) |

in-31.m | out-31.m | 8333.90 | 7494.64 | 39162.78 | >30h | 7419.98 | 7915.04 | 72994.38 | >30h | >30h | >30h | 12 | 74 | 87664 | knight walks ending at (-1,-1) |

in-32.m | out-32.m | 8392.92 | 7465.50 | 36957.55 | >30h | 7276.75 | 7938.35 | 74181.14 | >30h | >30h | >30h | 12 | 74 | 87664 | knight walks ending at (-1,1) |

in-33.m | out-33.m | 7754.31 | 8209.88 | 34464.58 | >30h | 8528.18 | 7808.53 | 72216.13 | >30h | >30h | >30h | 12 | 74 | 86952 | knight walks ending at (1,-1) |

in-34.m | out-34.m | 7436.03 | 10732.05 | 34075.48 | >30h | 10663.64 | 7756.55 | 71008.12 | >30h | >30h | >30h | 12 | 74 | 86952 | knight walks ending at (1,1) |

in-35.m | out-35.m | 9.75 | 9.86 | 12.99 | 28.70 | 9.71 | 9.85 | 7.47 | 34.83 | 16.11 | 992.58 | 4 | 9 | 3592 | quantum random walks ending at (0,0) |

in-36.m | out-36.m | 5.53 | 5.81 | 3.74 | 11.70 | 5.55 | 5.46 | 3.99 | 29.22 | 13.70 | 1852.53 | 3 | 9 | 3600 | quantum random walks ending at (1,0) |

in-37.m | out-37.m | 5.85 | 5.62 | 3.88 | 10.57 | 5.41 | 5.36 | 4.01 | 23.17 | 17.56 | 795.27 | 3 | 9 | 3520 | quantum random walks ending at (0,1) |

in-38.m | out-38.m | 6.89 | 6.22 | 6.93 | 19.09 | 5.77 | 5.35 | 3.89 | 23.13 | 11.19 | 834.99 | 3 | 11 | 4200 | quantum random walks ending at (-1,0) |

in-39.m | out-39.m | 6.00 | 6.62 | 4.92 | 13.98 | 6.61 | 6.33 | 5.44 | 70.45 | 45.13 | 2220.32 | 3 | 11 | 4056 | quantum random walks ending at (0,-1) |

in-40.m | out-40.m | 11.41 | 11.16 | 13.95 | 53.75 | 11.06 | 11.25 | 9.25 | 86.82 | 55.38 | 1995.38 | 4 | 13 | 5104 | quantum random walks ending at (-1,-1) |

in-41.m | out-41.m | 6.64 | 5.98 | 7.18 | 12.90 | 5.67 | 5.22 | 3.82 | 31.08 | 21.17 | 873.72 | 3 | 12 | 4432 | quantum random walks ending at (-1,1) |

in-42.m | out-42.m | 5.97 | 6.33 | 6.91 | 13.03 | 6.16 | 6.35 | 5.41 | 70.97 | 46.69 | 2069.46 | 3 | 12 | 4576 | quantum random walks ending at (1,-1) |

in-43.m | out-43.m | 10.26 | 10.37 | 9.80 | 35.48 | 10.05 | 10.56 | 7.52 | 33.66 | 23.73 | 1348.67 | 4 | 9 | 3592 | quantum random walks ending at (1,1) |

in-44.m | out-44.m | 3.48 | 2.61 | 2.59 | 24.55 | 0.36 | 0.40 | 0.59 | 0.63 | 0.60 | 470.72 | 3 | 5 | 3072 | diagonal of 3d-rooks |

in-45.m | out-45.m | 27.89 | 30.74 | 65.50 | 206.58 | 1.51 | 1.59 | 2.50 | 2.43 | 2.40 | 1538.17 | 4 | 21 | 15688 | diagonal of 3d-rooks (variation) |

in-46.m | out-46.m | 4838.00 | 4986.07 | 8019.37 | >30h | 543.69 | 580.05 | 479.52 | >30h | >30h | >30h | 6 | 45 | 49352 | diagonal of 3d-rooks (variation) |

in-47.m | out-47.m | 3.05 | 5.72 | 1.69 | 29.06 | 0.55 | 0.61 | 0.89 | 1.42 | 1.43 | 1120.87 | 3 | 10 | 3424 | diagonal of 3d-rooks (variation) |

in-48.m | out-48.m | 44.63 | 62.06 | 104.79 | 498.45 | 2.02 | 2.12 | 3.38 | 5.64 | 6.22 | 1888.39 | 4 | 30 | 22552 | diagonal of 3d-rooks (variation) |

in-49.m | out-49.m | 12109.41 | 12318.27 | 39640.75 | >30h | 1252.31 | 1452.92 | 1186.72 | >30h | >30h | >30h | 6 | 71 | 92552 | diagonal of 3d-queens |

in-50.m | out-50.m | 2.87 | 3.48 | 1.59 | 19.93 | 0.42 | 0.46 | 0.67 | 0.86 | 0.75 | 554.30 | 3 | 5 | 3072 | diagonal of 3d-rooks |

in-51.m | out-51.m | 31.38 | 28.82 | 42.98 | 182.97 | 1.40 | 1.53 | 2.39 | 2.39 | 2.72 | 2007.43 | 4 | 21 | 15688 | diagonal of 3d-rooks (variation) |

in-52.m | out-52.m | 5226.95 | 4912.46 | 8242.97 | >30h | 536.11 | 602.07 | 479.23 | >30h | >30h | >30h | 6 | 45 | 49352 | diagonal of 3d-rooks (variation) |

in-53.m | out-53.m | 5.76 | 3.03 | 6.69 | 24.42 | 0.39 | 0.44 | 0.64 | 0.85 | 0.89 | 551.16 | 3 | 10 | 3424 | diagonal of 3d-rooks (variation) |

in-54.m | out-54.m | 60.18 | 44.90 | 211.31 | 420.31 | 1.82 | 1.91 | 3.02 | 4.32 | 4.47 | 2233.15 | 4 | 30 | 22552 | diagonal of 3d-rooks (variation) |

in-55.m | out-55.m | 11797.54 | 11598.18 | 38827.70 | >30h | 1260.61 | 1453.87 | 1201.81 | >30h | >30h | >30h | 6 | 71 | 92552 | diagonal of 3d-queens |

in-56.m | out-56.m | 2.80 | 2.77 | 1.60 | 22.96 | 22.69 | 1.55 | 2.58 | 0.72 | 0.73 | 489.17 | 3 | 5 | 2952 | diagonal of 3d-rooks |

in-57.m | out-57.m | 31.48 | 32.27 | 42.25 | 198.90 | 201.68 | 42.73 | 31.40 | 2.56 | 2.42 | 1532.73 | 4 | 21 | 15352 | diagonal of 3d-rooks (variation) |

in-58.m | out-58.m | 5195.78 | 5082.99 | 8270.46 | >30h | 571.67 | 8262.68 | 5086.82 | >30h | >30h | >30h | 6 | 45 | 49048 | diagonal of 3d-rooks (variation) |

in-59.m | out-59.m | 6.47 | 5.91 | 7.53 | 28.93 | 0.54 | 0.71 | 0.88 | 1.40 | 1.43 | 1077.43 | 3 | 10 | 3424 | diagonal of 3d-rooks (variation) |

in-60.m | out-60.m | 65.59 | 61.88 | 208.60 | 1270.58 | 2.06 | 2.14 | 3.70 | 5.90 | 6.06 | 1904.77 | 4 | 30 | 22552 | diagonal of 3d-rooks (variation) |

in-61.m | out-61.m | 11805.01 | 11975.58 | 39649.67 | >30h | 1268.54 | 1416.92 | 1203.11 | >30h | >30h | >30h | 6 | 71 | 92552 | diagonal of 3d-queens |

in-62.m | out-62.m | 95.64 | 8.89 | 42.94 | 400.03 | 2.47 | 2.52 | 94.70 | 2.59 | 3.28 | 772.53 | 5 | 14 | 10128 | 2-1-1 diagonal of 3d-rooks |

in-63.m | out-63.m | >30h | 477.99 | 48682.20 | 86435.48 | 99.49 | 97.23 | 151.46 | 57.61 | 57.71 | 3486.60 | 8 | 60 | 87760 | 2-1-1 diagonal of 3d-rooks (variation) |

in-65.m | out-65.m | 709.68 | 3.91 | 30.30 | 57.40 | 0.54 | 0.58 | 0.87 | 1.59 | 1.24 | 1010.77 | 3 | 4 | 2712 | 2-1-1 diagonal of 3d-rooks (variation) |

in-66.m | out-66.m | >30h | 1166.03 | >30h | >30h | 183.22 | 183.68 | 242.61 | 111.48 | 75.88 | 6271.12 | 9 | 93 | 167008 | 2-1-1 diagonal of 3d-rooks (variation) |

in-68.m | out-68.m | 225.47 | 271.87 | 515.36 | 1517.94 | 259.35 | 204.20 | 1704.81 | 57583.62 | 61792.83 | >30h | 4 | 40 | 24936 | randomly generated input |

in-69.m | out-69.m | 216.96 | 243.07 | 466.36 | 1374.09 | 249.16 | 202.67 | 989.00 | 51819.31 | 54077.99 | >30h | 4 | 40 | 24360 | randomly generated input |

in-70.m | out-70.m | 221.35 | 45.30 | 436.84 | 1232.80 | 43.09 | 38.34 | 26.76 | 1133.48 | 200.49 | 6024.10 | 4 | 40 | 24144 | randomly generated input |

in-71.m | out-71.m | 37.21 | 238.57 | 447.21 | 3060.84 | 48.05 | 32.82 | 27.33 | 933.03 | 205.61 | 5548.89 | 4 | 40 | 24632 | randomly generated input |

in-72.m | out-72.m | 227.78 | 257.81 | 494.57 | 1357.72 | 263.46 | 210.24 | 1012.61 | 57982.13 | 64668.30 | >30h | 4 | 40 | 24752 | randomly generated input |

in-73.m | out-73.m | 36.77 | 231.64 | 351.79 | 2540.32 | 44.68 | 49.84 | 26.82 | 1487.15 | 225.44 | 5850.57 | 4 | 40 | 24120 | randomly generated input |

in-74.m | out-74.m | 225.23 | 242.18 | 438.31 | 1319.19 | 235.85 | 191.66 | 1185.38 | 51913.45 | 54944.86 | >30h | 4 | 40 | 24104 | randomly generated input |

in-75.m | out-75.m | 2.01 | 1.78 | 1.60 | 6.98 | 1.68 | 1.90 | 1.35 | 3.71 | 2.12 | 37.42 | 3 | 4 | 2776 | restricted 2D lattice walks with small steps |

in-76.m | out-76.m | 1.75 | 1.73 | 1.60 | 11.84 | 1.72 | 2.19 | 1.30 | 7.89 | 2.51 | 101.73 | 3 | 5 | 3392 | restricted 2D lattice walks with small steps |

in-77.m | out-77.m | 2.06 | 2.05 | 2.05 | 12.81 | 1.95 | 2.71 | 1.66 | 9.11 | 5.46 | 118.09 | 3 | 8 | 4680 | restricted 2D lattice walks with small steps |

in-78.m | out-78.m | 2.72 | 2.34 | 1.79 | 12.43 | 2.32 | 3.31 | 1.83 | 12.41 | 5.28 | 223.65 | 3 | 6 | 4000 | restricted 2D lattice walks with small steps |

in-79.m | out-79.m | 38.02 | 31.55 | 18.15 | 330.90 | 13.45 | 21.02 | 11.49 | 6.43 | 6.29 | 383.01 | 5 | 16 | 13368 | restricted 2D lattice walks with small steps |

in-80.m | out-80.m | 64.94 | 66.18 | 46.34 | 505.83 | 25.49 | 37.43 | 20.48 | 12.91 | 12.04 | 553.63 | 5 | 20 | 17680 | restricted 2D lattice walks with small steps |

in-81.m | out-81.m | 42.14 | 32.90 | 19.45 | 334.40 | 14.15 | 21.84 | 11.90 | 7.25 | 6.11 | 386.41 | 5 | 15 | 13120 | restricted 2D lattice walks with small steps |

in-82.m | out-82.m | 69.64 | 72.50 | 50.74 | 474.22 | 25.45 | 37.29 | 20.01 | 11.88 | 10.99 | 708.43 | 5 | 18 | 13688 | restricted 2D lattice walks with small steps |

in-83.m | out-83.m | 99.65 | 68.21 | 64.35 | 751.13 | 27.78 | 39.20 | 22.80 | 26.39 | 23.50 | 399.55 | 5 | 24 | 21328 | restricted 2D lattice walks with small steps |

in-84.m | out-84.m | 114.51 | 81.38 | 99.00 | 866.63 | 26.04 | 35.92 | 21.49 | 43.66 | 27.27 | 1132.38 | 5 | 24 | 21328 | restricted 2D lattice walks with small steps |

in-85.m | out-85.m | 36.02 | 23.29 | 35.90 | 192.23 | 6.38 | 6.97 | 4.94 | 5.65 | 5.34 | 190.02 | 5 | 15 | 12936 | restricted 2D lattice walks with small steps |

in-86.m | out-86.m | 57.41 | 45.51 | 26.76 | 250.46 | 18.21 | 19.23 | 13.63 | 11.99 | 10.68 | 322.24 | 5 | 18 | 15600 | restricted 2D lattice walks with small steps |

in-87.m | out-87.m | 109.70 | 65.87 | 71.00 | 775.62 | 29.81 | 42.23 | 24.04 | 21.39 | 15.43 | 381.87 | 5 | 24 | 21328 | restricted 2D lattice walks with small steps |

in-88.m | out-88.m | 136.23 | 79.23 | 101.06 | 863.12 | 27.99 | 39.03 | 23.40 | 27.83 | 17.88 | 625.52 | 5 | 24 | 21328 | restricted 2D lattice walks with small steps |

in-89.m | out-89.m | 30.51 | 22.46 | 30.68 | 162.53 | 6.03 | 6.76 | 4.85 | 5.51 | 5.31 | 198.36 | 5 | 16 | 13848 | restricted 2D lattice walks with small steps |

in-90.m | out-90.m | 51.72 | 43.52 | 25.63 | 228.03 | 17.87 | 19.34 | 13.61 | 11.66 | 10.22 | 347.50 | 5 | 19 | 16512 | restricted 2D lattice walks with small steps |

in-91.m | out-91.m | 396.82 | 363.07 | 11.36 | 66.08 | 6.29 | 6.85 | 3.91 | 5.19 | 5.03 | 768.18 | 3 | 4 | 2928 | restricted 2D lattice walks with small steps |

in-92.m | out-92.m | 0.21 | 0.27 | 0.46 | 1.68 | 0.13 | 0.21 | 0.38 | 0.14 | 0.18 | 47.81 | 2 | 3 | 2160 | diagonal of Szego's ratfun |

in-93.m | out-93.m | 8.99 | 0.81 | 9.03 | 40.33 | 0.83 | 0.85 | 1.06 | 1.02 | 1.08 | 241.80 | 4 | 10 | 7496 | 2-1-1 diagonal of Szego's ratfun |

in-94.m | out-94.m | 0.76 | 11.96 | 2.97 | 33.38 | 0.91 | 0.73 | 0.91 | 1.50 | 1.60 | 248.73 | 4 | 10 | 7496 | 1-2-1 diagonal of Szego's ratfun |

in-95.m | out-95.m | 200.42 | 7.82 | 103.93 | 510.25 | 9.78 | 7.54 | 10.41 | 4.57 | 4.31 | 401.69 | 6 | 15 | 16032 | 3-1-1 diagonal of Szego's ratfun |

in-96.m | out-96.m | 12.37 | 13.81 | 16.10 | 81.62 | 17.77 | 14.17 | 51.66 | 1627.46 | 1869.53 | >30h | 4 | 11 | 9464 | 2-2-1 diagonal of Szego's ratfun |

in-97.m | out-97.m | 6.01 | 206.22 | 17.49 | 579.80 | 10.30 | 10.10 | 10.62 | 4.78 | 4.78 | 419.63 | 6 | 15 | 16032 | 1-3-1 diagonal of Szego's ratfun |