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    "tree": "balanced_8",
    "Lambda1": 24.0,
    "Lambda2": 112.0,
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        "mu": 0.25,
        "tau": 1.5,
        "predicted": 61.0,
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        "observed": 112.23482185085675,
        "relative_error": 4.613461460484568e-05
      },
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        "mu": 1.0,
        "tau": 0.5,
        "predicted": 118.0,
        "observed": 117.99042463755437,
        "relative_error": 8.114713936975196e-05
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    "max_relative_error": 0.0065925974467937465
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  "roundoff_scaling": {
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        "u": 0.03125,
        "observed_mse": 0.015840501337322743,
        "predicted_mse": 0.015900059237790126,
        "ratio_obs_over_u2": 16.22067336941849,
        "relative_error": 0.003745765948206658
      },
      {
        "u": 0.0078125,
        "observed_mse": 0.0009827836305885073,
        "predicted_mse": 0.0009937537023618828,
        "ratio_obs_over_u2": 16.101927003562103,
        "relative_error": 0.011039024807960639
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        "u": 0.001953125,
        "observed_mse": 6.148304672196384e-05,
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      {
        "u": 0.00048828125,
        "observed_mse": 3.8823653113995034e-06,
        "predicted_mse": 3.881850399851105e-06,
        "ratio_obs_over_u2": 16.283820355064183,
        "relative_error": 0.00013264590217550311
      }
    ],
    "loglog_slope": 1.999089838367662,
    "predicted_slope": 2.0
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  "variance_topology_sweep": [
    {
      "ratio": 1,
      "balanced_cost": 24.0,
      "huffman_cost": 24.0,
      "predicted_gain_fraction": 0.0,
      "huffman_depth_high_variance": 3
    },
    {
      "ratio": 2,
      "balanced_cost": 27.0,
      "huffman_cost": 27.0,
      "predicted_gain_fraction": 0.0,
      "huffman_depth_high_variance": 3
    },
    {
      "ratio": 4,
      "balanced_cost": 33.0,
      "huffman_cost": 31.0,
      "predicted_gain_fraction": 0.06060606060606055,
      "huffman_depth_high_variance": 2
    },
    {
      "ratio": 8,
      "balanced_cost": 45.0,
      "huffman_cost": 35.0,
      "predicted_gain_fraction": 0.2222222222222222,
      "huffman_depth_high_variance": 1
    },
    {
      "ratio": 16,
      "balanced_cost": 69.0,
      "huffman_cost": 43.0,
      "predicted_gain_fraction": 0.37681159420289856,
      "huffman_depth_high_variance": 1
    },
    {
      "ratio": 32,
      "balanced_cost": 117.0,
      "huffman_cost": 59.0,
      "predicted_gain_fraction": 0.49572649572649574,
      "huffman_depth_high_variance": 1
    },
    {
      "ratio": 64,
      "balanced_cost": 213.0,
      "huffman_cost": 91.0,
      "predicted_gain_fraction": 0.5727699530516432,
      "huffman_depth_high_variance": 1
    }
  ],
  "heterogeneous_monte_carlo": [
    {
      "tree": "balanced",
      "predicted_cost": 69.0,
      "observed_cost": 68.61639584778041,
      "relative_error": 0.005559480466950578,
      "depths": {
        "0": 3,
        "1": 3,
        "2": 3,
        "3": 3,
        "4": 3,
        "5": 3,
        "6": 3,
        "7": 3
      }
    },
    {
      "tree": "huffman",
      "predicted_cost": 43.0,
      "observed_cost": 42.8624064748514,
      "relative_error": 0.0031998494220605274,
      "depths": {
        "7": 3,
        "1": 4,
        "2": 4,
        "3": 4,
        "4": 4,
        "5": 4,
        "6": 4,
        "0": 1
      }
    }
  ],
  "notes": [
    "All costs omit common nu*u^2 factors. Huffman minimizes sum sigma_i^2 depth_i among binary trees.",
    "The roundoff simulation uses the stated conditional Gaussian local-noise approximation."
  ]
}