## Scatter Plot: Lean-SMT vs. Ethos Checking Times ### Overview The image is a scatter plot comparing the checking times of Lean-SMT and Ethos. Both axes use a logarithmic scale. The plot includes a red dashed line representing y = x, which serves as a reference for comparing the performance of the two systems. The data points are clustered, showing the relationship between the checking times of the two methods. ### Components/Axes * **Title:** Lean-SMT vs. Ethos Checking Times * **X-axis:** Lean-SMT Checking Time (ms) - Logarithmic scale from 10^2 to 10^5 * **Y-axis:** Ethos Checking Time (ms) - Logarithmic scale from 10^2 to 10^5 * **Grid:** Light gray dashed grid lines are present. * **Legend:** Located in the top-left corner. * Red dashed line: y = x ### Detailed Analysis * **Data Points:** The data points are blue circles. * **y = x Line:** A red dashed line represents y = x. This line indicates where the checking times for Lean-SMT and Ethos are equal. * **Data Point Distribution:** * Most data points are clustered below the y = x line, indicating that Ethos generally has lower checking times than Lean-SMT. * There is a concentration of points in the lower-left corner, suggesting many instances where both systems have relatively low checking times. * There are some outliers above the y = x line, indicating cases where Ethos checking times are higher than Lean-SMT. * **X-Axis Markers:** 10^2, 10^3, 10^4, 10^5 * **Y-Axis Markers:** 10^2, 10^3, 10^4, 10^5 ### Key Observations * Ethos generally has lower checking times than Lean-SMT, as most data points are below the y = x line. * There is a wide range of checking times for both systems, spanning several orders of magnitude. * The clustering of points suggests a correlation between the checking times of the two systems. ### Interpretation The scatter plot provides a visual comparison of the checking times between Lean-SMT and Ethos. The fact that most data points lie below the y = x line suggests that Ethos is generally faster than Lean-SMT for the given set of checks. However, the presence of outliers above the line indicates that there are cases where Lean-SMT outperforms Ethos. The logarithmic scale highlights the wide range of checking times, and the clustering of points suggests a relationship between the performance of the two systems. The plot could be used to further investigate the specific conditions under which each system performs better.

\n ## Scatter Plot: Lean-SMT vs. Ethos Checking Times ### Overview This image presents a scatter plot comparing Lean-SMT checking times to Ethos checking times, both measured in milliseconds (ms). A dashed red line representing y=x is overlaid on the plot, serving as a reference for equal checking times. The plot displays a large number of data points, suggesting a substantial dataset. ### Components/Axes * **Title:** Lean-SMT vs. Ethos Checking Times * **X-axis:** Lean-SMT Checking Time (ms) - Logarithmic scale. Markers are approximately: 10^2, 10^3, 10^4, 10^5. * **Y-axis:** Ethos Checking Time (ms) - Logarithmic scale. Markers are approximately: 10^2, 10^3, 10^4, 10^5. * **Legend:** * `y = x` - Dashed red line. * **Data Points:** A large number of blue circular data points. ### Detailed Analysis The scatter plot shows a generally positive correlation between Lean-SMT checking time and Ethos checking time. However, the relationship is not strictly linear. * **Trend:** For Lean-SMT checking times below approximately 10^3 ms, the Ethos checking times are generally lower than the Lean-SMT checking times. As Lean-SMT checking time increases beyond 10^3 ms, the Ethos checking times tend to be higher, and the spread of data points increases significantly. * **Data Point Distribution:** * **Low Lean-SMT Times (10^2 ms):** Ethos checking times range from approximately 10^2 to 10^3 ms. * **Mid-Range Lean-SMT Times (10^3 ms):** Ethos checking times are widely distributed, ranging from approximately 10^2 to 10^4 ms. * **High Lean-SMT Times (10^4 - 10^5 ms):** Ethos checking times are also widely distributed, ranging from approximately 10^2 to 10^5 ms. * **y = x Line:** The dashed red line (y=x) represents the point where Lean-SMT and Ethos checking times are equal. Most data points fall *above* this line, indicating that Ethos checking times are generally longer than Lean-SMT checking times. * **Outliers:** There are a few data points with very high Ethos checking times (approaching 10^5 ms) even for relatively low Lean-SMT checking times. These represent potential outliers or cases where Ethos checking is significantly slower. ### Key Observations * Ethos checking times are generally longer than Lean-SMT checking times. * The relationship between the two checking times is not strictly linear, and the spread of Ethos checking times increases with increasing Lean-SMT checking times. * There are outliers where Ethos checking times are significantly longer than Lean-SMT checking times. * The logarithmic scale on both axes compresses the data, making it difficult to discern precise values without further analysis. ### Interpretation The data suggests that while there is a correlation between Lean-SMT and Ethos checking times, Ethos checking is often slower. The increasing spread of data points at higher Lean-SMT checking times indicates that the performance difference between the two systems becomes more variable as the complexity of the checking task increases. The outliers suggest that there are specific cases where Ethos checking is particularly slow, potentially due to specific features or complexities in the input. The y=x line serves as a benchmark. The fact that most points lie above it implies that Ethos is generally less efficient than Lean-SMT. The deviation from this line increases with the Lean-SMT checking time, suggesting that the performance gap widens for more complex tasks. Further investigation is needed to understand the reasons for the outliers and the increasing variability in Ethos checking times. This could involve analyzing the specific inputs that lead to these slower checking times and identifying potential optimizations for the Ethos system. The use of a logarithmic scale suggests that the data spans several orders of magnitude, and the relationship between the two checking times may be better understood by examining the data in more detail.

## Scatter Plot: Lean-SMT vs. Ethos Checking Times ### Overview The image is a log-log scatter plot comparing **Lean-SMT Checking Time** (x-axis) and **Ethos Checking Time** (y-axis), both measured in milliseconds (ms). A red dashed line labeled `y = x` serves as a reference for equality between the two metrics. Data points are represented as blue circles, with density varying across the plot. --- ### Components/Axes - **Title**: "Lean-SMT vs. Ethos Checking Times" (centered at the top). - **X-axis**: - Label: "Lean-SMT Checking Time (ms)". - Scale: Logarithmic, ranging from `10²` (100 ms) to `10⁵` (100,000 ms). - Ticks: `10²`, `10³`, `10⁴`, `10⁵`. - **Y-axis**: - Label: "Ethos Checking Time (ms)". - Scale: Logarithmic, ranging from `10²` (100 ms) to `10⁵` (100,000 ms). - Ticks: `10²`, `10³`, `10⁴`, `10⁵`. - **Legend**: - Position: Top-left corner. - Content: Red dashed line labeled `y = x`. - **Data Points**: - Color: Blue circles (varying opacity, denser in clusters). - Distribution: Concentrated in the lower-left (both times <10³ ms) and upper-right (both times >10³ ms) quadrants. --- ### Detailed Analysis 1. **Trend Verification**: - The red dashed line `y = x` diagonally bisects the plot, indicating where Lean-SMT and Ethos times are equal. - **Majority of data points** lie **below** the line, suggesting Ethos Checking Time is generally faster than Lean-SMT. - **Exceptions**: A sparse cluster of points above the line in the middle range (`10³`–`10⁴` ms) indicates Ethos is slower in these cases. 2. **Content Details**: - **Lower-left quadrant** (`x < 10³`, `y < 10³`): High density of points, showing both systems perform efficiently. - **Upper-right quadrant** (`x > 10³`, `y > 10³`): Moderate density, indicating slower performance for both systems. - **Outliers**: - A few points above `y = x` in the `10³`–`10⁴` ms range (e.g., `x ≈ 5×10³`, `y ≈ 1×10⁴`). - A single outlier at `x ≈ 10⁵`, `y ≈ 3×10⁴` (Ethos significantly faster than Lean-SMT). 3. **Key Observations**: - **Correlation**: A weak positive correlation exists; as Lean-SMT time increases, Ethos time tends to increase, but with significant variance. - **Performance Gap**: Ethos consistently outperforms Lean-SMT in most cases, except for specific mid-range inputs. - **Log-Scale Insight**: The logarithmic axes emphasize the wide range of input sizes and highlight performance disparities at extreme values. --- ### Interpretation - **What the Data Suggests**: - Ethos Checking Time is generally more efficient than Lean-SMT, particularly for small to moderate input sizes (`x < 10³` ms). - For large inputs (`x > 10⁴` ms), both systems degrade, but Ethos maintains a relative advantage. - The outliers above `y = x` suggest edge cases where Lean-SMT might outperform Ethos, possibly due to algorithmic differences or input-specific optimizations. - **Why It Matters**: - The plot provides empirical evidence for choosing between Lean-SMT and Ethos based on input size and performance requirements. - The log-log scale reveals scalability patterns, showing that Ethos scales better for large inputs compared to Lean-SMT. - **Notable Anomalies**: - The outlier at `x ≈ 10⁵`, `y ≈ 3×10⁴` warrants investigation: Why does Ethos perform so much better here? Could this indicate a bug, a special case, or an optimization in Ethos for extreme inputs? --- ### Spatial Grounding & Validation - **Legend Placement**: Top-left corner, clearly separated from the plot. - **Color Consistency**: All data points are blue, matching the legend’s reference (no mismatches). - **Axis Alignment**: Logarithmic scales ensure proportional representation of values across orders of magnitude. --- ### Final Notes The plot uses a log-log scale to effectively visualize data spanning four orders of magnitude. While Ethos generally outperforms Lean-SMT, the presence of outliers and mid-range exceptions highlights the need for further analysis of specific input types or system configurations.